# Periodically Forced Nonlinear Oscillatory Acoustic Vacuum

Makrina Agaoglou  Mathematical Institute of Slovak Academy of Sciences,
Štefánikova 49, 814 73 Bratislava, Slovakia  Department of Mechanical
Engineering, Faculty of Engineering, Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece  Michal Fečkan  Department of Mathematical Analysis
and Numerical Mathematics, Comenius University in Bratislava,
Mlynská dolina, 842 48 Bratislava, Slovakia  Mathematical Institute of Slovak
Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia  Michal
Pospíšil  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University in Bratislava,
Mlynská dolina, 842 48 Bratislava, Slovakia  Mathematical Institute of Slovak
Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia  Vassilis
Rothos  Department of Mechanical Engineering, Faculty of Engineering,
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece  Alexander F.
Vakakis  Department of Mechanical Science and Engineering, University of
Illinois at Urbana-Champaign, Urbana,IL 61801, USA

###### Abstract

In this work, we study the in-plane oscillations of a finite lattice of
particles coupled by linear springs under distributed harmonic excitation.
Melnikov-type analysis is applied for the persistence of periodic oscillations
of a reduced system.

nonlinear oscillatory acoustic vacuum; periodic oscillations; Melnikov
function;symmetry

##  I Introduction

We analytically study the persistence of periodic oscillations for certain
three-dimensional systems of ordinary differential equations (ODEs) with
periodic perturbations and a slowly-varying variable. The considered ODEs are
derived from a model of a finite lattice of particles coupled by linear
springs under distributed harmonic excitation, which is described in detail in
Section II. This model presents a low-energy nonlinear acoustic vacuum. We
refer the reader for more motivations, further details and applications to [1,
2]. Following the computations of [1], we consider just two modes in Section
III, and we postpone higher modes investigation to our future paper, since
another approach will be used. Melnikov analysis is demonstrated in Section IV
for finding conditions for the existence of periodic solutions for the
perturbed ODEs corresponding to two modes. More precisely, following [1], we
derive a three-dimensional periodically-perturbed system of ODEs with a
slowly-varying variable. Then, we analyze an unperturbed autonomous system of
ODEs to compute its family of periodic solutions by revising the results of
[1] in more detail. Since we are interested in the persistence of periodic
solutions for the perturbed ODEs, we compute the corresponding Melnikov
functions by [3]. Due to the difficulty of finding simple roots of these
Melnikov functions explicitly, we outline an asymptotic approach for the
location of some of them. Note that the simple roots of Melnikov functions
predict the persistence and location of periodic solutions for perturbed ODEs.
This is a novelty and a contribution of our paper. Section V outlines possible
future research along with summarizing our achievements in this paper.

##  II The Model

We consider a lattice consisting of $N$ identical particles coupled by
identical linear springs (they are un-stretched when the lattice is in the
horizontal position) and executing in-plane oscillations (see Figure 1). Fixed
boundary conditions and dissipative terms are imposed, and the transverse
harmonic forces are also applied. The equations of motion can be expressed as
follows,

$\begin{array}[]{lll}\displaystyle
m\frac{d^{2}u_{i}}{dt^{2}}+\left(T_{i}-\xi\frac{d\epsilon_{i}}{dt}\right)\cos\phi_{i}-\left(T_{i+1}-\xi\frac{d\epsilon_{i+1}}{dt}\right)\cos\phi_{i+1}&=&0\\\\[6.45831pt]
\displaystyle
m\frac{d^{2}v_{i}}{dt^{2}}+\left(T_{i}-\xi\frac{d\epsilon_{i}}{dt}\right)\sin\phi_{i}-\left(T_{i+1}-\xi\frac{d\epsilon_{i+1}}{dt}\right)\sin\phi_{i+1}-F_{i}&=&0\end{array}$
(1)

with $u_{i},v_{i}$ being the longitudinal and transversal displacements of the
$i$-th particle, respectively, $\phi_{i}$ the angle between the $i$-th spring
and the horizontal direction, $\xi$ the damping coefficient,
$\epsilon_{i}=l^{\prime}_{i}-l_{i}$ the deformation of the $i$-th spring,
$F_{i}$ the exciting transverse force and $m$ the mass of each particle of the
lattice. The tensile forces are proportional to the deformations of the
springs, and considering the geometry of the deformed state of the lattice
(see Figure 1), one may write:

$\begin{array}[]{lll}\displaystyle T_{i}&=&k(l^{\prime}_{i}-l_{i}),\\\
\displaystyle\epsilon_{i}=l^{\prime}_{i}-l_{i}&=&[(v_{i}-v_{i-1})^{2}+(l_{i}+u_{i}-u_{i-1})^{2}]^{1/2}-l_{i}\end{array}$
(2)

with $l_{i}$ being the equilibrium length of the $i$-th spring (each spring
has the same length) and $k$ the linear stiffness coefficient of each coupling
spring. Introducing $\delta_{i}=\epsilon_{i}/l_{i}$, $s_{i}=u_{i}/l_{i}$,
$w_{i}=v_{i}/l_{i}$, $c=\xi/(km)^{1/2}$, where $s_{i}$ and $w_{i}$ are the
normalized axial and transverse displacements, and the “slow” time scale
$\tau=(\frac{k}{m})^{1/2}t$, Equation (1) can be rewritten in normalized form:

$\begin{array}[]{lll}\displaystyle\frac{d^{2}s_{i}}{d\tau^{2}}&=&\delta_{i}\cos\phi_{i}-\delta_{i-1}\cos\phi_{i-1}+c\delta^{\prime}_{i}\cos\phi_{i}-c\delta^{\prime}_{i-1}\cos\phi_{i-1}\\\\[6.45831pt]
\displaystyle\frac{d^{2}w_{i}}{d\tau^{2}}&=&\delta_{i}\sin\phi_{i}-\delta_{i-1}\sin\phi_{i-1}+c\delta^{\prime}_{i}\sin\phi_{i}-c\delta^{\prime}_{i-1}\sin\phi_{i-1}+f_{i},\end{array}$
(3)

where:

$\begin{array}[]{rll}\displaystyle\delta_{i}&=&[(w_{i+1}-w_{i})^{2}+(1+s_{i+1}-s_{i})]^{1/2}-1,\\\\[6.45831pt]
\displaystyle\delta^{\prime}_{i}&=&\frac{(w_{i+1}-w_{i})(w_{i+1}^{\prime}-w_{i}^{\prime})+(1+s_{i+1}-s_{i})(s_{i+1}^{\prime}-s_{i}^{\prime})}{[(w_{i+1}-w_{i})^{2}+(1+s_{i+1}-s_{i})^{2}]^{1/2}}\end{array}$
(4)

and:

$\begin{array}[]{rrl}\displaystyle\cos\phi_{i}&=&\frac{1+s_{i+1}-s_{i}}{[(w_{i+1}-w_{i})^{2}+(1+s_{i+1}-s_{i})^{2}]^{1/2}},\\\\[6.45831pt]
\displaystyle\sin\phi_{i}&=&\frac{w_{i+1}-w_{i}}{[(w_{i+1}-w_{i})^{2}+(1+s_{i+1}-s_{i})^{2}]^{1/2}},\\\\[6.45831pt]
\displaystyle f_{i}&=&\frac{F_{i}}{k}.\end{array}$ (5)

Figure 1: Forced and damped lattice oscillating in the plane (see [2])

.

