# Fractional damping enhances chaos in the nonlinear Helmholtz oscillator

Adolfo Ortiz  Centro de Investigación en Micro y Nanotecnología, Facultad de
Ingeniería, Universidad Veracruzana, Calz. Ruiz Cortinez 455, CP 94294, Boca
del Río, Veracruz, México  Jianhua Yang  School of Mechatronic Engineering,
China University of Mining and Technology,
Xuzhou 221116, P. R. China<EMAIL_ADDRESS>Mattia Coccolo
Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física ,
Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain  Jesús
M. Seoane  Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento
de Física , Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid,
Spain  Miguel A.F. Sanjuán  Nonlinear Dynamics, Chaos and Complex Systems
Group, Departamento de Física , Universidad Rey Juan Carlos, Tulipán s/n,
28933 Móstoles, Madrid, Spain

###### Abstract

The main purpose of this paper is to study both the underdamped and the
overdamped dynamics of the nonlinear Helmholtz oscillator with a fractional
order damping. For that purpose, we use the Grünwald-Letnikov fractional
derivative algorithm in order to get the numerical simulations. Here, we
investigate the effect of taking the fractional derivative in the dissipative
term in function of the parameter $\alpha$. Our main findings show that the
trajectories can remain inside the well or can escape from it depending on
$\alpha$ which plays the role of a control parameter. Besides, the parameter
$\alpha$ is also relevant for the creation or destruction of chaotic motions.
On the other hand, the study of the escape times of the particles from the
well, as a result of variations of the initial conditions and the undergoing
force $F$, is reported by the use of visualization techniques such as basins
of attraction and bifurcation diagrams, showing a good agreement with previous
results. Finally, the study of the escape times versus the fractional
parameter $\alpha$ shows an exponential decay which goes to zero when $\alpha$
is larger than one. All the results have been carried out for weak damping
where chaotic motions can take place in the non-fractional case and also for a
stronger damping (overdamped case), where the influence of the fractional term
plays a crucial role to enhance chaotic motions. We expect that these results
can be of interest in the field of fractional calculus and its applications.

Nonlinear oscillations and Delay systems and Resonance and Fractional
derivatives

###### pacs:

05.30.Pr, 45.10.Hj, 82.40.Bj

##  I Introduction

Fractional calculus is an area of the mathematical analysis. It studies the
possibilities of taking real or even complex numbers as orders of the integral
and derivatives on a known or unknown function. Such operators are rather
useful in science and engineering. Also, fractional differential operators
involve defined integrals over a time domain, this poses significant memory
effects as shown in Ref. book:Yang  . For these reasons, the fractional
calculus has gained much attention and relevance in the past few years due to
its applications to several research fields such as control systems, nonlinear
oscillators, potential fields, diffusion problems, viscoelasticity and
rheology, synchronization, thermodynamics, biology among others Wang:Huang ;
Ezz ; Baleanu ; article:yang ; AMINA:2019 ; SHUO:2020 ; HeJi ; ionescu . In
the case of nonlinear oscillators, the research has been very extensive with
interesting results, especially in the study of the effect of introducing a
fractional derivative in the damping term NIU:2019 . In the last reference,
the Authors carried out the study of chaos detection in the Duffing oscillator
by using the Melnikov Method. On the other hand, the detection of chaos in
fractional systems by using SALI has been developed in Ref. HE:2019 .
Furthermore, the fractional Duffing equation in the presence of nonharmonic
external perturbations has been also studied with detail in Ref. JGV:2013 .
Recent and very complete reviews on the current research and real applications
of fractional calculus in science and engineering can be found in Refs.
REVIEW:APLICATIONS ; VAZQUEZ:2020 ; Yang ; Ouannas ; Wang ; antegana .

In our present work, due to the fact that the dissipation plays an important
role in the evolution of the systems, we focus our interest in analyzing the
effect of a fractional damping on a nonlinear oscillator dynamics. In fact, we
modify the equation of motion of the Helmholtz oscillator, which is being used
in many physical problems, in order to introduce a fractional differential
operator in the damping term instead of the classical derivatives. Then, we
focus our attention on the effect of the fractional operator on the dynamics
of the system by studying the behavior of the trajectories, this means, if the
asymptotic solutions fall inside or outside the potential well defined by the
Helmholtz potential. This is analyzed using the Grünwald-Letnikov fractional
derivative algorithm book:Yang ; ALGORITHM:1  for our numerical simulations,
in which we change certain initial condition, the fractional derivative and
the forcing amplitude for two values of the damping parameter. These two
damping values have been chosen in order to set the system in an underdamped
or in an overdamped case.

In fact, the Helmholtz oscillator has a very rich dynamical behavior for small
values of the damping parameter where both, chaotic and periodic motions can
take place and also escapes from the potential well are possible. On the other
hand, when the damping parameter grows, the system becomes more and more
predictable until all particles fall into the well. However, in the overdamped
case, the effect of the fractional derivative can generate different kind of
motions, including chaotic motion. That is why the underdamped case has been
the most interesting one to study in the nonlinear dynamics field in the last
decades, but the overdamped case can be interesting when fractional
derivatives are involved.

