# On the existence of a non-principal Euclidean ideal class in biquadratic
fields with class number two

Sunil kumar Pasupulati

###### Abstract.

We prove that a certain class of biquadratic fields have a Euclidean ideal.

##  1\. Introduction

For a given algebraic number field $K,$ we indicate its ring of integers by
$\mathcal{O}_{K},$ its discriminant by $d_{K},$ its ideal class group by
$Cl(K),$ its class number by $h_{K}$, and its multiplicative group of units by
$\mathcal{O}_{K}^{*}.$ In 1979, H.W. Jun. Lenstra [8] introduced the
definition of the Euclidean ideal which is a generalization of Euclidean
domain.

######  Definition 1.

Let $R$ be a Dedekind domain and $\mathbb{E}$ be the set of non zero integral
ideals of $R$. If $C$ is an ideal of $R,$ then it is called Euclidean if there
exists a function $\Psi:\mathbb{E}\to\mathbb{N}$, such that for every
$I\in\mathbb{E}$ and $x\in I^{-1}C\setminus C$ there exist a $y\in C$ such
that

$\Psi\left((x-y)IC^{-1}\right)<\Psi(I).$

Lenstra proved that if $C$ is a Euclidean ideal, then ideal class $[C]$
generates $Cl_{K},$ therefore $Cl_{K}$ is cyclic. Nevertheless the converse
need not be true. Imaginary quadratic fields $\mathbb{Q}(\sqrt{-d})$ , where
$d=19,43,67,163,$ have trivial class groups but it does not contain a
Euclidean ideal. He classified all number fields with a Euclidean ideal class
under the assumption of GRH. He proved that if $K$ is a number field with
$\mathrm{rank}(\mathcal{O}_{K}^{*})\geq 1$ then $Cl_{K}$ is cyclic if and only
if $K$ has a Euclidean ideal class. In other words, number fields except
imaginary quadratic fields have a cyclic class group if and only if it has
Euclidean class under the assumption of GRH.

H.Graves [5] proved a useful growth result which gives condition for the
existence of a Euclidean ideal in number fields, which removed appeal to GRH.
We state the result below and this is one of the main ingredient for the proof
of our theorem.

######  Theorem 1.

Suppose that K is a number field such that $|\mathcal{O}_{K}^{*}|=\infty$ and
that $C$ is a non-zero ideal of $\mathcal{O}_{K}$. If $[C]$ generates the
class group of $K$ and

$\Biggr{|}\\{\mathrm{prime\ ideals}\
\mathfrak{P}\mathcal{O}_{k}:N(\mathfrak{P})\leq x,\ [\mathfrak{P}]=[C],\
\pi_{p}\mathrm{\ is\ onto}\\}\Biggr{|}\gg\frac{x}{(\log x)^{2}}$

where $\pi_{p}$ is the canonical map from
$\mathcal{O}_{K}^{*}\to(\mathcal{O}_{K}/\mathfrak{P})^{*}$, then $[C]$ is a
Euclidean ideal class.

The following theorem is proved by Ram Murthy and Graves using the above
growth result. This theorem removes the assumption of GRH for a certain class
of algebraic number fields with a large unit rank.

######  Theorem 2.

Let $K$ be a number field, Galois over $\mathbb{Q}.$ If its Hilbert class
field $H(K)$ has an abelian Galois group over $\mathbb{Q}$ and if
$\mathrm{rank}(\mathcal{O}_{K}^{*})\geq 4,$ then

$Cl_{K}=\langle C\rangle\ \text{ if and only if }\ [C]\ \text{is Euclidean
class.}$

The above result was improved by Jhoshna, Deshouillers and Gun using sieve
theory results and they proved the following.

######  Theorem 3.

Let $K$ be a number field with the rank of $O_{K}^{*}$ at least 3 and the
Hilbert class field $H(K)$ is abelian over $\mathbb{Q}.$ Also suppose that the
conductor of $H(K)$ is $f$ and $\mathbb{Q}(\zeta_{f})$ over $K$ is cyclic.
Then $K$ has a Euclidean ideal class.

Graves [4] constructed an explicit biquadratic field
$\mathbb{Q}(\sqrt{2},\sqrt{35})$ which has a non-principal Euclidean ideal
class. The result by Graves was generalized by Hsu [6], Chattopadhyay and
Muthukrishnan [6]. Now recall few results regarding the existence of the
Euclidean ideal in the biquadratic field.

In [6], Hsu explicitly constructed a family of biquadratic field with a rank
of $O_{K}^{*}$ having a Euclidean ideal class, described below.

######  Theorem 4.

Suppose $K=\mathbb{Q}(\sqrt{q},\sqrt{kr})$. Then $K$ has a non-principal
Euclidean ideal class whenever $h_{K}=2.$ Here the integers $q,k,r$ are all
primes $\geq 29$ and are all congruent to $1$ modulo $4.$

The family of biquadratic field given Hsu [6] is extended by Chattopadhyay and
Muthukrishnan [1] and proved the following result.

######  Theorem 5.

Let $K=\mathbb{Q}(\sqrt{q},\sqrt{kr}),$ where $q\equiv 3$ and $k,r\equiv
1\pmod{4}$ are prime numbers. Suppose that $h_{K}=2$. Then $K$ has a Euclidean
ideal class.

In this article, we shall provide a new family of biquadratic fields having
Euclidean ideal class whenever the class number of biquadratic field is 2. Our
theorem is stated as follows.

######  Theorem 6.

Let $K=\mathbb{Q}(\sqrt{q},\sqrt{kr})$, where $q\geq 3$ and either $k$ or $r$
is congruent to $1$ modulo $4,$ are prime numbers not equal to $2,3,5,7$ and
$17$. Suppose that $h_{K}=2$. Then $K$ has a Euclidean ideal class.

Combining Theorems 4,5,6, we have the existence of Euclidean ideal in
biquadratic field $\mathbb{Q}(\sqrt{q},\sqrt{kr})$ with class number two
expect both $k,r$ are congruent to 3 modulo 4. It seems that it there are very
fewer biquadratic fields $\mathbb{Q}(\sqrt{q},\sqrt{kr})$ with $k,r\equiv
3\pmod{4}$ and having class number two.

For the real biquadratic field $K$ we have $rank(\mathcal{O}_{K}^{*})=3.$ But
we can’t conclude that K has a Euclidean ideal by using Theorem 3 because
$\mathbb{Q}(\zeta_{f})/K$ is not cyclic, where $f$ is the conductor of the
Hilbert class field of $K$.

##  2\. Preliminaries

######  Definition 2.

Let $L/\mathbb{Q}$ be finite abelian extension. The conductor of L is defined
to be the smallest even number $f$ such that $L\subset\mathbb{Q}(\zeta_{f}).$

######  Definition 3.

Let $K$ be a number field. The Hilbert class field $H(K)$ of $K$ is defined to
be the maximal unramified abelian extension of $K.$

The following lemma is a consequence of the Artin reciprocity law which
establishes the relation between Galois group of Hilbert class field and Class
group of the number field .

######  Lemma 7.

Let $K$ be a number field and let $H(K)$ be Hilbert class field of $K$. Then
the Galios group of $H(K)$ over $K$ is isomorphic to $Cl_{K}$ i.e

${\mathrm{Gal}}(H(K)/K)\cong Cl_{K}.$

By careful reading through [3] we state the following theorem.

######  Theorem 8.

If $p$ is a prime not equal to $2,3,5,7$ and $17,$ then there exists a prime
$q<p$ and $q\equiv 3\pmod{4}$ which is a quadratic residue modulo $p.$

We also recall the following theorem about the existence of quadratic non
residue modulo ${p}$ from [9].

######  Theorem 9.

For every prime $p\geq 5$, there is a prime quadratic non residue $q\equiv
3\pmod{4}$ with $q<p.$

######  Lemma 10.

([7] II, §4, Proposition 8) Let $L$ be a finite extension of discrete valued
field $K.$ If $L$ is unramified over $K$ and $K_{1}$ is a finite extension of
$K$, then $LK_{1}$ is unramified over $K_{1}.$

######  Lemma 11.

Let $L_{1},L_{2}$ are finite extensions of number field $K.$ Suppose prime $p$
in K is unramified in $L_{1}$. If $\mathfrak{P}_{2}$ be prime of $L_{2}$ lie
above $p$ then $\mathfrak{P}_{2}$ is unramified in $L_{1}L_{2}.$

###### Proof.

Let $L$ be the compositum of fields $L_{1}$ and $L_{2}.$ Consider $p$ adic
valuation on $K$ and extend the $p$ adic valuation to $L_{1},L_{2},L.$ Suppose
$\mathfrak{P}_{1},\mathfrak{P}_{2},\mathfrak{P}$ are primes of $L_{1},L_{2},L$
respectively which lie above prime $p$ of K. It is well known from the local-
global theory of ramification, if $E/F$ be an extension of number fields and
${\mathcal{Q}}$ be a prime of $E$ lie above prime $q$ of $F$, then
$e({\mathcal{Q}}/q)=e_{E_{q}/F_{q}},$ where $E_{q}$ and $F_{q}$ are $q$ adic
completion of $E$ and $F$ respectively.

${(L_{1}L_{2})}_{q}$${L_{1}}_{q}$${L_{2}}_{q}$$K_{q}$

Observe $e_{{L_{1}}_{p}/K_{p}}=e(\mathfrak{p}/p)=1$ then ${L_{1}}_{p}/K_{p}$
is unramified. So $L_{p}/{L_{2}}_{p}$ is unramified i.e
$e_{L_{p}/{L_{2}}_{p}}=1.$ So now using Lemma 10 we conclude that
$e(\mathfrak{P}/\mathfrak{P}_{2})=1.$ Hence $\mathfrak{P}_{2}$ is unramified
in $L_{1}L_{2}.$ ∎

We recall a result from class field theory.

######  Lemma 12.

Let $K\subset L$ be a Galois extension of number fields and $P$ be a prime of
$\mathcal{O}_{K}$ which is unramified in L. If $\mathfrak{P}$ is prime of
$\mathcal{O}_{L}$ containing $P,$ then there exist unique element $\sigma\in
Gal(L/K)$ such that for all $\alpha\in\mathcal{O}_{L},$

$\sigma(\alpha)\equiv\alpha^{N(P)}\mod{\mathfrak{P}},$

where $N(P)=|\frac{O_{K}}{P}|$ is the norm of $P.$

Now we look at one of the important objects of algebraic number theory, the
Artin symbol.

######  Definition 4.

The unique element $\sigma$ in above lemma is called Artin symbol and it is
denoted by $\left(\frac{L/K}{\mathfrak{P}}\right).$

Let us look at some useful properties of Artin symbol which will be useful in
later part of this article.