The normalized system (3) is referred to as the “exact lattice” in the
following sections.

According to the previous research [1], when we introduce this system (3)
without extra transverse force and damping terms, an interesting feature is
that in the low energy limit and under the assumption that the axial
displacements are assumed to be an order of magnitude smaller compared to the
transverse ones, it was shown that, correct to the leading order of
approximation, the transverse oscillations decouple from the axial ones and
are governed by the following reduced system of equations for predominantly
transverse oscillations of the particles:

$\begin{array}[]{ccc}\displaystyle\frac{d^{2}w_{i}}{d\tau^{2}}+2^{-1}(N+1)^{-1}\sum_{q=0}^{N}(w_{q+1}-w_{q})^{2}(2w_{i}-w_{i+1}-w_{i-1})=0,\\\
\displaystyle i=1,\cdots,N,\quad w_{0}(0)=w_{N+1}(0)=0.\end{array}$ (6)

Then, the nearly linear axial oscillations are driven by the transverse
responses (see [1, 2] for more details). Therefore, we focus our analysis like
in [1] just on Equation (6), which presents a low-energy nonlinear acoustic
vacuum, because in the absence of linear terms, it possesses zero speed of
sound in the context of classical linear acoustics. What is more, it is
notable that the existence of the strongly nonlocal multiplier
$2^{-1}(N+1)^{-1}\sum_{q=0}^{N}(w_{q+1}-w_{q})^{2}$ indicates that the
response of each particle is dependent (and hence, it is coupled) on the
responses of all other particles. Equation (6) admits _N_ exact nonlinear
standing waves, or nonlinear normal modes (NNMs), in the form:

$w_{i}(\tau)=A_{p}(\tau)\sin\frac{\pi pi}{N+1},\quad i=1,\cdots,N$

for the $p$-th NNM, $1\leq p\leq N$, where $A_{p}(\tau)$ denotes the $p$-th
modal amplitude. These, by construction, are mutually orthogonal, and there
are no other NNMs in this system, nor any NNM bifurcations [1]. Substituting
this NNM ansatz into Equation (6) yields a set of $N$ uncoupled nonlinear
equations governing the time-dependent amplitudes of the NNMs:

$A^{\prime\prime}_{p}(\tau)+(1/4)\omega_{p}^{4}A_{p}^{3}(\tau)=0$

with:

$\omega_{p}=2\sin\frac{\pi p}{2(N+1)},$

which is the $p$-th natural frequency of the corresponding linear system
Equation (6) and the prime denoting differentiation with respect to $\tau$.
The exciting force, which is applied on each particle in the transverse
direction, is expressed as:

$f_{i}=F_{p}\cos\omega_{p}\tau\sin\frac{pi\pi}{n+1}$

where $i=1,\cdots,N$, for the $p$-th NNM, $1\leq p\leq N$, and this exciting
force includes NNMs in the form $\sin\frac{pi\pi}{n+1},\ i=1,\cdots,N,$ for
the $p$-th NNM, $1\leq p\leq N$ and the $p$-th natural linear frequency
$\omega_{p}$.

The frequency of the $p$-th NNM is tunable with the force and energy, and it
also paves the way for nonlinear resonances between NNMs widely separated in
the nonlinear spectrum, given that their energies tune their frequencies to
satisfy certain rational relationships.

Summarizing, we can write (6) as:

$\frac{d^{2}w}{d\tau^{2}}+2^{-1}(N+1)^{-1}\langle Mw,w\rangle Mw=0,$

where $w=[w_{1},\cdots,w_{N}]\in\mathbb{R}^{N}$, $M$ is a symmetric matrix
given by

$M=\begin{pmatrix}2&-1&0&\cdots&0\\\ -1&2&-1&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&-1&2&-1\\\
0&\cdots&0&-1&2\end{pmatrix}$

and $\langle\cdot,\cdot\rangle$ is the standard scalar product on
$\mathbb{R}^{N}$. The eigenvectors of $M$ are
$\underline{\phi}_{p}=[\sin\frac{p\pi}{N+1},\cdots,\sin\frac{pN\pi}{N+1}]$
with the corresponding eigenvalues $\omega_{p}^{2}$, $1\leq p\leq N$.
Moreover, it holds (see [4], p. 37):

$\begin{gathered}\langle\underline{\phi}_{p},\underline{\phi}_{p}\rangle=\sum_{i=1}^{N}\sin^{2}\frac{pi\pi}{N+1}=\frac{N+1}{2},\\\
\langle\underline{\phi}_{p},\underline{\phi}_{k}\rangle=\sum_{i=1}^{N}\sin\frac{pi\pi}{N+1}\sin\frac{ki\pi}{N+1}\\\
=\frac{1}{2}\sum_{i=1}^{N}\left(\cos\frac{(p-k)i\pi}{N+1}-\cos\frac{(p+k)i\pi}{N+1}\right)=0,\quad
p\neq k.\end{gathered}$

The forced (6) has the form:

$\frac{d^{2}w}{d\tau^{2}}+2^{-1}(N+1)^{-1}\langle Mw,w\rangle
Mw=\sum_{p=1}^{N}F_{p}\cos\omega_{p}\tau\underline{\phi}_{p}.$ (7)

Therefore, considering the basis $\\{\underline{\phi}_{p}\\}_{p=1}^{N}$ of
$\mathbb{R}^{N}$, we take
$w(\tau)=\sum_{p=1}^{N}C_{p}(\tau)\underline{\phi}_{p}$ in (7) to get:

$C^{\prime\prime}_{p}(\tau)+\frac{1}{4}\left(\sum_{i=1}^{N}C_{i}^{2}(\tau)\omega_{i}^{2}\right)\omega_{p}^{2}C_{p}(\tau)=F_{p}\cos\omega_{p}\tau,\quad
1\leq p\leq N.$ (8)

Next, applying the coordinate transformation to (8):

$A_{p}(\tau)=\frac{\omega_{p}}{2}C_{p}(\tau),\quad 1\leq p\leq N,$

we get:

$A^{\prime\prime}_{p}(\tau)+\left(\sum_{i=1}^{N}A_{i}^{2}(\tau)\right)\omega_{p}^{2}A_{p}(\tau)=\frac{F_{p}\omega_{p}}{2}\cos\omega_{p}\tau,\quad
1\leq p\leq N.$

##  III Two-Mode System

In this paper, we consider just two modes: $k$ and $p$, so we study the
system:

$\begin{array}[]{lll}\displaystyle
A^{\prime\prime}_{k}(\tau)+[A^{2}_{k}(\tau)+A_{p}^{2}(\tau)]\omega_{k}^{2}A_{k}(\tau)+\epsilon\mu_{1}\cos(\omega_{k}\tau)&=&0\\\\[6.45831pt]
\displaystyle
A^{\prime\prime}_{p}(\tau)+[A^{2}_{k}(\tau)+A_{p}^{2}(\tau)]\omega_{p}^{2}A_{p}(\tau)+\epsilon\mu_{2}\cos(\omega_{p}\tau)&=&0,\end{array}$
(9)

for $\epsilon\neq 0$ small and parameters $\mu_{1}$, $\mu_{2}$. Using the
transformation:

$\begin{array}[]{lll}\displaystyle\psi_{1}(\tau)&=&A_{k}^{\prime}(\tau)+j\Omega
A_{k}(\tau)\equiv\zeta_{1}(\tau)e^{j\Omega\tau}\\\\[6.45831pt]
\displaystyle\psi_{2}(\tau)&=&A_{p}^{\prime}(\tau)+j\Omega
A_{p}(\tau)\equiv\zeta_{2}(\tau)e^{j\Omega\tau},\end{array}$

and performing an averaging approach like in [1], Equation (9) is modified to
the form:

$\begin{array}[]{lll}\displaystyle\zeta_{1}^{\prime}+\frac{j\Omega}{2}\zeta_{1}-\frac{j\omega_{k}^{2}}{8\Omega^{3}}(\zeta_{2}^{2}\zeta_{1}^{\ast}+2|\zeta_{2}|^{2}\zeta_{1}+3|\zeta_{1}|^{2}\zeta_{1})+\epsilon\mu_{1}\cos(\omega_{k}\tau)e^{-j\Omega\tau}&=&0\\\\[6.45831pt]
\displaystyle\zeta_{2}^{\prime}+\frac{j\Omega}{2}\zeta_{2}-\frac{j\omega_{p}^{2}}{8\Omega^{3}}(\zeta_{1}^{2}\zeta_{2}^{\ast}+2|\zeta_{1}|^{2}\zeta_{2}+3|\zeta_{2}|^{2}\zeta_{2})+\epsilon\mu_{2}\cos(\omega_{p}\tau)e^{-j\Omega\tau}&=&0.\end{array}$

Introducing $\zeta_{i}=a_{i}e^{j\beta_{i}}$ and $\Delta=\beta_{2}-\beta_{1}$,
we get:

$\begin{array}[]{rrl}\displaystyle
a_{1}^{\prime}+\frac{\omega_{k}^{2}}{8\Omega^{3}}a_{2}^{2}a_{1}\sin
2\Delta+\epsilon\mu_{1}\cos(\omega_{k}\tau)\cos(\Omega\tau+\beta_{1})&=&0\\\\[6.45831pt]
\displaystyle
a_{2}^{\prime}-\frac{\omega_{p}^{2}}{8\Omega^{3}}a_{1}^{2}a_{2}\sin
2\Delta+\epsilon\mu_{2}\cos(\omega_{p}\tau)\cos(\Omega\tau+\beta_{1}+\Delta)&=&0\\\\[6.45831pt]
\displaystyle\Delta^{\prime}-\frac{\omega_{p}^{2}}{8\Omega^{3}}(3a_{2}^{2}+a_{1}^{2}\cos
2\Delta+2a_{1}^{2})+\frac{\omega_{k}^{2}}{8\Omega^{3}}(3a_{1}^{2}+a_{2}^{2}\cos
2\Delta+2a_{2}^{2})\\\\[6.45831pt]
\displaystyle-\frac{\epsilon\mu_{2}}{a_{2}}\cos(\omega_{p}\tau)\sin(\Omega\tau+\beta_{1}+\Delta)+\frac{\epsilon\mu_{1}}{a_{1}}\cos(\omega_{k}\tau)\sin(\Omega\tau+\beta_{1})&=&0\end{array}$
(10)

where we consider $\beta_{1}$ as a constant parameter. Now, by introducing the
coordinate transformations $a_{1}=(\frac{\rho}{\omega_{p}})\sin\theta$ and
$a_{2}=(\frac{\rho}{\omega_{k}})\cos\theta$ into Equation (10), we get:

$\begin{array}[]{rrl}\displaystyle\rho^{\prime}=-\epsilon\mu_{1}\omega_{p}\sin\theta\cos(\omega_{k}\tau)\cos(\Omega\tau+\beta_{1})-\epsilon\mu_{2}\omega_{k}\cos\theta\cos(\omega_{p}\tau)\cos(\Omega\tau+\Delta+\beta_{1})\\\\[6.45831pt]
\displaystyle\theta^{\prime}+\frac{\rho^{2}}{16\Omega^{3}}\sin 2\theta\sin
2\Delta+\epsilon\mu_{1}\omega_{p}\frac{\cos\theta}{\rho}\cos(\omega_{k}\tau)\cos(\Omega\tau+\beta_{1})\\\\[6.45831pt]
\displaystyle-\epsilon\mu_{2}\omega_{k}\frac{\sin\theta}{\rho}\cos(\omega_{p}\tau)\cos(\Omega\tau+\Delta+\beta_{1})=0\\\\[6.45831pt]
\displaystyle\Delta^{\prime}-\frac{\rho^{2}}{8\Omega^{3}}\left[\frac{3\omega_{p}^{2}}{\omega_{k}^{2}}\cos^{2}\theta-\frac{3\omega_{k}^{2}}{\omega_{p}^{2}}\sin^{2}\theta-\cos
2\theta(2+\cos 2\Delta)\right]\\\\[6.45831pt]
\displaystyle-\frac{\epsilon\mu_{2}\omega_{k}}{\rho\cos\theta}\cos(\omega_{p}\tau)\sin(\Omega\tau+\Delta+\beta_{1})+\frac{\epsilon\mu_{1}\omega_{p}}{\rho\sin\theta}\cos(\omega_{k}\tau)\sin(\Omega\tau+\beta_{1})=0.\end{array}$

In the rest of the paper, we assume $\omega_{p}=\omega_{k}=P$ and $\Omega=kP$
for a natural number $k$, so we study the periodically-perturbed system:

$\begin{array}[]{rrl}\displaystyle\rho^{\prime}=-\epsilon\mu_{1}\sin\theta\cos(P\tau)\cos(kP\tau+\beta_{1})-\epsilon\mu_{2}\cos\theta\cos(P\tau)\cos(kP\tau+\Delta+\beta_{1})\\\\[6.45831pt]
\displaystyle\theta^{\prime}+\frac{\rho^{2}}{16k^{3}P^{3}}\sin 2\theta\sin
2\Delta+\epsilon\mu_{1}\frac{\cos\theta}{\rho}\cos(P\tau)\cos(kP\tau+\beta_{1})\\\\[6.45831pt]
\displaystyle-\epsilon\mu_{2}\frac{\sin\theta}{\rho}\cos(P\tau)\cos(kP\tau+\Delta+\beta_{1})=0\\\
\displaystyle\Delta^{\prime}-\frac{\rho^{2}}{4k^{3}P^{3}}\cos
2\theta\sin^{2}\Delta+\frac{\epsilon\mu_{1}}{\rho\sin\theta}\cos(P\tau)\sin(kP\tau+\beta_{1})\\\\[6.45831pt]
\displaystyle-\frac{\epsilon\mu_{2}}{\rho\cos\theta}\cos(P\tau)\sin(kP\tau+\Delta+\beta_{1})=0\end{array}$
(11)

where we scaled $P\mu_{i}\leftrightarrow\mu_{i}$, $i=1,2$. We may suppose:

$\mu_{1}^{2}+\mu_{2}^{2}=1.$

First, consider the unperturbed case where $\epsilon=0$, so the system:

$\begin{array}[]{rrl}\displaystyle\rho^{\prime}=0\\\
\displaystyle\theta^{\prime}+\frac{\rho^{2}}{16k^{3}P^{3}}\sin 2\theta\sin
2\Delta=0\\\\[6.45831pt]
\displaystyle\Delta^{\prime}-\frac{\rho^{2}}{4k^{3}P^{3}}\cos
2\theta\sin^{2}\Delta=0.\end{array}$ (12)

By introducing the temporal variable
$\tau_{2}=\frac{\rho^{2}}{8k^{3}P^{3}}\tau$ in Equation (12), we get:

$\begin{array}[]{rrr}\displaystyle\frac{d\theta}{d\tau_{2}}&=&-\frac{1}{2}\sin
2\theta\sin 2\Delta\\\\[6.45831pt]
\displaystyle\frac{d\Delta}{d\tau_{2}}&=&2\cos
2\theta\sin^{2}\Delta\end{array}$ (13)

which is fully integrable and gives us the first integral $I=\sin
2\theta\sin\Delta=K=const.$ of the degenerate slow flow. If we consider the
following initial conditions: $\theta(0)=\theta_{0}$, where
$0<\theta_{0}<\frac{\pi}{4}$ and $\Delta(0)=\pi/2$, then we get $K=\sin
2\theta_{0}\in(0,1)$. To find exact solutions of (13), we first derive:

$\begin{gathered}\frac{d\tau_{2}}{d\theta}=-\frac{2}{\sin 2\theta\sin
2\Delta}=-\frac{1}{K\cos\Delta}=\frac{1}{K\sqrt{1-\sin^{2}\Delta}}\\\
=\frac{1}{K\sqrt{1-\frac{K^{2}}{\sin^{2}2\theta}}}=\frac{1}{K}\frac{\sin
2\theta}{\sqrt{\sin^{2}2\theta-K^{2}}},\end{gathered}$

for $\tau_{2}>0$ small, since (13) gives $0<\theta<\frac{\pi}{2}$,
$0<\Delta<\pi$ (see Figure 2) and $\frac{d\Delta}{d\tau_{2}}(0)=2\cos
2\theta_{0}>0$, so for $\tau_{2}>0$ small, we have
$\Delta(\tau_{2})>\frac{\pi}{2}$.

By using the formula in [4] ((2.599.4) p. 205), we obtain:

$\tau_{2}=\int_{\theta_{0}}^{\theta}\frac{1}{K}\frac{\sin
2\theta}{\sqrt{\sin^{2}2\theta-K^{2}}}d\theta=\frac{1}{2K}\left(\frac{\pi}{2}-\sin^{-1}\left(\frac{\cos
2\theta}{\sqrt{1-K^{2}}}\right)\right),$

which gives:

$\theta(\tau_{2})=\frac{1}{2}\cos^{-1}\left(\sqrt{1-K^{2}}\cos(2K\tau_{2})\right),$
(14)

recalling $0<\theta<\frac{\pi}{2}$. Of course, Formula (14) holds for all
$\tau_{2}$, not just for small positive ones. Next, using (13) and (14), we
have:

$\frac{d\Delta}{d\tau_{2}}=2\sqrt{1-K^{2}}\cos(2K\tau_{2})\sin^{2}\Delta,$

which can be easily solved to arrive at:

$\Delta(\tau_{2})=\pi-\cot^{-1}\left(\frac{\sqrt{1-K^{2}}}{K}\sin(2K\tau_{2})\right).$

Figure 2: Top panel: Graph for $\theta$ for different initial values of
$\theta_{0}$. Bottom panel: Graph for $\Delta$ for different initial values of
$\theta_{0}$.

Hence, the exact solution of the system (13) is given as:

$\begin{gathered}\theta(\tau_{2})=\frac{1}{2}\cos^{-1}\left(\sqrt{1-K^{2}}\cos(2K\tau_{2})\right)\\\
\Delta(\tau_{2})=\pi-\cot^{-1}\left(\frac{\sqrt{1-K^{2}}}{K}\sin(2K\tau_{2})\right)\end{gathered}$

where the period is:

$T(\theta_{0})=\frac{2}{K}\int_{\theta_{0}}^{\frac{\pi}{2}-\theta_{0}}\frac{d\theta}{\left(1-\frac{\sin^{2}2\theta_{0}}{\sin^{2}2\theta}\right)^{1/2}}=\frac{\pi}{K}.$

It is easy to verify the following symmetry property (see Figure 3):

$\Delta\left(\tau_{2}+\frac{T}{2}\right)=\pi-\Delta(\tau_{2}),\quad\theta\left(\tau_{2}+\frac{T}{2}\right)=\frac{\pi}{2}-\theta(\tau_{2}).$

Figure 3: Orbits in the phase portrait of (13), where
$(\theta,\Delta)\in[0,\frac{\pi}{2}]\times[0,\pi]$.