By using the above algorithm we calculate the orbits in phase space and show,
in the parameter space, the attractor and the escape times, i.e., the final
state of the trajectories, inside or outside the potential well, and the time
to reach such state, respectively. The parameters that we change in the
simulations are: the damping term $\mu$, the fractional derivative $\alpha$,
the forcing $F$ and the initial condition $x_{0}$. For visualization purposes,
we have plotted in parameter space, both the attractor and the escape time
gradient (as can be seen along the manuscript). In these figures, it is
possible to appreciate the complexity of the relation between the fractional
parameter and the dynamics of the system. In fact, the attractor boundaries
show some sort of fractalization, which means that a small deviation in the
parameters value can lead to drastically different solutions, i.e., the system
asymptotic behavior can stay inside the well or leave it. It clearly means
that, in the fractal regions, errors in the parameters might cause different
final states, as is well known in chaotic systems. Indeed, these results show
that using the fractional parameter $\alpha$ as a control parameter, can
provide different dynamics for the dynamical system, either periodic or
chaotic motions and transitions between them. This is precisely one of the
main goals of the research work reported here by using phase-space
visualizations techniques. Among the goals, we can find a natural extension to
the fractional case the well-known results obtained for the non-fractional
case in both cases, the underdamped and the overdamped regimes.

The organization of this paper is as follows. We describe the fractional
damped Helmholtz oscillator, in Sec. II. The dependence of the trajectories in
function of the fractional parameter $\alpha$ for the underdamped case is
carried out in Sec. III. Section IV shows the distribution of the escape times
of the particles from the potential well versus $\alpha$ and also the analysis
of the bifurcation diagrams in function of $\alpha$ for an overdamped case.
Conclusions and a discussion of the main results of this paper are presented
in Sec. V.

##  II Fractional Damped Helmholtz Oscillator

Figure 1: Plot of the potential well defined through the function
$V(x)=\frac{1}{2}x^{2}-\frac{1}{3}x^{3}$. Note that for the interval
$x\in[-\frac{1}{2},1)$ in absence of forcing and friction the orbits are
bounded.

The Helmholtz oscillator is represented by a nonlinear second order
differential equation considering the presence of the potential defined as
$V(x)=\frac{1}{2}x^{2}-\frac{1}{3}x^{3}$, as shown in Fig. 1. It represents
the equation of motion of a unit mass particle in this potential under the
influence of both a periodic forcing and a dissipative force and is given by

$\displaystyle\ddot{x}+\mu\dot{x}+x-x^{2}=F\cos(\omega t),$ (1)

where $\mu$ is the damping parameter, $F$ and $\omega$ the forcing amplitude,
and the forcing frequency, respectively, taking all of them positive values.
Normally, the damping term, considered like the first derivative in Eq. (1),
has an impact proportional to the constant $\mu$ in the system. In what
follows, the first derivative of Eq. (1) is replaced by an order $\alpha$
fractional derivative, i.e, $\dot{x}(t)\rightarrow D^{\alpha}x(t)$, for which
we have the Fractional Helmholtz Oscillator. We consider this exchange because
of the interest in looking for a more general dynamics of the oscillator as a
consequence of considering the memory effects through the fractional operator.
As a matter of fact, the most relevant objective here is to investigate
changes over the whole dynamics taking fractional order derivatives in the
interval $[0,2]$ of $\alpha$ values, as is usually studied in the context of
fractional nonlinear oscillators. Then, the Fractional Helmholtz Oscillator
with fractional damping reads

$\displaystyle\ddot{x}+\mu D^{\alpha}x+x-x^{2}=F\cos(\omega t),$ (2)

with the additive property of the fractional derivative, one can get a new
system for the purpose of our numerical simulations, which can be expressed
as:

$\displaystyle D^{\alpha_{1}}D^{\alpha_{2}}x(t)=D^{\alpha_{1}+\alpha_{2}}x(t)$
(3)

Thus, the new system is composed by a set of three fractional differential
equations according to Eq. (3) and they read as follows:

$\displaystyle D^{\alpha}x$ $\displaystyle=$ $\displaystyle y$ (4)
$\displaystyle D^{1-\alpha}y$ $\displaystyle=$ $\displaystyle z$
$\displaystyle Dz$ $\displaystyle=$ $\displaystyle F\cos(\omega t)+x^{2}-x-\mu
y,$

where $z$ is a mathematical component coming from the transformation of the
model into a fractional order system. To perform the solution of Eqs. (4), we
use the Grünwald-Letnikov ALGORITHM:1  fractional derivative, for which the
algorithm to numerically solve this system is given by