######  Lemma 13.

Let $K\subset L$ be a Galois extension and let P be an unramified prime of
$K.$ Given a prime $\mathfrak{P}$ of $L$ containing $P,$ we have:

  1. (1)

If $\sigma\in Gal(L/K)$ then

$\left(\frac{L/K}{\sigma(\mathfrak{P})}\right)=\sigma\left(\frac{L/K}{\mathfrak{P}}\right)\sigma^{-1}$

  2. (2)

The prime $P$ splits completely in $L$ if and only if
$\left(\frac{L/K}{\mathfrak{P}}\right)=1.$

######  Remark 14.

Let $L/K$ is an abelian extension. For any two primes
$\mathfrak{P}_{1},\mathfrak{P}_{2}$ of $L$ which lie over prime $\mathfrak{p}$
of $K$ we have
$\left(\frac{L/K}{\mathfrak{P}_{1}}\right)=\left(\frac{L/K}{\mathfrak{P}_{2}}\right).$
Therefore for any prime $\mathfrak{P}$ which lie above $\mathfrak{p}$, the
Artin symbol denoted by $\left(\frac{L/K}{\mathfrak{p}}\right).$

######  Definition 5.

Let $K$ be a number field and $P_{K}$ be the set of all prime ideals of
$\mathcal{O}_{K}.$ If $\mathcal{S}$ is a subset of $\mathcal{P}_{K}$ then the
Dirichlet density of the set $\mathcal{S}$ is defined to be as

$\delta(\mathcal{S})=\mathrm{lim}_{s\to
1^{+}}\frac{\sum_{\mathfrak{p}\in\mathcal{S}}N(\mathfrak{p})^{-1}}{\sum_{\mathfrak{p}\in
P_{K}}N(\mathfrak{p})^{-1}}$

provided limit exist. Where $N(\mathcal{P})$ is order of residue field of
$\mathcal{P}$ which is same as the quantity $\mathcal{O}_{K}/\mathcal{P}.$

We recall the Chebotarev density theorem [Theorem 6.3.1 in [2]] which helps to
calculate the Dirichlet density of a set.

######  Theorem 15.

Let $L/K$ be a finite Galois extension of global fields and $\mathcal{C}$ be a
conjugacy class in ${\mathrm{Gal}}(L/K).$ Then the Dirichlet density of

$\Big{\\{}\mathfrak{p}\in P(K):\ \mathfrak{p}\ \text{ is unramified in $L$ and
}\left(\frac{L/K}{\mathfrak{p}}\right)=\mathcal{C}\Big{\\}}$

exists and is equal to $\frac{|\mathcal{C}|}{[L:K]}.$

We end this section by stating a lemma without proof, whose proof is evident
from Corollary 5.21 and Corollary 5.25 of [2].

######  Lemma 16.

Let $K\subset L$ be an extension of number field. Suppose $\mathfrak{P}$ and
$\mathfrak{p}$ are primes of $L\ and\ K$ respectively lie above rational prime
p. If $\mathfrak{P}/p$ has residue degree $2$ and $\mathfrak{p}/p$ has residue
degree $1$ then $\mathfrak{p}$ is non-principal ideal.

##  3\. Proof of the main theorem

We start this section by calculating the Hilbert class field of $K,$ which
plays a crucial role in the proof of our main theorem.

######  Lemma 17.

Let $K=\mathbb{Q}(\sqrt{q},\sqrt{kr})$ as in Theorem 6, then
$H(K)=\mathbb{Q}(\sqrt{q},\sqrt{k},\sqrt{r}).$

###### Proof.

Let $L=\mathbb{Q}(\sqrt{q},\sqrt{k},\sqrt{r}).$ The conductor of L is
$f(L)=\mathrm{lcm}(f(K_{1}K_{2}),f(K_{3}))=\mathrm{lcm}(4qk,r)=4qkr,$
therefore the only primes that might ramify in $L/K$ are primes that lying
above $2,q,k,r.$

Primes lie above 2:
We have either $k$ or $r$ is congruent to $1$ modulo ${4}$, so it is enough to
assume $r\equiv 1\pmod{4}.$ Observe that $r\equiv 1\pmod{4}$, so $2$ does not
ramify in $\mathbb{Q}(\sqrt{r}).$ Now Lemma 11 would imply that the primes lie
above $2$ in $K$ are unramified in $L.$

Primes lie above $k,q,r$:
Let $p$ be one of $k,q,r.$ It is clear that $p$ is unramified in either
$\mathbb{Q}(\sqrt{r})$ or $\mathbb{Q}(\sqrt{k}).$ We use Lemma 11 to deduce
that the primes above $p$ in $K$ are unramified in $L.$ Therefore all primes
of $K$ are unramified in $L$, Thus $L$ is an abelian unramified extension of
$K$. As a consequence of fact, Hilbert class field H(K) of $K$ is the maximal
abelian unramified extension of $K,$ we have $L\subset H(K).$ The Artin
reciprocity law (7) states that $Gal(H(K)/K)\cong Cl_{K}.$ Hence $H(K)=L.$ ∎

Graves had stated the following theorem in [4].

######  Theorem 18.

Let K be a totally real number field with conductor $f(K)$ and
$\\{e_{1},e_{2},e_{3}\\}$ be a multiplicatively independent set contained in
$\mathcal{O}_{K}^{*}.$ If $l=\mathrm{lcm}(16,f(K)),$ and if
$\mathrm{gcd}(u,l)=gcd(\frac{u-1}{2},l)=1$ for some integer u, then

$\displaystyle\Biggr{|}\Big{\\{}\mathrm{primes\ ideal\ of\ first\ degree}\ \
\Big{|}\ N(\mathfrak{P})\equiv u\pmod{l},\ N(\mathfrak{P})\leq
x,\langle-1,e_{i}\rangle\twoheadrightarrow(\mathcal{O}_{K}/\mathfrak{P})^{*}\Big{\\}}\Biggr{|}\gg\frac{x}{(\log
x)^{2}}$

for at least one i.

######  Lemma 19.

The primes which split completely in K but not in H(K) have density
$\frac{1}{8}.$

###### Proof.

Consider the following two sets of prime numbers $X_{K},X_{H(K)}$ which split
completely in $K,H(K)$ respectively.

$X_{K}=\Biggr{\\{}p:\ \text{ $p$ is prime and}\
\left(\frac{K/\mathbb{Q}}{p}\right)=1\Biggr{\\}},$ $X_{H(K)}=\Biggr{\\{}p:\
\text{ $p$ is prime and}\
\left(\frac{H(K)/\mathbb{Q}}{p}\right)=1\Biggr{\\}}.$

Observe that $K\subset H(K),$ which infers that if a prime splits completely
in $H(K)$ then it splits completely in K. Therefore $X_{H(K)}\subset X_{K}$.
The primes which split completely in $K,$ but do not split completely in
$H(K)$ are $X_{K}\setminus X_{H(K)}.$ The identity element in Galois group
alone account for conjugacy class. By the Theorem 15, the Dirichlet densities
of $X_{K},X_{H(K)}$ are 1/4 and 1/8. Since $X_{H(K)}\subset X_{K},$ Dirichlet
density of the set $X_{K}\setminus X_{H(K)}$ is
$\frac{1}{4}-\frac{1}{8}=\frac{1}{8}.$ Therefore $X_{K}\setminus X_{H(K)}$ is
infinite since $X_{K}\setminus X_{H(K)}$ has positive Dirichlet density.

∎

Proof of Theorem 6

Our goal is to apply Theorem 18 to $K=\mathbb{Q}(\sqrt{q},\sqrt{kr})$ in order
to obtain the growth result which is necessary to apply Theorem 1. Since K is
a totally real number field of degree 4, with signature $(4,0).$ We recall
Dirichlet’s unit theorem which asserts that the rank of the unit group is
$r_{1}+r_{2}-1.$ Therefore rank of $\mathcal{O}_{K}$ is 3, so there exists a
set $\\{e_{1},e_{2},e_{3}\\}$ which happens to be multiplicatively independent
in $\mathcal{O}_{K}^{*}.$

Our strategy in applying Theorem 18 is similar to [1] and [6] we find an
arithmetic progression with a common difference equal to the conductor of
$H(K).$

We need to find $u\in\mathbb{Z}$ satisfying following conditions to fit into
hypotheses of Theorem 18. Let
$l=\mathrm{lcm}(16,f(K))=\mathrm{lcm}(16,4qkr)=16qkr.$

  1. (1)

$(u,l)=1,$

  2. (2)

$(\frac{u-1}{2},l)=1.$
Since we are looking for non-principal prime ideals we demand $u$ to satisfy
one more condition.

  3. (3)

If prime $p\equiv u\pmod{4qkr}$, then $f(\mathfrak{p}/p)=1$ and
$f(\mathfrak{P}/p)=2,$ where $f(\mathfrak{p}/p)\ and\ f(\mathfrak{P}/p)$ are
residue degree of primes $\mathfrak{p}$ of K and $\mathfrak{P}$ of L lying
above $p.$

By Lemma 19, there are infinitely many primes satisfying condition 3. Observe,
prime $p$ splits completely in $K=\mathbb{Q}(\sqrt{q},\sqrt{kr})$ if and only
if $p$ splits completely $\mathbb{Q}(\sqrt{q}),\mathbb{Q}(\sqrt{kr}).$ This
translate to

$\left(\frac{q}{p}\right)=1\ and\ \left(\frac{kr}{p}\right)=1$

Similarly prime $p$ splits completely in
$H(K)=\mathbb{Q}(\sqrt{q},\sqrt{k},\sqrt{r})$ if and only if

$\left(\frac{q}{p}\right)=1,\left(\frac{k}{p}\right)=1\ and\
\left(\frac{r}{p}\right)=1.$

If $p\notin X_{H(K)}$ then $\left(\frac{q}{p}\right)=-1\ or\
\left(\frac{k}{p}\right)=-1\ or\ \left(\frac{r}{p}\right)=-1.$ Therefore

(3.1) $\displaystyle{}X_{K}\setminus X_{H(K)}=\Biggr{\\{}p:\ \text{ $p$ is
prime and}\ \left(\frac{q}{p}\right)=1\ and\
\left(\frac{k}{p}\right)=\left(\frac{r}{p}\right)=-1\Biggr{\\}}.$

We translate the conditions to be satisfied by $u$ into the simultaneous
congruences, then we obtain $u$ by Chinese reminder theorem. Note that the
condition (2) is equivalent to the following simultaneous congruences

$\displaystyle u\not\equiv 1\pmod{q};$ $\displaystyle u\not\equiv 1\pmod{k};$
$\displaystyle u\not\equiv 1\pmod{r};$ $\displaystyle u\not\equiv 1\pmod{4}.$

Now we are all set to produce a desired $u$. We demonstrate how to choose the
$u$ in two different cases.
Case-1 $q\equiv 1,k\equiv 3,r\equiv 1\pmod{4}$.
By Theorem 8 and Theorem 9, there exist prime numbers $p_{1},p_{2},p_{3}$ with
$p_{1}<q,p_{2}<k,p_{3}<r$ satisfying

$\left(\frac{p_{1}}{q}\right)=1=\left(\frac{p_{2}}{k}\right),\left(\frac{p_{3}}{r}\right)=-1\
and\ p_{1},p_{2},p_{3}\equiv 3\pmod{4}.$

Now consider the following system of simultaneous congruences.