Summarizing, the exact solution of the unperturbed (12) is the following:

$\begin{gathered}\rho(\tau)=const.\\\
\theta(\tau)=\frac{1}{2}\cos^{-1}\left(\sqrt{1-K^{2}}\cos\left(\frac{K\rho^{2}}{4k^{3}P^{3}}\tau\right)\right)\\\
\Delta(\tau)=\pi-\cot^{-1}\left(\frac{\sqrt{1-K^{2}}}{K}\sin\left(\frac{K\rho^{2}}{4k^{3}P^{3}}\tau\right)\right)\end{gathered}$

with the period:

$T=\frac{8k^{3}P^{3}\pi}{K\rho^{2}}.$

Consequently, for any:

$\rho>2kP^{2}\sqrt{k},$

taking:

$K(\rho)=\frac{4k^{3}P^{4}}{\rho^{2}}\in(0,1),$

Equation (12) has the $T=2\pi/P$-periodic solution:

$\begin{gathered}\theta(\rho,\tau)=\frac{1}{2}\cos^{-1}\left(\frac{\sqrt{\rho^{4}-16k^{6}P^{8}}}{\rho^{2}}\cos(P\tau)\right)\\\
\Delta(\rho,\tau)=\pi-\cot^{-1}\left(\frac{\sqrt{\rho^{4}-16k^{6}P^{8}}}{4k^{3}P^{4}}\sin(P\tau)\right).\end{gathered}$
(15)

##  IV Melnikov Analysis for Periodic Oscillations

Writing (11) as:

$\begin{array}[]{rrl}\displaystyle\theta^{\prime}+\frac{\rho^{2}}{16k^{3}P^{3}}\sin
2\theta\sin
2\Delta=\epsilon\rho^{-1}g_{1}(\rho,\theta,\Delta,\tau)\\\\[6.45831pt]
\displaystyle\Delta^{\prime}-\frac{\rho^{2}}{4k^{3}P^{3}}\cos
2\theta\sin^{2}\Delta=\epsilon\rho^{-1}g_{2}(\rho,\theta,\Delta,\tau)\\\\[6.45831pt]
\displaystyle\rho^{\prime}=\epsilon g_{3}(\rho,\theta,\Delta,\tau)\end{array}$

and using [5], (3.5.11), p. 111, with $\alpha=0$, and [6], Lemma 2.5, p. 283,
we compute the Melnikov function:

$M(\beta_{1},\rho)=\left(M_{1}(\beta_{1},\rho),M_{2}(\beta_{1},\rho)\right)$

as:

$\begin{gathered}M_{1}(\beta_{1},\rho)=\int_{0}^{T}\left(\frac{\partial
I}{\partial\theta}\rho^{-1}g_{1}+\frac{\partial
I}{\partial\Delta}\rho^{-1}g_{2}-\frac{\partial
I}{\partial\theta}\frac{\partial\theta}{\partial\rho}g_{3}-\frac{\partial
I}{\partial\Delta}\frac{\partial\Delta}{\partial\rho}g_{3}\right)d\tau\\\
=\int_{0}^{T}\left(\frac{\partial
I}{\partial\theta}\rho^{-1}g_{1}+\frac{\partial
I}{\partial\Delta}\rho^{-1}g_{2}\right)d\tau-\frac{dK}{d\rho}\int_{0}^{T}g_{3}d\tau\end{gathered}$
(16)

since differentiating by $\rho$ the identity:

$I(\theta(\rho,\tau),\Delta(\rho,\tau))=K(\rho),$

we get:

$\frac{\partial
I}{\partial\theta}\frac{\partial\theta}{\partial\rho}+\frac{\partial
I}{\partial\Delta}\frac{\partial\Delta}{\partial\rho}=\frac{dK}{d\rho},$

which is independent of $\tau$, and:

$M_{2}(\beta_{1},\rho)=\int_{0}^{T}g_{3}d\tau.$ (17)

Formulas (16) and (17) are similar to [3], (2.7). We are looking for a simple
zero of $M$, which is equivalent to considering:

$\begin{gathered}\bar{M}(\beta_{1},\rho)=\left(\bar{M}_{1}(\beta_{1},\rho),\bar{M}_{2}(\beta_{1},\rho)\right)\\\
\bar{M}_{1}(\beta_{1},\rho)=\int_{0}^{T}\left(\frac{\partial
I}{\partial\theta}g_{1}+\frac{\partial
I}{\partial\Delta}g_{2}\right)d\tau,\quad\bar{M}_{2}(\beta_{1},\rho)=\int_{0}^{T}g_{3}d\tau.\end{gathered}$

Since:

$g_{i}=\mu_{1}g_{i1}+\mu_{2}g_{i2},\quad i=1,2,3$

for:

$\begin{array}[]{l}\displaystyle
g_{11}(\rho,\theta,\Delta,\tau)=-\cos\theta\cos(P\tau)\cos(kP\tau+\beta_{1})\\\\[6.45831pt]
\displaystyle
g_{12}(\rho,\theta,\Delta,\tau)=\sin\theta\cos(P\tau)\cos(kP\tau+\Delta+\beta_{1})\\\\[6.45831pt]
\displaystyle
g_{21}(\rho,\theta,\Delta,\tau)=-\frac{\cos(P\tau)}{\sin\theta}\sin(kP\tau+\beta_{1})\\\\[6.45831pt]
\displaystyle
g_{22}(\rho,\theta,\Delta,\tau)=\frac{\cos(P\tau)}{\cos\theta}\sin(kP\tau+\Delta+\beta_{1})\\\\[6.45831pt]
\displaystyle
g_{31}(\rho,\theta,\Delta,\tau)=-\sin\theta\cos(P\tau)\cos(kP\tau+\beta_{1})\\\\[6.45831pt]
\displaystyle
g_{32}(\rho,\theta,\Delta,\tau)=-\cos\theta\cos(P\tau)\cos(kP\tau+\Delta+\beta_{1})\\\
\end{array}$

we get:

$\bar{M}_{i}(\beta_{1},\rho)=\mu_{1}\bar{M}_{i1}(\beta_{1},\rho)+\mu_{2}\bar{M}_{i2}(\beta_{1},\rho),\quad
i=1,2,$

for

$\bar{M}_{1j}(\beta_{1},\rho)=\int_{0}^{T}\left(\frac{\partial
I}{\partial\theta}g_{1j}+\frac{\partial
I}{\partial\Delta}g_{2j}\right)d\tau,\quad\bar{M}_{2j}(\beta_{1},\rho)=\int_{0}^{T}g_{3j}d\tau,\quad
j=1,2.$

To solve:

$\mu_{1}\bar{M}_{i1}(\beta_{1},\rho)+\mu_{2}\bar{M}_{i2}(\beta_{1},\rho)=0,\quad
i=1,2,$

we first solve the scalar equation:

$\tilde{M}(\beta_{1},\rho)=\bar{M}_{11}(\beta_{1},\rho)\bar{M}_{22}(\beta_{1},\rho)-\bar{M}_{12}(\beta_{1},\rho)\bar{M}_{21}(\beta_{1},\rho)=0$
(18)

to get its root $\beta_{1,0}$ and $\rho_{0}$. Then, we look for $\mu_{1,0}$
and $\mu_{2,0}$ with $\mu_{1,0}^{2}+\mu_{2,0}^{2}=1$ such that:

$\begin{gathered}\mu_{1,0}\bar{M}_{i1}(\beta_{1,0},\rho_{0})+\mu_{2,0}\bar{M}_{i2}(\beta_{1,0},\rho_{0})=0,\quad
i=1,2\\\
\det\begin{pmatrix}\mu_{1,0}\nabla\bar{M}_{11}(\beta_{1,0},\rho_{0})^{\top}+\mu_{2,0}\nabla\bar{M}_{12}(\beta_{1,0},\rho_{0})^{\top}\\\
\mu_{1,0}\nabla\bar{M}_{21}(\beta_{1,0},\rho_{0})^{\top}+\mu_{2,0}\nabla\bar{M}_{22}(\beta_{1,0},\rho_{0})^{\top}\end{pmatrix}\neq
0.\end{gathered}$ (19)

Summarizing, we have the following result.