$\displaystyle x(t_{k})$ $\displaystyle=$ $\displaystyle
y(t_{k-1})h^{\alpha}-\sum_{j=\upsilon}^{k}c_{j}^{(\alpha)}x(t_{k-j})$ (5)
$\displaystyle y(t_{k})$ $\displaystyle=$ $\displaystyle
z(t_{k-1})h^{1-\alpha}-\sum_{j=\upsilon}^{k}c_{j}^{(1-\alpha)}y(t_{k-j})$
$\displaystyle z(t_{k})$ $\displaystyle=$ $\displaystyle\Psi
h-\sum_{j=\upsilon}^{k}c_{j}^{(1)}z(t_{k-j}),$

where $\Psi=F\cos(\omega t_{k})+x^{2}(t_{k})-x(t_{k})-\mu y(t_{k})$ and $h$ is
the discrete time step. The coefficients $c_{j}^{\alpha}$ are the binomial
coefficients derived in the numerical scheme implemented, $c_{0}^{\alpha}=1$
and

$\displaystyle c_{j}^{\alpha}=(1-\frac{\alpha+1}{j})c_{j-1}^{\alpha}.$ (6)

In the next subsection, we numerically analyze the dynamics of our model with
the fractional algorithm for $\alpha=1$ and with a non-fractional algorithm.
Then, we compare the results.

In order, and with the only purpose, to test the numerical solutions of the
fractional numerical scheme, a comparison with a fourth order Runge-Kutta
method is carried out. This comparison only is possible for $\mu=0$ or
$\alpha=1$ which corresponds with the non-fractional case.

Figure 2: Comparison between solutions provided by distinct methods for the
damped Helmholtz oscillator at the zero initial conditions with $\mu=0.1$ and
$F=0.1$ in panels (a) and (b), $\mu=0.8$ and $F=0.46$ in panels (c) and (d).
Panels (a) and (c) show the trajectories of the solution of the fourth-order
Runge-Kutta scheme. Panels (b) and (d) show the trajectories of the solution
of the fractional numerical scheme with $\alpha=1$.

In Fig. 2, we show the resulting orbits starting at zero initial conditions,
which means $(x_{0},y_{0},z_{0})=(0,0,0)$, for both cases the underdamped and
the overdamped, the other parameter values are shown in the caption. In Figs.
2(a)-(c) the trajectories are calculated by using the Runge-Kutta method and
in Figs. 2(b)-(d) with the Grünwald-Letnikov method with $\alpha=1$. In this
case, the fractional method scheme gives us the same trajectories as the one
for the Runge-Kutta method, showing bounded motions of the particle inside the
potential well. This is a first proof of the adequate response of the
fractional algorithm. Now, we focus on an interesting point that we
investigate, the existence of orbits that escape from the potential well,
defined above, and the time that the particles need to leave the potential
well, namely the escape time. In Fig. 3 we represent an extra comparison
between the two methods by plotting the escape times of the orbits varying the
initial conditions inside the well, for parameter values for which the
particles escape from the well. For that purpose, we set $F=0.46$ and
$\omega=1$ with $\mu=0.1$ in Figs. 3(a)-(b) and with $\mu=0.8$ in Figs.
3(c)-(d). Then, we vary the initial conditions in the interval $x\in(0.85,1)$.
In the non-fractional Helmholtz oscillator, the effects of dissipation help to
keep the particles inside the well. For that reason, when the damping term
grows bigger and the system is in an overdamped regime, the dynamics of the
system becomes more predictable since for all initial conditions the particles
need more time to escape the well and more initial conditions fall into the
bottom of the potential well. On the other hand, in the case of weak
dissipation, the dynamics of the system are very rich and the escape times
become smaller. In fact, in the panel it is possible to appreciate that the
escape time follow that trend, since for $\mu=0.8$ the particle needs more
time to leave the potential well. As a summary of our comparison, also here,
in the left figures of the panel we have used the Runge-Kutta integrator,
while for the others the fractional numerical scheme. In general, the orbits
take more time to escape in the fractional scheme, as supported by Fig. 3, due
to the characteristics of the Grünwald-Letnikov method of integration Scherer
. However, for the sake of the results of this paper, we want to stress out
that, in both numerical schemes, the escape time curves follow the same trend.
This last affirmation can be proved trough the comparison of the fitting
curves, that appear in the textboxes of each figure. Starting from Figs.
3(a)-(b), we can see that the fitting curve is exponential for both
algorithms, but also that the parameter values of the first one falls within
the $95\%$ confidence bounds, that are the numbers inside the parenthesis of
the other one, and vice versa. The same thing happens for the fitting curves
in Figs. 3(c)-(d). Therefore, we can confidently affirm that, in both cases,
the fitting curves of the escape times calculated with the Runge-Kutta method
follow the same equation of the ones calculated with the Grünwald-Letnikov
algorithm and their parameter match with a confidence at least of the $95\%$.
This gives us another good proof of the reliability of the fractional
algorithm. That settled, we can start our analysis of the system.