(3.2) $\displaystyle x\equiv p_{1}\pmod{q};$ (3.3) $\displaystyle x\equiv
p_{2}\pmod{k};$ (3.4) $\displaystyle x\equiv p_{3}\pmod{r};$ (3.5)
$\displaystyle x\equiv 3\pmod{4}.$

The Chinese reminder theorem promises a unique solution modulo $4qkr$ to the
above system of congruence, say $x_{0}.$ Since $\mathrm{gcd}(x_{0},4qkr)=1$,
by the Dirichlet’s Theorem on primes in arithmetic progression, there exist
infinitely many prime numbers $p$ satisfying $p\equiv x_{0}\pmod{4qkr}.$ We
pick such a prime number $p$ and call it $u.$ It is easy to observe that $u$
satisfies conditions (1) and (2).

Now we use quadratic reciprocity law to show that $u$ satisfies 3.1. We have

$\left(\frac{q}{u}\right)=(-1)^{\frac{u-1}{2}\frac{q-1}{2}}\left(\frac{u}{q}\right)=\left(\frac{p_{1}}{q}\right)=1.$

Also,

$\left(\frac{k}{u}\right)=(-1)^{\frac{k-1}{2}\frac{u-1}{2}}\left(\frac{u}{k}\right)=(-1)\left(\frac{p_{2}}{k}\right)=-1,$

and

$\left(\frac{r}{u}\right)=(-1)^{\frac{r-1}{2}\frac{u-1}{2}}\left(\frac{u}{r}\right)=\left(\frac{p_{3}}{r}\right)=-1.$

Case-2 $q\equiv 3,k\equiv 3,r\equiv 1\pmod{4}$. By Theorem 8 and Theorem 9,
there exist prime numbers $p_{1},p_{2},p_{3}$ with $p_{1}<q,p_{2}<k,p_{3}<r$
satisfying

$\left(\frac{p_{1}}{q}\right)=-1=\left(\frac{p_{3}}{r}\right),\left(\frac{p_{2}}{k}\right)=1\
and\ p_{1},p_{2},p_{3}\equiv 3\pmod{4}.$

Now consider the following system of simultaneous congruences.

(3.6) $\displaystyle x\equiv p_{1}\pmod{q};$ (3.7) $\displaystyle x\equiv
p_{2}\pmod{k};$ (3.8) $\displaystyle x\equiv p_{3}\pmod{r};$ (3.9)
$\displaystyle x\equiv 3\pmod{4}.$

The Chinese remainder theorem promises a unique solution modulo $4qkr$ to the
above system of congruence, say $x_{0}.$ Since $\mathrm{gcd}(x_{0},4qkr)=1$,
by the Dirichlet’s Theorem on primes in arithmetic progression, there exist
infinitely many prime numbers $p$ satisfying $p\equiv x_{0}\pmod{4qkr}.$ We
pick such a prime number $p$ and call it $u.$ It is easy to observe that $u$
satisfies conditions (1) and (2)..

Now we use quadratic reciprocity law to show that $u$ satisfies 3.1. We have

$\left(\frac{q}{u}\right)=(-1)^{\frac{u-1}{2}\frac{q-1}{2}}\left(\frac{u}{q}\right)=-\left(\frac{p_{1}}{q}\right)=1.$

Also,

$\left(\frac{k}{u}\right)=(-1)^{\frac{k-1}{2}\frac{u-1}{2}}\left(\frac{u}{k}\right)=-\left(\frac{p_{2}}{k}\right)=-1,$

and

$\left(\frac{r}{u}\right)=(-1)^{\frac{r-1}{2}\frac{u-1}{2}}\left(\frac{u}{r}\right)=\left(\frac{p_{3}}{r}\right)=-1.$

Now it is clear that $u$ satisfies the conditions (1) , (2) and (3.1) and also
for any prime $p$ satisfying $p\equiv u\pmod{4qkr},$ we have $\mathfrak{p}$
lying above $p$ which does not split in $H(K)$ therefore
$f(\mathfrak{P}/\mathfrak{p})=2$. By Lemma 16, $\mathfrak{p}$ is not a
principal ideal. Hence $[\mathfrak{p}]$ is generator of $Cl_{K}$.

$\left|\begin{Bmatrix}&N(\mathfrak{P})\leq x,\\\ \mathrm{prime\ ideals}\
\mathfrak{P}\mathcal{O}_{k}\Biggr{|}&[\mathfrak{P}]=[C],\\\ &\pi_{p}\mathrm{\
is\ onto}\end{Bmatrix}\right|>>\left|\begin{Bmatrix}&N(\mathfrak{P})\equiv
u\pmod{l},\\\ \mathrm{primes\ ideal\ of}\Biggr{|}&N(\mathfrak{P})\leq x,\\\
\text{\ first\
degree}&\langle-1,e_{i}\rangle\twoheadrightarrow(\mathcal{O}_{K}/\mathfrak{P})^{*}\end{Bmatrix}\right|\gg\frac{x}{(\log
x)^{2}}$

Now by applying Theorem 1, we conclude that $K$ has a non-principal Euclidean
ideal.

##  4\. Examples

There are plenty of biquadratic fields $\mathbb{Q}(\sqrt{q},\sqrt{kr})$ with
class number two. Here we present a few examples
$\mathbb{Q}(\sqrt{q},\sqrt{kr})$ which satisfy conditions in Theorem 6. All
these computations are carried out in SageMath.

$(q,k,r)$ |  $h_{\mathbb{Q}(\sqrt{q},\sqrt{kr})}$ |  $(q,k,r)$ |  $h_{\mathbb{Q}(\sqrt{q},\sqrt{kr})}$
---|---|---|---
(11,19,13) |  2 |  (11,67,13) |  2
(11, 23,13) |  2 |  (11, 71,13) |  2
(11,31 ,13) |  2 |  (19,11,13) |  2
(11,43 ,13) |  2 |  (19,23,13) |  2
(11,47,13) |  2 |  (19,31,13) |  2
(11,59 ,13) |  2 |  (19,43,13) |  2

Table 1: Examples of $\mathbb{Q}(\sqrt{q},\sqrt{kr})$ with $q\equiv 3,k\equiv
3,q\equiv 1\pmod{4}$ and having class number 2.

$(q,k,r)$ |  $h_{\mathbb{Q}(\sqrt{q},\sqrt{kr})}$ |  $(q,k,r)$ |  $h_{\mathbb{Q}(\sqrt{q},\sqrt{kr})}$
---|---|---|---
(13,11,37) |  2 |  (37,11,29) |  2
(13,11,41) |  2 |  (37,11,61) |  2
(29,11,37) |  2 |  (37,11,109) |  2
(29,11,41) |  2 |  (41,11,13) |  2
(29,11,73) |  2 |  (41,11,29) |  2
(37,11,13) |  2 |  (41,11,53) |  2

Table 2: Examples of $\mathbb{Q}(\sqrt{q},\sqrt{kr})$ with $q\equiv 1,k\equiv
3,q\equiv 1\pmod{4}$ and having class number 2.

##  5\. Concluding Remarks

In [8], Lenstra also defined norm Euclidean ideals, which are the Euclidean
ideals with the Euclidean algorithm as the norm on a number field.

######  Definition 6.

If K is a number field and C be a fractional ideal of $O_{K}$, then C is norm
Euclidean if for all $x\in K$ there exists $y\in C$ such that

$Nm(x-y)<Nm(C)$

We do not know if K has norm Euclidean ideal.

## Acknowledgments

I would like to thank Dr.Srilakshmi Krishnamoorty for valuable suggestions and
constant support. I thank IISER Tiruvananthapuram for providing excellent
working conditions. I also wish to thank Dr.Viji Z Thomas for useful
discussions.

## References

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  * [2] David A. Cox, _Primes of the form $x^{2}+ny^{2}$_, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989, Fermat, class field theory and complex multiplication. MR 1028322
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# Emerging chimera states under non-identical counter-rotating oscillators

K. Sathiyadevi,1, V. K. Chandrasekar,2, and M. Lakshmanan1 1Department of
Nonlinear Dynamics, School of Physics, Bharathidasan University,
Tiruchirappalli - 620 024, Tamil Nadu, India.
1Department of Physics, Centre for Nonlinear Science & Engineering, School of
Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur -613
401, Tamil Nadu, India.

###### Abstract

Frequency plays a crucial role in exhibiting various collective dynamics in
the coexisting co- and counter-rotating systems. To illustrate the impact of
counter-rotating frequencies, we consider a network of non-identical and
globally coupled Stuart-Landau oscillators with additional perturbation.
Primarily, we investigate the dynamical transitions in the absence of
perturbation, demonstrating that the transition from desynchronized state to
cluster oscillatory state occurs through an interesting partial
synchronization state in the oscillatory regime. Followed by this, the system
dynamics transits to amplitude death and oscillation death states.
Importantly, we find that the observed dynamical states do not preserve the
parity($P$) symmetry in the absence of perturbation. When the perturbation is
increased one can note that the system dynamics exhibits a new kind of
transition which corresponds to a change from incoherent mixed synchronization
to coherent mixed synchronization through chimera state. In particular,
incoherent mixed synchronization and coherent mixed synchronization states
completely preserve the $P$-symmetry, whereas the chimera state preserves the
$P$-symmetry only partially. To demonstrate the occurrence of such partial
symmetry breaking (chimera) state, we use basin stability analysis and
discover that partial symmetry breaking exists as a result of the coexistence
of symmetry preserving and symmetry breaking behavior in the initial state
space. Further, a measure of the strength of $P$-symmetry is established to
quantify the $P$-symmetry in the observed dynamical states. Subsequently, the
dynamical transitions are investigated in the parametric spaces. Finally, by
increasing the network size, the robustness of the chimera state is also
inspected and we find that the chimera state is robust even in networks of
larger sizes. We also show the generality of the above results in the related
phase reduced model as well as in other coupled models such as the globally
coupled van der Pol and Rössler oscillators.