###### Theorem 1

If there are $\beta_{1,0}\in[0,2\pi)$, $\rho_{0}$ satisfying (18), $\mu_{1,0}$
and $\mu_{2,0}$ with $\mu_{1,0}^{2}+\mu_{2,0}^{2}=1$ solving (19), then for
any $\mu_{1}$ near $\mu_{1,0}$ and $\mu_{2}$ near $\mu_{2,0}$ with
$\mu_{1}^{2}+\mu_{2}^{2}=1$ and $\epsilon\neq 0$ small, there are
$\beta_{1}(\epsilon)$ near $\beta_{1,0}$ and $\rho(\epsilon)$ near $\rho_{0}$
such that (11) with $\beta_{1}=\beta_{1}(\epsilon)$ and $\rho=\rho(\epsilon)$
has a $T=2\pi/P$-periodic solution near (15) with $\rho=\rho(\epsilon)$.

Note:

$\begin{gathered}\bar{M}_{11}(\beta_{1},\rho)=-\int_{0}^{T}\Big{(}2\cos(2\theta(\rho,\tau))\sin\Delta(\rho,\tau)\cos\theta(\rho,\tau)\cos(P\tau)\cos(kP\tau+\beta_{1})\\\
+2\cos\theta(\rho,\tau)\cos\Delta(\rho,\tau)\cos(P\tau)\sin(kP\tau+\beta_{1}))\Big{)}d\tau\\\
\bar{M}_{12}(\beta_{1},\rho)=\int_{0}^{T}\Big{(}2\cos(2\theta(\rho,\tau))\sin\Delta(\rho,\tau)\sin\theta(\rho,\tau)\cos(P\tau)\cos(kP\tau+\Delta(\rho,\tau)+\beta_{1})\\\
+2\sin\theta(\rho,\tau)\cos\Delta(\rho,\tau)\cos(P\tau)\sin(kP\tau+\Delta(\rho,\tau)+\beta_{1})\Big{)}d\tau\\\
\bar{M}_{21}(\beta_{1},\rho)=-\int_{0}^{T}\sin\theta(\rho,\tau)\cos(P\tau)\cos(kP\tau+\beta_{1})d\tau\\\
\bar{M}_{22}(\beta_{1},\rho)=-\int_{0}^{T}\cos\theta(\rho,\tau)\cos(P\tau)\cos(kP\tau+\Delta(\rho,\tau)+\beta_{1})d\tau.\end{gathered}$

Next, taking $\rho\to\infty$ in (15), we obtain:

$\begin{gathered}\theta(\infty,\tau)=\begin{cases}\displaystyle\frac{P\tau}{2}\quad\tau\in\left[0,\frac{\pi}{P}\right],\\\
\\\\[-8.61108pt]
\displaystyle\pi-\frac{P\tau}{2}\quad\tau\in\left[\frac{\pi}{P},\frac{2\pi}{P}\right],\end{cases}\\\
\Delta(\infty,\tau)=\begin{cases}\displaystyle\pi\quad\tau\in\left(0,\frac{\pi}{P}\right),\\\
\\\\[-8.61108pt] \displaystyle
0\quad\tau\in\left(\frac{\pi}{P},\frac{2\pi}{P}\right),\\\ \\\\[-8.61108pt]
\displaystyle\frac{\pi}{2}\quad\tau\in\left\\{0,\frac{\pi}{P},\frac{2\pi}{P}\right\\},\end{cases}\\\
\frac{\partial\theta}{\partial\rho}(\infty,\tau)=0,\quad\frac{\partial\Delta}{\partial\rho}(\infty,\tau)=0.\end{gathered}$
(20)

Hence:

$\begin{gathered}\bar{M}_{11}(\beta_{1},\infty)=\int_{0}^{\frac{2\pi}{P}}2\cos\frac{P\tau}{2}\cos(P\tau)\sin(kP\tau+\beta_{1})d\tau\\\
=\frac{16k(4k^{2}-5)\cos\beta_{1}}{(16k^{4}-40k^{2}+9)P}\\\
\bar{M}_{12}(\beta_{1},\infty)=\int_{0}^{\frac{2\pi}{P}}2\sin\frac{P\tau}{2}\cos(P\tau)\sin(kP\tau+\beta_{1})\Big{)}d\tau\\\
=-\frac{8(4k^{2}+3)\sin\beta_{1}}{(16k^{4}-40k^{2}+9)P}\\\
\bar{M}_{21}(\beta_{1},\infty)=-\int_{0}^{\frac{2\pi}{P}}\sin\frac{P\tau}{2}\cos(P\tau)\cos(kP\tau+\beta_{1})d\tau\\\
=\frac{4(4k^{2}+3)\cos\beta_{1}}{(16k^{4}-40k^{2}+9)P}\\\
\bar{M}_{22}(\beta_{1},\infty)=\int_{0}^{\frac{2\pi}{P}}\cos\frac{P\tau}{2}\cos(P\tau)\cos(kP\tau+\beta_{1})d\tau\\\
=-\frac{8k(4k^{2}-5)\sin\beta_{1}}{(16k^{4}-40k^{2}+9)P}.\end{gathered}$

Then, (18) gives as $\rho\to\infty$,

$\tilde{M}(\beta_{1},\infty)=-\frac{16\sin 2\beta_{1}}{(4k^{2}-9)P^{2}}$

with asymptotic solutions $\beta_{1,0}^{\infty}$ satisfying either
$\sin\beta_{1,0}^{\infty}=0$ or $\cos\beta_{1,0}^{\infty}=0$. The asymptotic
equation of (19) is as follows:

$\begin{gathered}0=\mu_{1,0}\bar{M}_{11}(\beta_{1,0}^{\infty},\infty)+\mu_{2,0}\bar{M}_{12}(\beta_{1,0}^{\infty},\infty)\\\
\\\\[-8.61108pt]
=\mu_{1,0}\frac{16k(4k^{2}-5)\cos\beta_{1,0}^{\infty}}{(16k^{4}-40k^{2}+9)P}-\mu_{2,0}\frac{8(4k^{2}+3)\sin\beta_{1,0}^{\infty}}{(16k^{4}-40k^{2}+9)P},\end{gathered}$

which has solutions: $\mu_{1,0}^{\infty}=0$ and $\mu_{2,0}^{\infty}=\pm 1$
when $\sin\beta_{1,0}^{\infty}=0$ or $\mu_{1,0}^{\infty}=\pm 1$ and
$\mu_{2,0}^{\infty}=0$ when $\cos\beta_{1,0}^{\infty}=0$. However,
$\frac{\partial\bar{M}_{ij}}{\partial\rho}(\beta_{1},\infty)=0$ for
$i,j\in\\{1,2\\}$, so Theorem 1 cannot be applied directly. Note we consider
just the first asymptotic equation of (19), since the second one is a scalar
multiple of the first one.

On the other hand, following the method of [5], p. 111, we get the following
result.

###### Corollary 1

For any $\rho>0$ sufficiently large and $\epsilon\neq 0$ sufficiently small,
there is $\mu_{1}(\rho,\epsilon)$, $\mu_{2}(\rho,\epsilon)$ with
$\mu_{1}^{2}(\rho,\epsilon)+\mu_{2}^{2}(\rho,\epsilon)=1$ and
$\beta_{1}(\rho,\epsilon)$ such that (11) has a $T$-periodic solution near
(20) for $\epsilon\neq 0$ small and either $\mu_{1}(\infty,0)=0$,
$\mu_{2}(\infty,0)=\pm 1$, $\sin\beta_{1}(\infty,0)=0$ or
$\mu_{1}(\infty,0)=\pm 1$, $\mu_{2}(\infty,0)=0$, $\cos\beta_{1}(\infty,0)=0$.

###### Proof 1

The bifurcation equation has the form (see [5], p. 111):

$\mu_{1}\bar{M}_{i1}(\beta_{1},\rho)+\mu_{2}\bar{M}_{i2}(\beta_{1},\rho)=O(\epsilon),\quad
i=1,2,\quad\mu_{1}^{2}+\mu_{2}^{2}=1$

i.e.,

$\sin\Gamma\bar{M}_{i1}(\beta_{1},\rho)+\cos\Gamma\bar{M}_{i2}(\beta_{1},\rho)=O(\epsilon),\quad
i=1,2$ (21)

for $\mu_{1}=\sin\Gamma$ and $\mu_{2}=\cos\Gamma$. For $\rho=\infty$ and
$\epsilon=0$, (21) takes the form:

$\begin{gathered}\frac{16k(4k^{2}-5)}{(16k^{4}-40k^{2}+9)P}\sin\Gamma\cos\beta_{1}-\frac{8(4k^{2}+3)}{(16k^{4}-40k^{2}+9)P}\cos\Gamma\sin\beta_{1}=0\\\
\frac{4(4k^{2}+3)\cos\beta_{1}}{(16k^{4}-40k^{2}+9)P}\sin\Gamma\cos\beta_{1}-\frac{8k(4k^{2}-5)}{(16k^{4}-40k^{2}+9)P}\cos\Gamma\sin\beta_{1}=0.\end{gathered}$
(22)

The determinant of (22) is $-\frac{32}{(4k^{2}-9)P^{2}}\neq 0$, so (22) has
the only solutions:

$\sin\Gamma_{\infty}=\sin\beta_{1,0}^{\infty}=0,\quad\cos\Gamma_{\infty}=\cos\beta_{1,0}^{\infty}=0.$

The determinants of Jacobians of (22) at these zeros are as follows:

$\begin{gathered}\det\begin{pmatrix}\frac{16k(4k^{2}-5)}{(16k^{4}-40k^{2}+9)P}\cos\Gamma_{\infty}\cos\beta_{1,0}^{\infty}&-\frac{8(4k^{2}+3)}{(16k^{4}-40k^{2}+9)P}\cos\Gamma_{\infty}\cos\beta_{1,0}^{\infty}\\\
\frac{4(4k^{2}+3)\cos\beta_{1,0}^{\infty}}{(16k^{4}-40k^{2}+9)P}\cos\Gamma_{\infty}\cos\beta_{1,0}^{\infty}&-\frac{8k(4k^{2}-5)}{(16k^{4}-40k^{2}+9)P}\cos\Gamma_{\infty}\cos\beta_{1,0}^{\infty}\end{pmatrix}\\\
=-\frac{32\cos^{2}\Gamma_{\infty}\cos^{2}\beta_{1,0}^{\infty}}{(4k^{2}-9)P^{2}}\neq
0\end{gathered}$

and:

$\begin{gathered}\det\begin{pmatrix}\frac{8(4k^{2}+3)}{(16k^{4}-40k^{2}+9)P}\sin\Gamma_{\infty}\sin\beta_{1,0}^{\infty}&-\frac{16k(4k^{2}-5)}{(16k^{4}-40k^{2}+9)P}\sin\Gamma_{\infty}\sin\beta_{1,0}^{\infty}&\\\
\frac{8k(4k^{2}-5)}{(16k^{4}-40k^{2}+9)P}\sin\Gamma_{\infty}\sin\beta_{1,0}^{\infty}&-\frac{4(4k^{2}+3)\cos\beta_{1,0}^{\infty}}{(16k^{4}-40k^{2}+9)P}\sin\Gamma_{\infty}\sin\beta_{1,0}^{\infty}\end{pmatrix}\\\
=\frac{32\sin^{2}\Gamma_{\infty}\sin^{2}\beta_{1,0}^{\infty}}{(4k^{2}-9)P^{2}}\neq
0,\end{gathered}$

respectively. Hence, the zeroes $\Gamma_{\infty}$ and $\beta_{1,0}^{\infty}$
are simple, so we can apply the implicit function theorem to get the result.
The proof is complete.

##  V Discussion

Melnikov analysis is applied for the persistence of periodic oscillations for
periodically-perturbed systems of ODEs with a slowly-varying variable. The
ODEs are obtained from a model of a finite lattice of particles coupled by
linear springs under distributed harmonic excitation, which presents a low-
energy nonlinear acoustic vacuum (see [1]), but we consider just two modes. We
extend the study of [1] to a problem with small exciting harmonic forces. We
apply an analytical method based on derivation of Melnikov functions and then
on the location of their simple roots. Melnikov functions are derived by using
the approach of [5, 3] developed for slowly-varying ODEs. Since the Melnikov
functions are rather complicated, we follow an asymptotic way for solving the
corresponding Melnikov equations. It would be nice to solve these Melnikov
equations numerically for finding other simple roots, which is postponed to
our next research. These roots determine and locate periodic solutions of the
periodically-perturbed systems of ODEs derived from the two-mode low-energy
nonlinear acoustic vacuum system. Moreover, our next investigation will be
also to consider higher numbers of modes represented by a system of ODEs in
(8). The method used will be different from that in this paper, since it will
be based on the results of Section 3.3 of [5]. Note that higher modes of (3)
were numerically studied in [2], which is another challenge for our study.