Figure 3: Comparison between escape times of solutions provided by distinct
methods for the damped Helmholtz oscillator at $x_{0}\in(0.85,1)$ with
$\mu=0.1$ in panels (a) and (b), $\mu=0.8$ in panels (c) and (d). In all the
panels $F=0.46$ and $\omega=1$. Panels (a) and (c) show the escape times
calculated with the Runge-Kutta method. Panels (b) and (d) show the escape
times calculated with the fractional numerical scheme and $\alpha=1$. In the
textboxes we can see the fitting equations and their parameter. Near the
parameter, in the parenthesis, we show the $95\%$ confidence bounds. This is
important to check that the fitting equations and their parameters values of
each couple of panels, (a)-(b) and (c)-(d), are compatible within each other.

##  III Dynamics of the Fractional Helmholtz Oscillator in the underdamped
case

Here, we show numerical simulations in order to understand the dynamics of the
fractional Helmholtz oscillator involving a fractional order damping for
different values of the damping parameter $\mu$. We fix the parameters of the
system for the underdamped case as $\mu=0.1$, $F=0.1$ and $\omega=1$ which are
a very convenient choice for this study. In this physical situation, as
previously mentioned, the dynamics of the non-fractional Helmholtz oscillator
is very rich. In this case, the fractional damping term will add more
complexity in both the dynamics and the topology of the system as we show now.

At first, we focus our attention to the final state of the system, whether the
trajectories stay inside the potential or leave it. Then, we study the
variation of the escape times of the particles from the well. The escape time,
$t$, is the time that the particle spends inside the well before crossing the
boundary ($x=1$ in this model) escaping from it. In fact, we consider that the
particle has escaped when it crosses the point $x>1$ with a positive velocity,
$\dot{x}>0$. For that purpose, we start with the analysis of the fractional
dynamics by changing the $\alpha$ and the $x_{0}$ initial condition values and
find critical values where the behavior of the system changes abruptly. This
analysis keep the other initial condition equal to zero and the same values of
the rest of the constant parameters as previously stated.

Figure 4: Figure (a) shows the final state of the system depending on the
value of $\alpha$, from $0.1$ to $2$, and the variation of the first initial
condition $x_{0}$ from $-3$ to $3$. White color (yellow on line) implies that
the particle remains inside the well and black color (blue on line) that
particle escapes. In (a) it is possible to see the areas where the boundaries
show the fractalization. Figure (b) shows the gradient of the particle escape
times from the potential well depending on the $\alpha$-value and the
variation of the first initial condition for the same values of figure (a).
Figures (c) and (d) show an example of the trend of the escape time in
function of $\alpha$, for different initial conditions, $x_{0}=-2$ and
$x_{0}=-3$ respectively.  Figure 5: Figure (a) shows the final state of the
system taking $\alpha=[0.1,1.8]$ and $F=[0,0.3]$ and keeping constant the rest
of the parameters. Gray color (yellow on line) implies the particle that
remains inside the potential well and black color (blue on line) that particle
escapes from it. Figure (b) shows, for the same range of $F$ and $\alpha$ and
the same parameter as in figure (a), the gradient escape times. The other
figures show slices of the figure (b), for different $\alpha$ values, such as
$0.25,1,1.25,1.39$, respectively.  Figure 6: Behavior of the fractional
damping Helmholtz oscillator with $\mu=0.1$, $F=0.1$ and $\omega=1$. The
figures represent the trajectories of the Helmholtz oscillator $z$ vs $x$ and
the oscillations of the position $x$ in time with different $\alpha$ values at
different initial conditions $x_{0}$. In particular $\alpha=0.39$, $x_{0}=0.4$
(a) and (b), $\alpha=0.4$, $x_{0}=0.4$ (c) and (d), $\alpha=0.41$, $x_{0}=0.5$
(e) and (f). In figures (a) and (b) the plots represent the typical bounded
orbit inside the potential well. In figures (c) and (d) the plots show a
faster escaping trajectory. In figures (e) and (f) the plots represent a
slower escaping trajectory.

Thus, we can see in Fig. 4(a) the final state of the system by varying the
values of the factional derivative $\alpha$ and the initial condition $x_{0}$.
That means that we have plotted the attractors, being an attractor a set of
numerical values toward which a system tends to evolve, for a wide variety of
starting conditions of the system. So that, it is possible to appreciate two
recognizable attractors, the gray one (yellow online) that represent the case
when the particle remains inside the well and black one (blue online) that
represent when the particles escape. It is, also, possible to appreciate the
fractalization of the boundaries between the different region of the figure.
In Fig. 4(b) we show the gradient of the escape time. Here, it is possible to
appreciate that near the boundaries that separate the two attractors defined
in Fig. 4(a) the escape times are larger and, in particular, in the zones that
show fractalization. Also, we depicted the trend of the escape times in
function of $\alpha$ to visualize how they change along the boundaries of the
attractor. In order to avoid fluctuations in the escape times, we decided to
move along the smoother boundary of the attractor, i.e., for $x_{0}<0$ and
$\alpha>0.5$, so we have fixed the initial conditions, $x_{0}=-2$, Fig. 4(c),
and $x_{0}=-3$, Fig. 4(d) and varied $\alpha$. It is possible to see that the
two curves do not share the same trend, as in the first one the exponential
curve does not fit the data very well, $R^{2}\approx 0.8$, while in the second
case it works perfectly, $R^{2}\approx 0.99$. The same thing happens for a
fixed $\alpha$ and varying $x_{0}$. So, we can say that for this value of
$\mu$ the trend of the escape times depends on the parameter values, chosen on
the boundaries of the attractor. Therefore, it is not possible to find a
common decay law for the escape times, just that the farther from the
attractor the lower the escape time.