##  I Introduction

Coupled nonlinear oscillators are effective tools for studying the collective
behaviors of many complex dynamical systems in nature. The interaction of
coupled elements, properties of an isolated system, and coupling topologies
result in a variety of collective dynamical states such as desynchronization,
synchronization, clusters, traveling waves, solitary states, and oscillation
death states syn1 ; syn2 ; coh1 ; sat_vdp ; adod . Among all these states, the
coexistence of spatially coherent and incoherent dynamical behavior known as
“chimera” has been investigated vigorously during the past couple of decades.
It is a fascinating hybrid dynamical state having a strong resemblance with
many real-world phenomena, including uni-hemispheric slow-wave sleep uni1 ;
uni2 ; uni3 , bump states in neural networks pumb_neu , power grid blackouts
power1 ; power2 , social networks social , etc. The chimera state is also
related to various brain diseases such as Parkinson’s disease, epileptic
seizures, Alzheimer’s disease, schizophrenia, and brain tumors chim .
Originally, the coexistence of coherent and incoherent oscillatory dynamics
has been reported in a network of nonlocally coupled oscillators by Kuramoto
chim1 . Later, it was named as a chimera by Abrams and Strogatz chim2 .
Eventually, the chimera state has also been realized in locally and globally
coupled oscillators in various numerical models chim_rev1 ; ac_local ;
chim_global ; prema_glob . Also, a variety of chimera states including
breathing chimera bc , amplitude chimera sat_ac , frequency chimera prema_glob
, traveling chimera tc1 ; tc2 , and imperfect chimera ic , etc. have been
reported through different models. Also, besides numerical studies, various
experimental evidences for chimera states exist, such as optical coupled-map
lattices exp_cml , coupled chemical oscillatorsexp_co , metronomes ic ;
exp_met , and squid meta-materials exp_squid1 ; exp_squid2 , etc.

On the other hand, coexisting co- and counter-rotating dynamical activity has
a wide range of applications in diverse fields, including physical phy1 ; phy2
, biological bio ; bio1 , and fluid dynamics flu1 ; flu2 ; flu3  contexts. For
instance, counter-rotating spirals can be found in biological media, such as
Physarum plasmodium protoplasm bio ; bio1 . In physical systems, such as
magnetohydrodynamics of plasma flow plasma , and Bose-Einstein condensates BEC
, counter-rotating vortices are purposefully formed for practical reasons diba
. As a consequence, examining the collective behaviors induced by counter-
rotation becomes intriguing. The coexistence of co- and counter-rotating
oscillators was initially identified by Tabor tabor . This is followed by a
series of works that have been done in co- and counter-rotating oscillators.
The existence of anti-synchronization has been reported in coupled chaotic
Lorentz systems as . Subsequently, the mixed synchronization and its universal
occurrence were investigated in both limit-cycle and chaotic oscillators
awadhes1 . Later, the realizations of mixed synchronization were also revealed
in both experimental and theoretical systems cr_exp1 . Further, the breaking
of rotational symmetry induced dynamical effects were investigated in counter-
rotating oscillators with symmetry preserving and symmetry breaking couplings
awadhes2 . Recently, the aging transitions have also been investigated in
counter-rotating oscillators aging . Therefore, competing co- and counter-
rotating frequencies play an essential role in exhibiting a distinct set of
collective behaviors including mixed synchronization and different oscillation
quenching states. However, the effects of frequency and additional
perturbation on the onset of the chimera states in counter-rotating
oscillators are unclear and they have not been explored explicitly to the best
of our knowledge. Therefore, in this paper, we aim to investigate whether
coexistence of co- and counter-rotating oscillators with additional
perturbation implies chimera behavior (partial symmetry breaking dynamics)
among the two groups of oscillators.

Motivated by the above, in this paper we consider a network of nonidentical
and globally coupled Stuart-Landau oscillators with additional perturbation.
Importantly, the influence of frequencies is demonstrated by splitting the
network into two groups and distributing the frequencies around a specific
threshold range. To start with, the dynamical behaviors of the nonidentical
counter-rotating frequencies are investigated in the absence of additional
perturbation. We show that the oscillatory dynamics transits from the
desynchronized state ($DS$) to the cluster oscillatory state ($COS$) via a
partial synchronization ($PS$) state with increasing coupling strength. The
system dynamics then attains oscillation quenching states such as amplitude
death ($AD$) and oscillation death ($OD$) at larger coupling strengths.
Further, we find the observed dynamical states, namely $DS$, $PS$, $COS$, and
$OD$ states, break the parity$(P)$-symmetry in the absence of perturbation.
When the additional perturbation is increased we find that symmetry breaking
$DS$ and $COS$ become symmetry preserving incoherent mixed synchronization
($IMS$) and coherent mixed synchronization ($CMS$) states. Importantly, the
symmetry breaking partial synchronization state turns into a partial symmetry
broken state where a partial set of oscillators satisfies the $P-$symmetry
while the others do not. Such partial symmetry broken state is designated here
as the chimera ($CH$) state which is realized for the first time in non-
identical counter-rotating oscillators. The co-existence of symmetry
preserving and symmetry breaking oscillators in the chimera state is then
confirmed through the basin of attraction studies. In addition, the new
measure, namely the strength of $P-$symmetry, is used to validate the partial
symmetry breaking state and its transitions. Interestingly, we find that
partial symmetry breaking occurs only when the perturbation is increased at
the $DS$ and $PS$ states. Followed by this, the global dynamical transitions
are analyzed in the parametric spaces. We can note that the increasing of the
threshold increases the collective dynamical states, namely the
$DS,~{}PS,~{}CH,~{}IMS$, and $AD$ states. In particular, the regions for the
$DS,~{}CH,~{}IMS,~{}AD$ and $OD$ states are increased with the decrease in
$CMS$ and $PS$ regions. Furthermore, we also discover that increasing the
strength of additional perturbation suppresses the regions of
$DS,~{}PS,~{}COS$ and $AD$ regions completely with the onset of $IMS,~{}CH,$
and $CMS$ states. Finally, the robustness of the chimera is examined by
increasing the size of the network at the chimera state. Eventually, the
generality of the observed oscillatory states and their transition are studied
in other dynamical models such as the coupled van der Pol (vdP) and Rössler
oscillators as well as in the phase reduced model. Notably, we find that the
globally coupled vdP and Rössler oscillators also exhibit similar kinds of
partial synchronisation and chimera states.

The structure of the article is as follows: In Sec. II, we introduce the model
of globally coupled Stuart-Landau oscillators with additional perturbation.
The dynamical transitions in the absence and in the presence of perturbation
are discussed in the Secs. III and IV, respectively. Then the mechanism and
the characterization of the partial symmetry breaking state are detailed in
Secs. V and VI, respectively. Further, the global dynamical transitions are
discussed in Sec. VII and the robustness of the chimera state is described in
Sec. VIII. Finally, we summarize our major findings in Sec. IX.

Figure 1: Spatio-temporal plots of globally coupled Stuart-Landau oscillators
distributed with nonidentical counter-rotating frequencies in the absence of
perturbation ($\varepsilon=0.0$) for (a) desynchronized state ($K=0.05$), (b)
partial synchronization state ($K=0.82$), (c) cluster oscillatory state
($K=1.3$), (d) amplitude death state ($K=1.86$), and (e) oscillation death
state ($K=2.0$). The corresponding snapshots in terms of $x_{j}$ variables are
displayed in (f)-(j), and the $y_{j}$ variables are shown in (k)-(o). Other
parameters are as follows: $\lambda=1.0$, $\Delta=0.5$, $\omega=1.0$, and
$N=100$.

##  II  The model

Limit cycle oscillations are often found in various physical, chemical, and
biological phenomena including neuronal excitations, circadian rhythms,
respiratory cycles, chemical oscillations, vibrations in bridges, etc. Thus,
we consider a simple, prototype, self-excitatory model of Stuart-Landau (SL)
limit-cycle oscillators. Such limit cycle oscillators can be used to model a
variety of nonlinear oscillators near Hopf bifurcation stu1 ; stu2 ; stu3 . In
particular, to investigate the effect of additional perturbation on non-
identical counter-rotating oscillators, we consider a system of globally
coupled Stuart-Landau oscillators whose governing equation is expressed as

$\displaystyle\dot{z}_{j}=(\lambda+i\omega_{j}-|z_{j}|^{2})z_{j}+\frac{K+\varepsilon}{N}\sum_{k=1}^{N}Re({z_{k}}-\,{z_{j}})$
$\displaystyle+i\frac{K-\varepsilon}{N}\sum_{k=1}^{N}Im({z_{k}}-\,{z_{j}}),\quad
j=1,2,...,N,$ (1)

where $z_{j}=x_{j}+iy_{j}\in\mathbb{C}$, $x_{j}$ and $y_{j}$ are the state
variables of the SL oscillators. $\lambda$ is the Hopf bifurcation parameter
and $\omega$ is the system frequency. $K$ is the coupling strength and
$\varepsilon$ is the additional perturbation in the coupling. In the absence
of additional perturbation ($\varepsilon=0$), the system preserves the
rotational symmetry and Eq. (1) remains invariant under the gauge
transformation $z_{j}\rightarrow z_{j}e^{i\theta}$. For $\varepsilon\neq 0$,
the coupled system loses this rotational symmetry. In an earlier report, the
effect of bimodal distribution was investigated in a system of globally
coupled phase oscillators without time delay rev2 . Then the influence of
time-delay induced dynamics was further investigated in a large population of
globally coupled phase oscillators using a bimodal frequency distribution rev1
. Further, asymmetry in bimodal frequency induced symmetry breaking has also
been reported in bimodal . With reference to the above studies, we partitioned
the network(1) into two groups of $N/2$ oscillators each (sub-populations)
designated as $z_{j}^{(1)}$ and $z_{j}^{(2)}$ and the corresponding
frequencies are $\omega_{j}^{(1)}$ and $\omega_{j}^{(2)}$, respectively.
Specifically, the first group is distributed with a uniform frequency
distribution in the range $\omega_{j}^{(1)}=(\omega-\Delta,~{}\omega+\Delta)$
for $j=1,2,...,N/2$, while the second group is distributed uniformly in the
frequency range $\omega_{j}^{(2)}=(-\omega-\Delta,~{}-\omega+\Delta)$ for
$j=N/2+1,...,N$. Depending on the positive and negative signs infront of
$\omega$, the individual oscillator rotates either in the clockwise or
anticlockwise direction. For numerical simulations, we used the Runge-Kutta
$4^{th}$ order scheme with a step size of $h=0.01$.