Then, to extend our study of the behavior of the system we have also decided
to plot the final state and the escape time gradient in function of the
variation of the $\alpha$ value and the amplitude of the forcing $F$, for the
zero-initial condition, plotted in Figs. 5(a)-(b), respectively. It is
interesting to see that for $\alpha=1$, the particle escapes the potential
well for, approximately, all the forcing amplitude values bigger than
$F\approx 0.2$, a well known result SH:2008 . Moreover, it is interesting to
stress out that $F\approx 0.2$ seems to be the maximum value of the forcing
amplitude for which we can have bounded trajectories independently of the
values of $\alpha$. Also, some details in the figures are interesting, such as
the escape region delimited by $0<\alpha<0.5$ and by $0.05<F<0.1$ and the gap
between $F\approx 0.1$ and $F\approx 0.17$, for all the values of $\alpha$.
Also in this case, as in Fig. 4, the higher escape times are related with the
fractalization of the boundaries between the different basins of attraction.
In order to have a better insight of Fig. 5(b), we have plotted Figs 5 (c-f).
These figures have been computed for the zero-initial conditions and different
values of the fractional parameter, $\alpha=0.25$, $\alpha=1$, $\alpha=1.25$
and $\alpha=1.39$, respectively. Here again, it is not possible to establish a
trend of the escape time in function of $F$ for a fixed $\alpha$ value,
neither it is possible in function of $\alpha$ for a fixed $F$ value, due to
the fractalization of the boundaries and the shape of the basins.

To complete our investigation, we have plotted the Fig. 6. Here we can find
three trajectories in the ${x-z}$ plane and the time series of $x(t)$. In
Figs. 6(a)-(b) we represent a bounded trajectory with its related time series.
In Figs. 6(c)-(d), we represent an escaping trajectory and its related time
series. In Figs. 6(e)-(f) also an escaping trajectory and its time series with
a larger escape time. This last trajectory has been chosen in a fractal
boundary where the escape time are, in effect, larger. In the plot of these
last figures, we have chosen the parameter values reported in the captions of
the panel by using the data extrapolated by the precedent figures, Figs. 4 and
5, to help visualize the dynamics of the system while the parameters,
$\alpha,F,x_{0}$ change and the $\mu$ parameter is fixed.

##  IV Dynamics of the Fractional Helmholtz Oscillator in the overdamped case

In this section, we investigate the effect of large values of the damping
parameter $\mu$ on the dynamics of the system. So that, now, we study the case
with a bigger damping value, $\mu=0.8$, the overdamped case, of the fractional
Helmholtz oscillator. Notice that, as already discussed, in the non-fractional
case, the overdamped situation has not physical interest since almost all
particles with initial condition inside the well are trapped into the
potential well. For that reason, the role of the fractional parameter $\alpha$
will be crucial to enhance chaotic or regular motions and also escapes from
the potential well.

For the reader’s convenience, we decide to follow the same path as in the
previous case and started to plot the final state of the system and the escape
times gradient for the simultaneous variation of $\alpha$ and the first
initial conditions $x_{0}$, maintaining fixed the value of the amplitude of
the forcing $F=0.46$ and its frequency $\omega=1$. Notice that, with respect
to the underdamped case, while the frequency does not change, the forcing
amplitude that we use is bigger. The results of the computational experiments
are shown in Figs. 7(a-c) and Figs. 8(a-c). The first panel shows the final
state of the system with the purpose to know whether the particle remains
inside the well or escapes from it. Gray color (yellow online) has been used
to represent the case when the particle remains inside the well and black
color (blue on line) when it escapes. It is possible to appreciate the
fractalization of the basins in Figs. 7(b)-(c). In particular, in the second
one we can see that the fractalization starts for $\alpha\approx 1.36$.

Then, Figs. 8(a-c) show the gradient of the escape time. Here, it is possible
to appreciate that, as in the previous case, near the boundaries defining the
non-escaping and escaping regions the escape times are larger and in
particular, in the zones that show fractalization.

In Figs. 9(a)-(b), the escape time of the particle is shown for fixed $\alpha$
and different initial condition $x_{0}$. In Fig. 9 we have set the value of
$\alpha$ smaller than $\alpha\approx 1.36$, for which value we previously have
found where the fractalization of the attractor boundaries begins, as shown in
Fig 7(c). So that, as in the underdamped case, we use the existence of smooth
boundaries and focus our study on finding a general decay law for the escape
time in function of the variation of the initial condition $x_{0}$ and the
$\alpha$ parameter. Thus, we analyze the effects of the higher damping
coefficient on the escape time near the attractor boundaries. We started, in
Figs 9(a) and 9(b), by varying the initial conditions and fixing the $\alpha$
parameter at $\alpha=1.1$ and $\alpha=1.3$, respectively. It is possible to
appreciate in both figures the exponential decay law for the time escape in
function of the initial condition $x_{0}$ where the best fitting is
$t(x_{0})=ae^{bx_{0}}+ce^{dx_{0}}$. For that reason, we have decided to
explore the variation of the exponential parameters in the decay law in
function of the $\alpha$ parameter.

First of all, we found for $\alpha=1.36$, some fluctuations in the escape time
before the monotonic exponential decay, as shown in Fig. 10. In this last
panel, we show the change in the behavior of the escape time of the system,
when the $\alpha$ value passes from $1.35$ (Fig. 10(a)) to $1.36$ (Fig.
10(b)).Through the figures analysis, we can determine that the appearance of
the peaks (Fig. 10(b)), due to the fractalization of the attractor boundaries
Scherer , inhibited us to keep going on to larger values of $\alpha$ then
$1.36$ for the study of the exponential parameter variation.

The plot of the exponential parameter $b,d$ in function of the parameter
$\alpha$ can be found in Figs. 11(a)-(b), respectively. It is important to say
that, all the curves from which we have got those values of $b$ and $d$ show a
$R^{2}>0.99$. Differently from the underdamped case, we can find here a common
trend for the escape time distribution all along the attractor boundary,
possibly an effect of the high damping parameter value. In Fig. 11(a), we can
observe some fluctuations for $0.8<\alpha<1.4$. However, in Fig. 11(b), a
nonlinear decrease of $d$ versus $\alpha$ is observed. The corresponding terms
of this decay law are depicted in Fig. 11(c). In that figure, a polynomial
decay law fits the data very well since the regression coefficient
$R^{2}\approx 0.99$. Therefore, we can write the mathematical expression of
this curve as follows:

$d(\alpha)=p_{1}\alpha^{2}+p_{2}\alpha+p_{3},$ (7)

with $p_{1}=-14$, $p_{2}=25.56$ and $p_{3}=-13$.

The polynomial fitting of $d$ versus $\alpha$ connects with the exponential
fitting shown in Fig. 9 and in Fig. 12. For that reason, we can say that the
two figures are related. In fact, the curves in Fig. 9(b) and in Fig. 12 cross
each other in the following points: $t=1.1$ for $x_{0}=0.5$ in the first one
and for $\alpha=1.3$ in the second one. Furthermore, the trend of the
parameter $d$ versus $\alpha$ is as we expected since the curve of the
distribution of the escape time $t$ in function of $x_{0}$, reads:

$t(x_{0})=ae^{bx_{0}}+ce^{dx_{0}}$ (8)

in which the first term is negligible because of the parameter values $a,b$.
Therefore, the main term of the regression curve can be written as follows:

$t(x_{0})\approx ce^{dx_{0}}.$ (9)

Finally, we substitute $d$ as Eq. (7), for values of the initial condition
$x_{0}>0$ obtaining an exponential decay law similar to the one shown in Fig.
12.

Finally, in the latter figure, the study of the average escape times, by
varying $\alpha$ for a fixed value $x_{0}=0.5$, shows that the fractional
damping term can be used as a control parameter for the escape of the
particles from the potential well. In this sense, when we move from points
which are on the smooth boundaries of the attractor, the escape times of the
particles decrease insofar the values of $\alpha$ are going far away from the
attractor, as shown in Fig. 12. It can be seen that all of the escape times
cases exposed are contained in Figs. 7 and 8.

Figure 7: Final state of the system depending on the $\alpha$-value and the
variation of the first initial condition $x_{0}$ from $-3$ to $3$. White color
(yellow on line) implies the particle remains inside the well and black color
(blue on line) that particle escapes. In figure (a) We can see in this figure
a defined area representing the last state of the particle as time goes on
depending on the initial position $x_{0}$ and on the $\alpha$ variation,
keeping constant the rest of the parameters of the model. In figures (b) and
(c) it is possible to appreciate the zoom of the figure (a) for values of
$\alpha=(0.4,0.44)$ and $\alpha=(1.35,1.5)$, respectively. These last ones are
the areas where the boundaries show the fractalization. Figure 8: Figure (a)
shows the gradient of the particle escape times from the potential well
depending on the variation of the $\alpha$ value and of the first initial
condition $x_{0}$ from $-3$ to $3$. Figures (b) and (c) present a zoom of the
most interesting areas of figure (a), i.e., values of $\alpha=(0.4,0.44)$ and
$\alpha=(1.35,1.5)$, respectively. As stated in the previous panel, the areas
where the boundaries show the fractalization.  Figure 9: (a) Average escape
times from the well for $\alpha=1.1$ versus different initial conditions
inside the well. (b) Represents the same for $\alpha=1.3$. It is possible to
observe the exponential decay law. Figure 10: (a) Average escape times from
the well for $\alpha=1.35$ and different initial conditions inside the well.
(b) The same for $\alpha=1.36$. It is possible to observe that from values of
$\alpha\approx 1.36$ the escape times, before to show the monotonic decay,
have fluctuations. Figure 11: The figures (a) and (b) show, respectively, the
dynamics of the decay law exponentials $(b,d)$, calculated as in figures 9 in
function of the $\alpha$ values smaller than $\alpha=1.36$, value for which
the boundaries fluctuations of the escape times begin. In figure (c) a
regression curve for the dynamics of parameter $d$ in function of the
fractional derivative parameter $\alpha$, shown in figure (b), is proposed.
Figure 12: Average escape times from the well for fixed $x_{0}=0.5$ and from
the first $\alpha$ value for which the trajectories escape from the well, from
$\alpha\approx 1.0995$,to $1.5$.