##  III Dynamical transitions in the absence of additional perturbation
($\varepsilon=0.0$) and parity symmetry

Figure 2: Spatio-temporal plots of globally coupled Stuart-Landau oscillators
distributed with nonidentical counter-rotating frequencies in the presence of
perturbation ($\varepsilon=0.1$) for (a) incoherent mixed synchronization
state ($K=0.05$), (b) chimera state ($K=0.82$), (c) coherent mixed
synchronization state ($K=1.3$), (d) oscillation death state ($K=1.86$), and
(e) oscillation death state ($K=2.0$). The shaded areas in Figs.2(f)-2(h) and
Figs.2(k)-2(m) illustrate the $P-$symmetry preserving oscillators. The
corresponding snapshots in terms of $x_{j}$ variables are displayed in figures
(f)-(j) and the $y_{j}$ variables are shown in figures (k)-(o). Other
parameters are as follows: $\lambda=1.0$, $\Delta=0.5$, $\omega=1.0$, and
$N=100$.

At first, we examine the dynamical transitions in globally coupled SL
oscillators in the absence of additional perturbation ($\varepsilon=0.0$). To
show the dynamical transitions, the spatio-temporal behavior (in the upper
panel) and the corresponding snapshots of $x_{j}$ and $y_{j}$ variables (in
the middle and the lower panels, respectively) are plotted in Fig. 1 by fixing
$\Delta=0.5$, $\omega=1.0$ and for different values of coupling strength
($K$). If the coupling strength is low (for $K=0.05$), we notice the existence
of a desynchronized state ($DS$) as in Fig. 1(a). In the desynchronized state,
all the oscillators in the first and second groups are distributed randomly as
evident from the snapshots in Figs. 1(f) and 1(k). When the coupling strength
is increased to $K=0.82$, there exists the coexistence of coherent and
incoherent behaviors (see Fig. 1(b)) in each of the groups and it is further
referred to as partial synchronization ($PS$) state ps1 ; ps2 ; ps3 . Co-
existence of coherent and incoherent patches in the partial synchronization
state is also clear from the snapshots of $x_{j}$ and $y_{j}$ variables in
Figs. 1(g) and 1(l). Upon increasing the coupling strength to $K=1.3$, we find
the emergence of cluster oscillatory state ($COS$). Here we observed two
cluster states, where the first group (co-rotating oscillators) forms one
group and the corresponding second set (counter-rotating oscillators) forms
the other group as shown in Figs. 1(c), 1(h), and 1(m). In particular,
individual clusters exist as a travelling wave, and on comparing both the
clusters (of $y_{j}$ variables), we find that they are antiphase with each
other rev1 ; rev5 ; io . Increasing the coupling strength further ($K=1.86$),
all the oscillators in the network attain the homogeneous steady state (see
Fig. 1(d)) namely amplitude death ($AD$) state. As a result, all the
oscillators in the network acquire the trivial steady state in the $x_{j}$ and
$y_{j}$ variables (see Figs. 1(i), and 1(n)). Further, at even larger coupling
strength ($K=2.0$), the oscillators in the network split into two groups as in
the cluster oscillatory state and populate into two different inhomogeneous
steady states constituting the oscillation death ($OD$) state which is shown
in Figs. 1(e), 1(j) and 1(o). Thus, it is clear that the transitions from
desynchronized state to cluster oscillatory state through the partial
synchronization state occur first as the coupling strength increases. Then,
the transition to oscillation quenching states such as amplitude death and
oscillation death states are observed at larger coupling strengths.

On the other hand, it has been revealed from the earlier reports that the pair
of counter-rotating oscillators preserve a set of permutation and permutation-
parity symmetries in the $x_{j}$ and $y_{j}$ variables of the same system.
Such combined symmetry preservation causes mixed synchronization among the
oscillators awadhes1 . In our case, according to the distribution of counter-
rotating frequencies, the considered network is designed to have the following
symmetries in the $x_{j}$ and $y_{j}$ variables,

$\displaystyle x_{j}{{}^{(1)}}=x_{N-j+1}{{}^{(2)}},$ $\displaystyle
y_{j}{{}^{(1)}}=-y_{N-j+1}{{}^{(2)}}.$ (2)

Here the superscripts ${(1)}$ and ${(2)}$ stand for the groups corresponding
to co- and counter-rotating oscillators, respectively. We simply call this
symmetry as parity symmetry or $P-$symmetry throughout the text. However, by
comparing the snapshots of $x_{j}$ and $y_{j}$ variables in Fig.1, we notice
that the observed dynamical states do not preserve the above mentioned
$P-$symmetry in the absence of additional perturbation. Furthermore,
heterogeneity or asymmetry in time-delay, frequency, and coupling strength in
globally coupled oscillators has been extensively investigated rev3 ; rev4 ;
rev5 . For instance, heterogeneity in time-delay induced synchronization has
been described in rev3 . Then the asymmetry in frequency and coupling strength
associated with macroscopic traveling waves has also been studied in the
Kuramoto model rev4 . Motivated by these studies, we implement asymmetry in
the coupling via additional perturbation to analyze the symmetry of the
dynamical states. Thus, in the following, we investigate the dynamical
transitions for the same set of parameters with an additional perturbation
($\varepsilon\neq 0$ in Eq. (1)), to understand whether the observed symmetry
breaking dynamical states can preserve the $P-$symmetry due to the additional
perturbation.

##  IV Dynamical transitions in the presence of additional perturbation
($\varepsilon=0.1$)

In order to understand the effect of the additional perturbation on the
dynamical states, we plotted the spatio-temporal behavior and the
corresponding snapshots of $x_{j}$ and $y_{j}$ variables in Figs. 2 by fixing
the strength of the additional perturbation as $\varepsilon=0.1$. The shaded
areas in the snapshots in Figs.2(f)-2(h) and Figs.2(k)-2(m) denote the regions
containing the $P-$symmetry preserving oscillators. For $K=0.05$, we observed
the incoherent mixed synchronization $(IMS)$ state as in Fig. 2(a). Such an
incoherent mixed synchronization state preserves the $P-$symmetry as described
below. The first and second groups preserve the symmetry
$x_{j}{{}^{(1)}}=x_{j}$, and $x_{N-j+1}{{}^{(2)}}=x_{j}$ in $x_{j}$ variables
as clear from Fig. 2(f). Simultaneously, the corresponding $y_{j}$ variables
of the first and second groups preserve $y_{j}{{}^{(1)}}=y_{j}$, and
$y_{N-j+1}{{}^{(2)}}=-y_{j}$ symmetry as shown in Fig. 2(k). Though the
oscillators are having random values in the $x_{j}$ and $y_{j}$ variables, the
oscillators in the network preserve the above $P$-symmetry. Hence the state in
Figs.2(a) or 2(f) and 2(k) is referred to as an incoherent mixed
synchronization $(IMS)$ state. By increasing the coupling parameter to
$K=0.82$, we found coexisting coherent and incoherent patches in each group as
seen in Fig. 2(b). Compared to the partial synchronization state which
completely breaks the $P-$symmetry (see Figs. 1(b),(g),(l)), here we notice
that the oscillators in the coherent patches of the first and second groups
preserve the $P-$symmetry in $x_{j}$ and $y_{j}$ variables. As a consequence,
coexisting symmetry preserving and symmetry breaking oscillators are noticed
in Figs. 2(g) and Fig. 2(l). The shaded regions correspond to symmetry
preserving oscillators while the unshaded regions represent the symmetry
breaking oscillators. Such a partial symmetry breaking state is interesting
and it is referred to as chimera $(CH)$ state here. Thus, even though the
oscillators in the network exhibit spatially coexisting coherent and
incoherent behaviors, if they completely break the $P-$symmetry or completely
preserve the $P-$symmetry, we referred to such a dynamical state as partial
synchronization; otherwise, if the coherent and incoherent patches are
partially symmetry broken, the state is designated as a chimera. We believe
that this kind of chimera state in nonidentical counter-rotating oscillators
has been observed for the first time and it has not yet been realized in the
earlier literature on counter-rotating oscillators. When $K=1.3$, all the
oscillators in both the first and second groups preserve the $P-$symmetry. As
a result, we find a coherent cluster oscillatory state which is further
represented as a coherent mixed synchronization state $(CMS)$. The coherent
mixed synchronization state among the two groups is clearly depicted in Figs.
2(c), 2(h), and 2(m). Further, the additional perturbation suppresses the
amplitude death state. As a result, we observe oscillation death alone at the
larger coupling strengths $\varepsilon=1.86$ and $\varepsilon=2.0$. From Figs.
2(d), 2(i), 2(n) and Figs. 2(e), 2(j), 2(o), it is clear that the observed
oscillation death states are symmetry breaking. Therefore, we observed the
transition from incoherent mixed synchronization $(IMS)$ to coherent mixed
synchronization $(CMS)$ through chimera $(CH)$ state and finally to
oscillation death $(OD)$ state when the additional perturbation is included.

The above observed oscillatory patterns are confirmed further using the phase
reduced model described in Appendix A.

##  V Mechanism for partial symmetry breaking

In order to understand the mechanism for the co-existence of multistable
states, i.e. coexisting partial symmetry preserving and symmetry breaking
states, we have performed a basin stability analysis in the following. Due to
multistability, a partial set of oscillators in the network alone preserves
the $P-$symmetry, resulting in the coexistence of symmetry preserving and
symmetry breaking behavior in the dynamical states. In order to show the role
of multistability among the symmetry preserving and symmetry breaking
dynamical states in the initial state space, we consider a minimal network of
two coupled Stuart-Landau oscillators with counter-rotating frequencies. The
corresponding basin of attraction is depicted in Fig. 3, by fixing
$x_{1}(0)=0.90$ and $x_{2}(0)=0.901$ and by varying the state variables
$y_{1}(0)$ and $y_{2}(0)$ for $K=1.0$. Figure 3(a) is plotted for the absence
of perturbation, namely $\varepsilon=0.0$. We can note that the system
dynamics preserves $P-$symmetry only for a specific initial state (i.e. when
the initial states are equal and opposite), otherwise the initial state space
is filled with a symmetry-breaking state.

Figure 3:  Basin of attraction for two coupled oscillators by fixing the state
variables $x_{1}(0)=0.9$ and $x_{2}(0)=0.9001$ and by varying the state
variables $y_{1}(0)$ and $y_{2}(0)$ for different perturbation strengths, (a)
$\varepsilon=0.0$, (b) $\varepsilon=0.0025$, (c) $\varepsilon=0.003$, and (d)
$\varepsilon=0.1$. The grey and red colors represent the basin for symmetry
preserving state and symmetry breaking state, respectively. Other parameters:
$\lambda=1.0$, $K=1.0$, $\omega=1.0$ and $N=2$.  Figure 4: Mixed
synchronization error estimated using Eq. (3) at coupling strength $K=0.82$
for specific perturbation strengths (a) $\varepsilon=0.0$, (b)
$\varepsilon=0.1$, and (c) $\varepsilon=0.2$.

The symmetry preserving state observed in Fig. 3(a) is unstable and it exists
only for specific asymmetric initial conditions. Here, the asymmetric initial
condition corresponds to the case where $y_{1}(0)$ is positive while
$y_{2}(0)$ is negative or vice versa. When the additional perturbation in the
coupling is increased to $\varepsilon=0.0025$, we find from the asymmetric
initial state that either $y_{1}(0)$ or $y_{2}(0)$ positive while the other
one is negative favors the symmetry preserving behavior and it is clearly
depicted in Fig. 3(b). As a result, we find that for some initial conditions,
the system dynamics preserves $P-$symmetry, while for others, it breaks
$P-$symmetry. In this manner, the coexistence of symmetry preserving and
symmetry breaking states are realized. Upon increasing the perturbation
further to $\varepsilon=0.003$, an increase in the symmetry preserving state
basin is observed as in Fig. 3(c). On increasing the perturbation further one
finds that the entire basin is occupied by a symmetry preserving state (see
Fig. 3(d)). Thus, from the basin stability analysis, one can confirm the
presence of multistability among the symmetry preserving and symmetry breaking
states. Because of the multistable nature, we find the partial symmetry
breaking states (co-existence of symmetry preserving and symmetry breaking
dynamical behavior) in the larger network. In the following section, we
discuss the characterization of the partial symmetry breaking state using the
notion of the strength of $P-$symmetry in some detail.

##  VI Characterization of partial symmetry breaking states

In order to check whether the observed dynamical states preserve $P-$symmetry,
we first measure the error of $P-$symmetry preservation by finding the
difference and sum of $x_{j}$ and $y_{j}$ variables which can be expressed as

$\displaystyle PS_{i}=\sqrt{(x_{j}-x_{N-j+1})^{2}+(y_{j}+y_{N-j+1})^{2}},$ (3)
$\displaystyle j=1,2,...,N,~{} ~{}i=1,2,...,N/2,$

where $PS_{i}$ is the parity symmetry error for the $i^{th}$ pair. Note that
here the index $i$ represents the $P-$ symmetry error between the pair of
first and second sub-populations. If the oscillators in the first and second
sub-populations preserve the $P-$symmetry, the error takes the null value,
otherwise, the system does not preserve the $P-$symmetry. The error estimated
from Eq. (3) is plotted in Fig. 4 for $K=0.82$. In the absence of perturbation
$\varepsilon=0.0$, we find that the system does not preserve $P-$symmetry (see
Fig. 4(a)), since all the $PS_{i}$’s take the values between $0<PS_{i}\leq 1$.
On increasing the perturbation, one finds that a partial set of oscillators
take null values while the remaining set takes non-zero values (see Fig.
4(b)). Symmetry preserving oscillators are represented by the shaded regions.
As a consequence, it is observed that the partial symmetry breaking state
(coexistence of partial symmetry preserving and partial symmetry breaking
dynamical states) exhibits the error in synchronization as in Fig. 4(b). Upon
increasing the perturbation strength further it is observed that all the
oscillators are showing zero error, that is the system dynamics completely
preserves $P-$symmetry as depicted in Fig. 4(c). Thus, we observe the
transition from symmetry breaking state to symmetry preserving state through
partial symmetry breaking state.

In addition, to quantify $P-$symmetry in each dynamical state and their
transition as a function of parameters, we introduced a new measure that
quantifies the strength of the $P-$symmetry. The corresponding expression for
the strength of $P-$symmetry is

$\displaystyle S_{PS}=\frac{\sum_{i=1}^{N/2}H_{PS_{i}}}{N/2}, \qquad
H_{PS_{i}}=\Theta(\delta-PS_{i})$ (4)

where $\delta$ is a small threshold (which is taken as $0.001$ for this
study), $\Theta$ is the Heaviside step function, and $PS_{i}$ is the error of
$P-$symmetry. If $PS_{i}$ is less than $\delta$, $H_{PS_{i}}$ takes the value
unity, otherwise, it is zero. Then the measure estimates the ratio of
$P-$symmetry in each dynamical state. If the strength of the $P-$symmetry
$S_{PS}$ is unity, the dynamical state completely preserves the $P-$symmetry
and if it is zero then it represents a completely symmetry broken state. The
strength of the $P-$symmetry between $0<S_{PS}<1$ represents the partial
symmetry broken state.

To understand the effect of additional perturbation on different dynamical
states as observed in Figs. 1 and 2, we have plotted the strength of
$P-$symmetry ($S_{PS}$) as a function of perturbation ($\varepsilon$) in Fig.
5. The filled circles represent the range of the strength of $P-$symmetry
($S_{PS}$) corresponding to the additional perturbation $\varepsilon$. Figure
5(a) is plotted for the dynamical transition at the desynchronized state (for
$K=0.05$) which shows that $S_{PS}$ is zero in the absence of perturbation
indicating a symmetry broken ($SB$) state. By increasing the perturbation, the
symmetry broken state transits to a symmetry preserving ($SP$) state through
partial symmetry broken ($PSB$) state. In the partial symmetry broken state
the range of $S_{PS}$ takes the value between $0<S_{PS}<1$ and the unit value
of $S_{PS}$ indicates the maximum strength of $P-$symmetry which is a symmetry
preserving state. The shaded region denotes the partial symmetry broken state.
A similar transition is also observed in the partial synchronization state
plotted in Fig. 5(b) for $K=0.82$. In comparison with Fig. 5(a), the region
for $PSB$ state is seen to widen in Fig. 5(b). Further, in the cluster
oscillatory state, initially, the system is in a symmetry breaking state then
it directly transits to symmetry preserving state by slightly increasing the
perturbation strength (see Figure 5(c) for $K=1.3$). In contrast, by fixing
the parameter in the amplitude death state and increasing the perturbation,
the value of $S_{PS}$ transits from $1$ to $0$ as a function of perturbation,
that is the transition from symmetry preserving to symmetry breaking (see Fig.
5(d)) state occurs. Importantly, here, the observed symmetry breaking state is
the oscillation death state. Furthermore, increasing the perturbation at the
oscillation death state always shows a null value of $S_{PS}$ (which has not
been displayed here) which indicates the symmetry breaking state for all
values of further perturbation. Thus, from the observations, it is to be noted
that partial symmetry breaking exists only in the originally desynchronized
and partial synchronization states. Next, the global dynamical transitions are
delineated as a function of frequency thresholds and perturbation strengths in
the following.

Figure 5:  The strength of mixed synchronization symmetry $(S_{PS})$ as a
function of perturbation strength $\varepsilon$ by fixing the coupling
parameters at (a) desynchronized state ($K=0.05$), (b) partial synchronization
state ($K=0.82$), (c) cluster oscillatory state ($K=1.3$), and (d) amplitude
death state ($K=1.86$). $SB$, $PSB$ and $SP$ are the symmetry breaking,
partial symmetry breaking and symmetry preserving dynamical transitions,
respectively. Other parameters: $\lambda=1.0$, $\Delta=0.5$, $\omega=1.0$, and
$N=100$.

##  VII Global dynamical transitions

The global dynamical transitions are analyzed by using the measure of the
strength of incoherence as discussed in sic1 ; sic2  as well as the strength
of $P-$symmetry. Primarily, to show the dynamical transitions in the
parametric space, we have plotted the two-parameter diagrams in the
$(K,\Delta)$ space for $\omega=1.0$. Figure 6(a) is plotted for the absence of
perturbation ($\varepsilon=0.0$), and it clearly depicts that at lower values
of threshold, there exists a direct transition from desynchronized state to
cluster oscillatory state. The region for the desynchronized state becomes
increased with an increase in the threshold. Further, a moderate value of
threshold exhibits the transition from desynchronized state to cluster
oscillatory state through the partial synchronization state. On increasing the
threshold further, the region for cluster oscillatory state gets decreased and
suppressed completely at larger thresholds. At larger coupling strengths, we
find that the existence of oscillation death state is confirmed and that the
region for oscillation death state gets increased with the increase in the
threshold. In addition, while increasing the threshold, we notice a new
additional branch of amplitude death state between the cluster oscillatory
state and oscillation death state.

Figure 6: Two parameter diagrams of globally coupled nonidentical counter-
rotating oscillators in $(K,\Delta)$ space with different perturbations, (a)
$\varepsilon=0.0$, and (b) $\varepsilon=0.1$. $DS$, $PS$, and $COS$ are the
desynchronized state, partial synchronization state and cluster oscillatory
state, respectively. $IMS$, $CH$ and $CMS$ are the incoherent mixed
synchronization, chimera and coherent mixed synchronization states,
respectively. $AD$ and $OD$ correspond to the amplitude death and oscillation
death states, respectively. Other parameters: $\lambda=1.0$, $\omega=1.0$, and
$N=100$.

Furthermore, Fig. 6(b) is plotted for the additional perturbation
$\varepsilon=0.1$. At lower values of coupling strength, we observe the
incoherent mixed synchronization state, and the region for incoherent mixed
synchronization state is increased up to a certain threshold and then it again
decreases. As a function of coupling strength, the incoherent mixed
synchronization state transits to coherent mixed synchronization state through
the chimera state (partial symmetry breaking state). The region for chimera
state increases with a decrease in coherent mixed synchronization state
region. Further, oscillation death state is noticed at still larger coupling
values and the spread of the $OD$ region gets increased as a function of
threshold. Also, we find that the additional perturbation suppresses the
amplitude death state completely. .