Now, in order to follow the same path as in the underdamped case, we show the
final state of the system in Fig. 13(a), and the escape times gradient in Fig.
13(b), in function of the simultaneous variations of $F$ and $\alpha$. In Fig.
13(a), the parameter values for which the particles do not escape are colored
in gray (yellow online) and the black color (blue online) is used for the
different case. It is interesting the semi parabolic shape in both Figs.
13(a)-(b). Again, as shown in Fig. 13(b), the closer the values of $F$ and
$\alpha$ are to the values on the boundaries for which there is no escape, the
higher the time of the escape.

Figure 13: In figure (a) the panel shows the final state of the system taking
$\alpha=[0.35,1.5]$ and $F=[0,0.7]$ and keeping constant the rest of the
parameters. Gray color (yellow on line) implies the particle remains inside
the potential well and black color (blue on line) that particle escapes from
it. In figure (b) the panel shows, for the same range of $F$ and $\alpha$ and
the same parameter as in figure (a), the gradient escape times. A semi
parabolic shape denotes the behaviour of the particle as $\alpha$ and $F$
vary.

Then, we considered distributions of the escape times, $t$, in function of the
periodic forcing $F$ to study the behavior of the escape times from another
point of view. Different results have been obtained and are shown in Figs.
14(a-d). For the numerical simulations, we choose the values of
$\alpha=0.5,1,1.25,1.39$, respectively, and we vary $F$ for the zero-initial
conditions. Figure 14(a) represents the case when $\alpha=0.25$. In this first
plot we can see the fluctuations of the escape time for the first values in
the interval, but from $F\approx 0.6$ we obtained a smooth curve due to the
strong effect of the high values of $F$. In the next plot, we have repeated
the same experiment with $\alpha=1$ and we show the results in Fig. 14(b). For
this $\alpha$ values, differently from the underdamped case, the figure
illustrates a whole smooth curve for every value of $F$ and, as the
fractalization is not present, very short escape times in comparison with Fig.
5(d). This is a counterintuitive result, as we have commented along the paper,
the higher the dissipation the longer the escape times. Here, on the other
hand, the escape times diminish in the overdamped case, since the higher value
of the damping term makes the boundaries of the attractors smooth and so the
escape times become smaller. This is a different behavior, compared with the
above case, $\alpha=0.25$. This shows that the fractalization induced by the
fractional term has a deep impact on the dynamics of the system. The next
experiments with $\alpha=1.25,1.39$, are respectively shown in Figs.
14(c)-(d). The trend of these two cases is similar. Again, we obtained
fluctuations in the escape times for the first values of $F$ and near $F=0.6$
the curves become smooth. Therefore, as we already know from Fig. 13, the
first value of $F$ for which the particle escape depend on the value of
$\alpha$, and the dependence is not linear. As it happened in the underdamped
case, the fluctuations, due to the fractalization of the boundaries, impede us
to study a general decay law for the escape times as we did for the
$\\{\alpha-x_{0}\\}$ plane, where a smooth boundary was present. All this
analysis suggests that the variation of the parameter $\alpha$ along with the
forcing amplitude $F$ can be responsible of deep changes in the dynamics of
the system, due to the induction of the fractalization of the parameter space.
This means that $\alpha$ is a suitable control parameter for the behavior of
the system.

Figure 14: The panel shows the particle escape times from the potential well
with $\alpha=0.25,1,1.25,1.39$, by reading the figures from (a) to (d), and
varying $F$.  Figure 15: The panel shows the behavior of the fractional
damping Helmholtz oscillator with different values of $\alpha$ at zero-initial
conditions and $\mu=0.8$, $F=0.46$ and $\omega=1$. Figures (a), (c) and (d)
show the trajectories of Helmholtz oscillator $x$ vs $z$, while figures (b),
(d) and (f) the oscillations of the position $x$ with time, for $\alpha=0.5$,
$\alpha=1.39$, $\alpha=1.75$, respectively. Figures (a) and (b) represents the
typical bounded orbit inside the potential well. Figures (c) an (d) plot the
trajectory that escape the potential well. The last figures plot a trajectory
with a shorter escape time.