Figure 7: Two parameter diagrams of globally coupled nonidentical counter-
rotating oscillators in $(K,\varepsilon)$ parametric space by fixing different
thresholds for (a) $\Delta=0.0$, (b) $\Delta=0.2$, (c) $\Delta=0.5$, and (d)
$\Delta=0.8$. Other parameters are same as in Fig. 6

Additionally, the two-parameter diagrams are displayed in the
$(K,\varepsilon)$ parametric space in Fig.7 for four distinct values of the
threshold to provide a better understanding of the dynamical transitions as a
function of the perturbation. Figure 7(a) is depicted for the identical
counter-rotating frequencies, i.e. by fixing the threshold $\Delta=0.0$. We
discover that the identical counter-rotating oscillators display a direct
transition from $CMS$ to $OD$ and that the region for $CMS$ state decreases
with increasing $OD$ region. By increasing the threshold to $\Delta=0.2$, we
find that the onset of distinct dynamical states at the lower coupling
strengths (see Fig. 7(b)). Importantly, we find the $DS$ state at the lower
values of coupling strength and perturbation. Then, it transits to $IMS$ state
through $CH$ state while increasing the perturbation. In the absence of
perturbation, increasing the coupling exhibits a transition from $DS$ state to
$OD$ state via $PS$ and $COS$ states (not explicitly displayed in the two
parameter diagrams). Further, when the perturbation strength is slightly
increased, the $COS$ state becomes a $CMS$ state. Comparing Fig. 7(a), the
$CMS$ state region is shortened in Fig. 7(b). At larger coupling, there exists
the $OD$ region and it increases as a function of perturbation. On increasing
the value of the threshold even more to $\Delta=0.5$ and $\Delta=0.8$, we
notice that the regions for $DS$, $CH$, $IMS$, and $OD$ states get increased
with decreasing $CMS$ region. Additionally, we notice the onset of a new
branch of amplitude death state at lower values of perturbation before the
onset of oscillation death state, and its region is enlarged at a larger
threshold. In addition, the AD region vanishes completely while increasing the
perturbation to larger values. In the next section, we examine whether the
observed chimera is robust in larger network sizes.

##  VIII Robustness of chimera at larger sizes of the network

Figure 8: Spatio-temporal behavior and error of $P-$symmetry for two groups of
non-identical counter-rotating oscillators for the network size (a)-(b)
$N=1000$ and (c)-(d) $N=2500$ by fixing the coupling strength $K=0.82$. The
inset in (b) and (d) denote zoomed views of the symmetry breaking oscillators
in the chimera state. Other parameters: $\lambda=1.0$, $\Delta=0.5$,
$\omega=1.0$, and $\varepsilon=0.1$.

The robustness of the observed chimera state is also investigated at $K=0.82$,
using spatio-temporal behavior and the corresponding error of $P-$symmetry in
Fig. 8 by fixing the network sizes as $N=1000$ and $N=2500$. From the spatio-
temporal plots in Figs. 8(a) and 8(c), it is clear that the existence of the
chimera state with coexisting coherent and incoherent patches of oscillators
for the network sizes $N=1000$ and $N=2500$. Further, the partial symmetry
breaking in the chimera state is also confirmed through the error of
$P-$symmetry measure in Figs. 8(b) and 8(d) which clearly depict that some of
the oscillators are symmetry breaking ($PS_{i}>0$) while the remaining
oscillators are symmetry preserving ($PS_{i}=0$). The inset of Figs. 8(b) and
8(d) denote the zoomed view of the symmetry-breaking oscillators. Thus, it is
clear that the structure of the observed chimera state is also unaffected by
increasing the size of the network and it is independent of the number of the
oscillators in the network.

In addition, the generality of the above observed dynamical states and their
transitions will be confirmed for two other different dynamical models, namely
globally coupled van der Pol and Rössler oscillators in Appendix B. We will
explicitly demonstrate the existence of partially synchrnonized as well as
chimera states in these systems also to prove their generic nature.

##  IX Conclusions

Co-existence of co- and counter-rotation can be found in many physical,
mechanical, and biological systems. In particular, frequency plays a
significant role in generating co- and counter-rotation in a dynamical system.
Hence, in the present work, we have explored the dynamical transitions by
distributing the non-identical counter-rotating frequencies. To exemplify
this, we considered a network of non-identical and globally coupled Stuart-
Landau oscillators with additional perturbation. Primarily, the dynamical
transitions are investigated in the absence of perturbation and it is observed
that transitions from desynchronized state to cluster oscillatory state via
the partial synchronization state in the oscillatory regime and amplitude
death to oscillation death in the oscillation quenching regime do occur.
Further, we find that the observed dynamical states are breaking the
parity$(P)-$symmetry in the absence of perturbation. When the perturbation is
increased the system preserves $P-$symmetry and exhibits incoherent and
coherent mixed synchronization states. Interestingly, we have identified the
partial symmetry breaking state between the incoherent and coherent mixed
synchronization states. Such a partial symmetry breaking state is referred to
as a chimera state. Thus, when the entire set of oscillators in the coherent
and incoherent groups either preserve or break the $P-$symmetry as a whole,
then it is referred to as the partial synchronization state. Otherwise, if
only a partial set of the oscillators break the $P-$symmetry while the
remaining oscillators preserve the $P-$symmetry then it is called as the
chimera state. The mechanism for the coexistence of partial symmetry breaking
and symmetry preserving dynamics in the chimera state is further confirmed
through the basin stability analysis. Followed by this, we introduced a
statistical measure, namely the strength of $P-$symmetry, to quantify the
partial symmetry breaking state. Using this measure, the existence of the
partial symmetry breaking state is validated and it exists only when fixing
the coupling parameter at the desynchronized and partial synchronization
states. Moreover, the global dynamical transitions are also investigated in
the presence and in the absence of perturbation as well as by fixing different
frequency thresholds. Increasing the threshold enlarges the regions for
desynchronized state, incoherent mixed synchronization, chimera, and
oscillation death states. Ultimately, raising the threshold range decreases
the region of coherent mixed synchronization state. The onset of amplitude
death is also observed for increasing the thresholds and it gets suppressed
while increasing the perturbation. Finally, we have also examined the
robustness of the chimera state by increasing the network size. We find that
the observed chimera state is structurally similar and robust at larger sizes
of the network as well. In summary, we have revealed for the first time the
occurrence of novel chimera states in a network of globally coupled non-
identical counter-rotating Stuart-Landau oscillators in this work. We further
examined the generality of the observed oscillatory states and their
transitions in other dynamical systems as well. It is discovered that the
newly observed partial synchronization and chimera states, as well as their
dynamical transitions, are robust in globally coupled van der Pol and Rössler
oscillators also. We believe that our findings will shed further light on the
dynamical transitions in counter-rotating oscillators in the science and
engineering disciplines.

## Acknowledgments

KS thank the DST-SERB, Government of India, for providing National Post
Doctoral fellowship under the Grant No. PDF/2019/001589. The work of VKC is
supported by the SERBDST-MATRICS Grant No. MTR/2018/000676 and DSTCRG Project
under Grant No. CRG/2020/004353 and VKC wish to thank DST, New Delhi for
computational facilities under the DST-FIST programme (SR/FST/PS- 1/2020/135)
to the Department of Physics. M.L. wishes to thank the Department of Science
and Technology for the award of DST-SERB National Science Chair.

##  Appendix A  Phase reduced model: Co-existence of coherent and incoherent
dynamical behaviors and their dynamical transitions

In addition to the results discussed in the main text, the observed
oscillatory states are validated here in this Appendix using a reduced phase
model, which can be derived by substituting $z_{j}=r_{j}e^{i\theta_{j}}$ in
Eq. (1). Then by separating the real and the imaginary parts and by
considering $r_{j}=r_{k}=1$, we can get the reduced phase model expression as
given below

$\displaystyle\dot{\theta_{j}}=\omega_{j}-\frac{K+\varepsilon}{N}\sum_{k=1}^{N}(\cos(\theta_{k})-\cos(\theta_{j}))\sin{\theta_{j}}$
$\displaystyle+\frac{K-\varepsilon}{N}\sum_{k=1}^{N}(\sin(\theta_{k})-\sin(\theta_{j}))\cos{\theta_{j}}.$
(5)

Figure 9: Phases of oscillators are plotted using Eq. (5) and by fixing
$\varepsilon=0.0$ for (a) $K=0.1$ (desynchronized state), (b) $K=0.65$
(partial synchronization state), and (c) $K=1.5$ (cluster oscillatory state).
(d)-(e) are the corresponding dynamical behaviors at the perturbation strength
$\varepsilon=0.3$. Other parameters: $\Delta=0.5$, $\omega=1.0$, and $N=100$.
Figure 10: Mean-field ($\Omega$) and mean-ensemble frequency ($f$) as a
function of coupling strength ($K$) for (a) $\varepsilon=0.0$ and (b)
$\varepsilon=0.3$. Red and black linepoints indicate the range of mean-field
and mean-ensemble frequencies. Other parameters are same as in Fig. 9.

Using Eq. (5), the phases of the oscillators are plotted in Fig. 9. Figures
9(a)-9(c) in the top panel correspond to the absence of perturbation, that is
for $\varepsilon=0.0$, and Figs. 9(d)-9(f) in the lower panel depict the
dynamical states for the perturbation strength $\varepsilon=0.3$. In Fig. 9(a)
for $K=0.1$, we can see that the phases of the oscillators are randomly
distributed, with the first group (co-rotating oscillators) having negative
random phases and the second group (counter-rotating oscillators) showing
positive random phases. Such a state with incoherent random phases corresponds
to a desynchronized state. When the coupling strength is increased to $K=1.1$,
some of the oscillators in the first and second groups form coherent patches
with coherent phases while the remaining oscillators portray incoherent
phases, as shown in Fig. 9(b). Such coexistence of coherent and incoherent
phases in a dynamical state in globally coupled oscillators is known as
partial synchronization. On increase of the coupling strength to even higher
value (for example, $K=2.0$), the phases of co- and counter-rotating
oscillators display two independent clusters resulting in the cluster
oscillatory state as shown in Fig. 9(c). Thus, it is clear that the phases of
the oscillators also show a dynamical transition from the desynchronized state
to the cluster oscillatory state through the partial synchronization state.

In addition, the oscillatory states are also analyzed in the presence of the
additional perturbation by fixing $\varepsilon=0.3$. From Figs. 9(d)-9(f), we
can note that the system (5) exhibits a similar dynamical bahaviors even in
the presence of additional perturbation as seen in Figs. 9(a)-9(c). However,
in this case, all of the oscillators in Figs. 9(d) show symmetry breaking,
resulting in a desynchronized state. It is to be noted that the coherent
patches in Fig. 9(e) are symmetry preserving while the incoherent patches are
symmetry breaking. Therefore Fig. 9(e) is a partial symmetry broken state that
is analogous to a chimera state. Similarly, the coherent patches in Figs. 9(f)
preserve symmetry, hence it is equivalent to incoherent mixed synchronization.
In this manner, we validated our results using the phase-reduced model as
well. Particularly, the existence of partial synchronization and chimera
states have been confirmed.

Furthermore, as discussed in ref. rev3 ; rev4 , we calculated the mean-field
frequency ($\Omega$) and mean-ensemble frequency ($f$). The corresponding
diagrams are illustrated in Figs. 10(a) and 10(b) as a function of the
coupling strength $(K)$ without and with additional perturbations
$\varepsilon=0.0$ and $\varepsilon=0.3$. We observe the disordered values of
the mean-field frequency in the dynchronized ($DS$) and partial
synchronization ($PS$) states in the absence of perturbation
($\varepsilon=0.0$), as shown in Fig. 10(a) by the red linepoints.
Additionally, the cluster oscillatory state or steady states show the presence
of ordered coherent mean-field frequencies. We also noticed that on increasing
the additional perturbation to $\varepsilon=0.3$, the disorderliness (see Fig.
10(b)) of the mean-field frequencies observed at the $DS$ and $PS$ states
decreases. It is also important to notice that in both the cases of additional
perturbations, the system(5) shows a lower range of coherent mean-ensemble
frequencies.

##  Appendix B Robustness of observed dynamical states and their transitions
in other dynamical models

Figure 11:  Spatio-temporal plots of globally coupled van der Pol oscillators
in the absence of perturbation ($\varepsilon=0.0$) for (a) $K=0.05$
(desynchronized state), (b) $K=0.65$ (partial synchronization state), (c)
$K=0.8$ (cluster oscillatory state), and (d) $K=1.0$ (oscillation death).
(e)-(h) are the snapshots of $x_{j}$ variables and (i)-(l) are the
corresponding parity symmetry error $(MS_{i})$. Other parameters: $b=2.0$,
$\Delta=0.5$ and $N=100$. Figure 12: Spatio-temporal plots of globally coupled
van der Pol oscillators in the presence of perturbation ($\varepsilon=0.3$)
for (a) $K=0.05$ (incoherent mixed synchronization state), (b) $K=0.65$
(chimera state), (c) $K=0.8$ (chimera state), and (d) $K=1.0$ (oscillation
death). (e)-(h) are the snapshots of $x_{j}$ variables and (i)-(l) are the
corresponding parity symmetry error $(MS_{i})$. Other parameters are same as
in Fig. 11.

In order to confirm the generality of the observed dynamical states in other
dynamical models, we choose the globally coupled van der Pol and Rössler
nonidentical counter-rotating systems with additional perturbation. The
corresponding dynamical transitions are detailed in the following subsections.

###  B.1 Globally coupled van der Pol oscillators

The dynamical equations for the ring network of globally coupled van der Pol
oscillators are expressed as

$\displaystyle\dot{x_{j}}$ $\displaystyle=$
$\displaystyle\omega_{j}y_{j}+\frac{K+\varepsilon}{N}\sum_{k=1}^{N}({x_{k}}-\,{x_{j}}),$
$\displaystyle\dot{y_{i}}$ $\displaystyle=$ $\displaystyle
b(1-x_{j}^{2})y_{i}+\frac{K-\varepsilon}{N}\sum_{k=1}^{N}({y_{k}}-\,{y_{j}}),$
(6)

where $b$ is the damping parameter which is chosen as $b=2$. $\omega_{j}$ are
the system frequencies that are distributed with uniform counter-rotating
frequencies as detailed in Sec. II. $K$ is the coupling strength and
$\varepsilon$ is the additional perturbation. The dynamical states observed
from Eq. (6) are portrayed in Figs. 11 and 12 in the absence and in the
presence of additional perturbation. Primarily, the dynamical states and their
transitions are shown by fixing $b=2.0$, $\varepsilon=0.0$, and $\Delta=0.5$.

At lower coupling strength $K=0.05$, the existence of desynchronized state is
observed as shown in Figs. 11(a) and 11(e). On increasing the coupling
strength to $K=0.65$, the oscillators in the system (6) get split into
coherent and incoherent groups and exhibit a partial synchronization state as
shown in Figs. 11(b) and 11(f). Here the incoherent behavior is observed at
the edges and middle of the network. Further increment of $K$ to $K=0.8$
results in the entire set of oscillators to form two independent clusters
manifesting into a cluster oscillatory state. In this state, the co-rotating
oscillators form one cluster and the corresponding counter-rotating
oscillators form the other cluster (see Figs. 11(c) and 11(g)). Furthermore,
at strong coupling strength $K=2.0$, we find that the co- and counter-rotating
oscillators populate into two different cluster oscillation death states as
shown in Figs. 11(d) and 11(h).

In addition, we illustrate the parity symmetry error of each dynamical state
in Figs. 11(i)-11(l). By observing all the dynamical states in the absence of
additional perturbation, $\varepsilon=0.0$ (see Figs.11(a) and 11(d)), we
discover that each pair of oscillators yield a non-zero value of parity
symmetry error indicating that the observed dynamical states including
$DS,~{}PS,~{}COS$ and $COD$ states are of $P-$symmetry broken nature.
Subsequently, we perform a similar analysis in the presence of additional
perturbation by fixing $\varepsilon=0.3$. As compared to the desynchronized
state, here we find that each pair of the oscillators in the network preserves
the parity symmetry (see Figs. 12(a) and 12(e) for $K=0.05$) and results in
the incoherent mixed synchronization state. Therefore all the pairs of the
oscillators show null value of $P-$symmetry error $MS_{i}$ in Fig. 12(i).
Further, the occurrence of chimera state is revealed while increasing $K$ to
$K=0.65$ (see Figs. 12(b) and 12(f)) and $K=0.8$ (see Figs. 12(c) and 12(g)).
In the chimera state, some of the pairs of oscillators in the coherent and
incoherent patches preserve the $P-$symmetry ($MS_{i}=0$) while the others
break the $P-$symmetry ($MS_{i}\neq 0$) as illustrated in Figs. 12(j) and
12(k). Furthermore, we discover similar kind of cluster oscillation death in
Figs. 12(d), 12(h) and 12(l) at $K=2.0$ as observed in Figs. 11(d), 11(h) and
11(l). As a result, our analysis clearly shows that the network of globally
coupled van der Pol limit cycle oscillators also shows similar kind of
dynamical states as that of the globally coupled Stuart-Landau oscillators.

Furthermore, to provide additional support for our findings, we examine the
dynamical behaviors using a chaotic system, specifically Rössler oscillators,
in the following.

###  B.2 Globally coupled Rössler oscillators

In order to show the existence of partial synchronization and chimera states
in a chaotic system, we consider the ring network of globally coupled Rössler
oscillators which can be written as

$\displaystyle\dot{x_{j}}$ $\displaystyle=$
$\displaystyle-\omega_{j}y_{j}-z_{j}+\frac{K+\varepsilon}{N}\sum_{k=1}^{N}({x_{k}}-\,{x_{j}}),$
$\displaystyle\dot{y_{j}}$ $\displaystyle=$
$\displaystyle\omega_{j}x_{j}+ay_{j}+\frac{K-\varepsilon}{N}\sum_{k=1}^{N}({y_{k}}-\,{y_{j}}),$
$\displaystyle\dot{z_{j}}$ $\displaystyle=$ $\displaystyle b+z_{j}(x_{j}-c)$
(7)

Figure 13: Spatio-temporal plots of globally coupled Rössler oscillators with
non-identical counter-rotating frequencies in the absence of additional
perturbation $(\varepsilon=0.0)$ for (a) $K=0.05$ (desynchronized state), (b)
$K=0.1$ (partial synchronization state), (c) $K=0.15$ (amplitude death state),
(d)-(f) are the snapshots of $x_{j}$ variables, (g)-(i) are the corresponding
parity symmetry error $(MS_{i})$. Other parameters:
$a=0.2,~{}b=0.2,~{}c=10.0,~{}\Delta=0.1$ and $N=100$.  Figure 14: Spatio-
temporal plots of globally coupled Rössler oscillators with non-identical
counter-rotating frequencies in the presence of additional perturbaton
$(\varepsilon=0.05)$ for (a) $K=0.05$ (desynchronized state), (b) $K=0.1$
(chimera state), (c) $K=0.15$ (coherent mixed synchronization state), (d)-(f)
are the snapshots of $x_{j}$ variables, (g)-(i) are the corresponding parity
symmetry error $(MS_{i})$. Other parameters are same as in Fig. 13

where, $a,~{}b$, and $c$ are the constant parameters and they are chosen as
$a=b=0.2$, and $c=10$. $K$ is the coupling strength, $\varepsilon$ is the
strength of additional perturbation, and $\omega_{j}$ are the system
frequency. The co- and counter-rotating frequencies are distributed with
uniform distribution as detailed in Sec. II. Primarily, the dynamical
behaviors are first analyzed for $\varepsilon=0.0$ and by fixing the coupling
strength at different values. For $K=0.01$, one can note from Figs. 13(a) and
13(d) that the oscillators are randomly distributed in the desynchronized
state. Then the existence of partial synchronization state with coexistence of
coherent and incoherent behaviors is revealed from Figs. 13(b) and 13(e) for
$K=0.2$. Upon increasing the coupling strength, the system exhibits amplitude
death state as seen in Figs. 13(c) and 13(f). From the figures of error of
parity symmetry given in Figs. 13(g) and 13(h), we can note that the
desynchronized state and partial synchronization state are complete
$P-$symmetry broken where all the oscillators take non-zero values of error.
Due to trivial values of system variables, the $P-$symmetry error of amplitude
death state takes the null value as in Fig. 13(i).

Furthermore, the effect of additional perturbation is explored by fixing
$\varepsilon=0.05$ in Figs. 14. For $K=0.05$, we observed a desynchronized
state as in Figs. 14(a)-14(d). The corresponding $P-$symmetry error takes non-
zero values of error which implies that the $P-$symmetry is broken in the $DS$
state, Fig. 14(g). On increasing the coupling strength further, chimera states
with the co-existence of coherent and incoherent behaviors are observed which
is depicted in Figs. 14(b) and 14(e). In such a state, some of the oscillators
preserve the $P-$symmetry while the others do not preserve the $P-$symmetry.
As a result, the symmetry preserving pairs of oscillators in the network show
zero error while the symmetry broken pairs show non-zero values of error as
shown in Fig. 14(h). On further increasing the coupling strength ($K=0.15$),
we restore the oscillatory dynamics from amplitude death due to the influence
of the additional perturbation. As a consequence, we observed a symmetry
preserving cluster state that is a coherent mixed synchronization state (see
Figs. 14(c),14(f), and 14(i).

Thus, from the above discussion, one can clearly observe that the results
presented in the main text are more general and valid in all classes of limit
cycle and chaotic systems. Interestingly, we identified that the additional
perturbation leads to the onset the chimera behavior in nonidentical counter-
rotating systems.

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