In order to have a better understanding of the dynamics of the system, we plot
the trajectories with different values of the $\alpha$ parameter in Fig. 15 by
fixing the value of the damping $\mu=0.8$, the forcing amplitude $F=0.46$, the
frequency $\omega=1$ and the zero-initial conditions. In Figs. 15(a)-(b) we
take $\alpha=0.5$ and we can see the closed orbit and the evolution with time
of the trajectory described by the system. Next case we consider is
$\alpha=1.39$ which is depicted in Figs. 15(c)-(d). This value of $\alpha$
generates escapes from the potential well even considering zero-initial
conditions. This abrupt change in the dynamics is an interesting issue that
reveals, again, the dependence on the fractional damping term. Another
important fact is that this result confirms the importance to study the effect
of taking a specific value of $\alpha$ and measure the escape times depending
on different initial conditions. Thus, Fig. 15(c) shows that the trajectory is
initially contained in the well but at certain time it escapes from it. In
fact, in Fig. 15(d), we show the escaping from the potential well other than
the oscillations before. We also consider the case of $\alpha=1.75$ another
case for which the trajectory is escaping from the well, according to the
numerical results shown in Figs. 15(e) and 11(f). The difference is that for
this value of $\alpha$ the oscillator behaviors before the escape die out
faster than the case for $\alpha=1.39$. So that, indeed $\alpha$ can be
considered as a control parameter of the system, which confirms its relevance
in this work.

Now, to stress out this crucial role of the $\alpha$ parameter, in the
overdamped case, we have considered important to analyze the dynamics of the
system in function of the $\alpha$ variation with a bifurcation diagram, see
Figs. 16(a)-(b). These figures show two different bifurcation diagrams with
just a little difference on their initial conditions. In both figures the
dynamics related with escapes have been left white, for an easier reading of
the graphics. Figure 16(a) corresponds to initial conditions
$(x_{0},y_{0},z_{0})=(0,0,0)$ and $\alpha$ has been considered from $1.3$ to
$1.5$. Figure 16(b) has been drawn with a slightly different initial
conditions, $(x_{0},y_{0},z_{0})=(0.01,0,0)$. The figures show the
effectiveness of $\alpha$ as a control parameter. In fact, the results, that
match with the previous ones of Fig. 7 and Fig. 8, confirm that the $\alpha$
parameter can induce chaos in the system, even in the overdamped case, and
that its appearance is robust for small variation of the initial conditions.

Figure 16: Bifurcaction diagram with $x_{0}=0.0$, $\mu=0.8$, $F=0.46$ and
$\omega=1$(a). Bifurcaction diagram with $x_{0}=0.01$, $\mu=0.8$, $F=0.46$ and
$\omega=1$(b)

##  V Conclusions

We have studied, using the Grünwald-Letnikov integrator, the dynamics of the
fractional Helmholtz oscillator with fractional term in the underdamped and
the overdamped cases. Since the system allows the particle to escape the
potential well, we used the attractor and the escape time plots as tools to
analyze the impact of the fractional parameter on the dynamics of the system.
In the underdamped case, the dynamics of the system are already rich and all
kind of behavior are possible. However, changing the fractional parameter has
a big impact on the system dynamics. On the other hand, the second case that
is normally not so interesting because the high dissipation makes the system
more predictable, becomes more interesting when the fractional derivative is
introduced. In fact, in the overdamped case is where the crucial role of the
fractional parameter becomes evident. When we vary the initial conditions and
the fractional parameter, all kind of behavior of the system are possible,
even chaotic. All this variety of the system dynamics appears, in the
attractor and escape time gradient plots, as a fractalization of the parameter
space in function of the variation of the fractional parameter. All this makes
the fractional parameter a suitable candidate to control the asymptotic
behaviors of the system. Moreover, in both cases, the escape time plots showed
us that near the boundaries of the attractor they are higher. However, in the
overdamped case, the escape time distribution of the particles show an
exponential-like decay law. Finally, in the overdamped case we have studied
the bifurcation diagrams in function of $\alpha$ to contrast the previous
results. Also, we have seen that the appearance of chaotic behavior is robust
for small changes, inside certain interval, of the initial condition, and the
$\alpha$ parameter.

To summarize, chaotic and periodic regions appearance in the parameter space
depends on the fractional parameter. So that we can say that it acts as a
control parameter of the dynamics of the Helmholtz oscillator. In fact, the
computation of the basins of the final state and the gradient of the escape
times of our model in the parameter space corroborates the previous
conclusions. We expect these results to be useful for a better understanding
of fractional calculus in chaotic systems since the memory effects are
relevant on their dynamics.

##  VI ACKNOWLEDGMENTS

This work has been supported by the Spanish State Research Agency (AEI) and
the European Regional Development Fund (ERDF, EU) under Projects No.
FIS2016-76883-P and No. PID2019-105554GB-I00, and the National Natural Science
Foundation of China (Grant No. 11672325).

## Conflict of Interest:

The authors declare that they have no conflict of interest.

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