Authors Contributions: The authors contributed equally to this work.

Funding:M.A. is supported by the National Scholarship Programme of the Slovak
Republic for the Support of Mobility of Students, Ph.D. Students, University
Teachers, Researchers and Artists. M.F. and M.P. are supported by the Slovak
Research and Development Agency (Grant Number APVV-14-0378) and the Slovak
Grant Agency VEGA(Grant Numbers 2/0153/16 and 1/0078/17).

## References

  * [1] Manevich, L.I.; Vakakis, A.F. Nonlinear oscillatory acoustic vacuum. SIAM J. Appl. Math. 2014, _74_ , 1742–1762.
  * [2] Zhang, Z.; Manevitch, L.I.; Smirnov, V.; Bergman, L.; Vakakis, A.F. Extreme nonlinear energy exchanges in a geometrically nonlinear lattice oscillating in the plane. J. Mech. Phys. Solids 2018, _110_ , 1–20.
  * [3] Wiggins, S.; Holmes, P. Periodic orbits in slowly varying oscillators. SIAM J. Math. Anal. 1987, _18_ , 592–611.
  * [4] Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, 7th ed.; Academic Press: Cambridge, MA, USA, 2007.
  * [5] Fečkan, M. Topological Degree Approach to Bifurcation Problems; Springer: Berlin, Germany, 2008.
  * [6] Hale, J.K. Ordinary Differential Equations, 2nd ed.; Robert E. Krieger Publishing Company, Inc.: Malabar, FL, USA, 1980.


11institutetext: Chair of Data Science, University of Passau, 94036 Passau,
Germany
11email: {sahib.julka<EMAIL_ADDRESS>

# Knowledge distillation with Segment Anything (SAM) model for Planetary
Geological Mapping

Sahib Julka 11 0000-0003-3566-5507 Michael Granitzer 11 0000-0002-2952-5519

###### Abstract

Planetary science research involves analysing vast amounts of remote sensing
data, which are often costly and time-consuming to annotate and process. One
of the essential tasks in this field is geological mapping, which requires
identifying and outlining regions of interest in planetary images, including
geological features and landforms. However, manually labelling these images is
a complex and challenging task that requires significant domain expertise and
effort. To expedite this endeavour, we propose the use of knowledge
distillation using the recently introduced cutting-edge Segment Anything (SAM)
model. We demonstrate the effectiveness of this prompt-based foundation model
for rapid annotation and quick adaptability to a prime use case of mapping
planetary skylights. Our work reveals that with a small set of annotations
obtained with the right prompts from the model and subsequently training a
specialised domain decoder, we can achieve satisfactory semantic segmentation
on this task. Key results indicate that the use of knowledge distillation can
significantly reduce the effort required by domain experts for manual
annotation and improve the efficiency of image segmentation tasks. This
approach has the potential to accelerate extra-terrestrial discovery by
automatically detecting and segmenting Martian landforms.

###### Keywords:

Segment Anything Model (SAM) Semantic Segmentation Knowledge Distillation
Geological Mapping

##  1 Introduction

We have recently witnessed a paradigm shift in AI with the advent of
_foundation models_ utilising astronomical amounts of data. The fields of
natural language processing and multi-modal learning have been revolutionised
with the emergence of ChatGPT and the like [19, 22]. The very first foundation
models such as CLIP [23], ALIGN [13], and DALLE [24], have focused on pre-
training approaches but are not suited to image segmentation. However,
recently, Segment Anything (SAM) [18] was released, which is a large vision
transformer ViT-based [6] model trained on the large visual corpus (SA-1B)
containing more than 11 million images and one billion masks. SAM is designed
to generate a valid segmentation result for any prompt. However, SAM is
trained on general world case scenarios with popular structures. Recent
studies have revealed that SAM can fail on typical medical image segmentation
tasks [5, 9] and other challenging scenarios [4, 25, 12, 11]. Since SAM’s
training set mainly contains natural image datasets, it may not be directly
transferable to niche tasks on data such as magnetic resonance (MRI), or
HiRISE imaging 111“High-Resolution Imaging Science Experiment” is camera
aboard the Mars Reconnaissance Orbiter (MRO) spacecraft, which is designed to
capture high-resolution images of the Martian surface and provide detailed
information about the planet’s geology and atmosphere., amongst other
specialised data formats. Nonetheless, SAM is still a powerful tool that has a
powerful image encoder and its prompt functionality can significantly boost
the efficiency of manual annotation. In the planetary science domain, where
vast amounts of remote sensing data are gathered, annotation is an intensive
task. An approach that reduces the effort on the domain experts’ end is highly
desired. In these scenarios, active learning [17, 15] and knowledge
distillation [7] via training a specialised model with relatively fewer
samples can be highly valuable.

###  1.1 Segment Anything Model

SAM utilises a vision transformer-based [10] approach to extract image
features and prompt encoders to incorporate user interactions for segmentation
tasks. The extracted image features and prompt embeddings are then processed
by a mask decoder to generate segmentation results and confidence scores.
There are four 222Text prompt is currently not released. types of prompts
supported by SAM, namely _point_ , _text_ , _box_ , and _mask_ prompts.

For the points prompt, SAM encodes each point with Fourier positional encoding
and two learnable tokens that specify foreground and background. The bounding
box prompt is encoded by using the point encoding of its top-left and bottom-
right corners. SAM employs a pre-trained text encoder in CLIP for encoding the
free-form text prompt. The mask prompt has the same spatial resolution as the
input image and is encoded by convolution feature maps.

Finally, SAM’s mask decoder consists of two transformer layers with a dynamic
mask prediction head and an Intersection-over-Union (IoU) score regression
head. The mask prediction head generates three downscaled masks, corresponding
to the whole object, part, and subpart of the object. SAM supports three main
segmentation modes: fully automatic, bounding box, and point mode.

Figure 1: Overview of our deployed approach by extending SAM. It consists of
SAMs image encoder that learns an embedding of the image, and a specialised
decoding unit to learn the domain-specific semantics. SAMs prompt encoder and
mask decoder, represented within the orange bounding box are utilised only for
annotating incrementally the $\mathbf{\Delta}(\mathbb{N})$ training samples.
While training the domain decoder, the image encoder is frozen so as not to
update its weights.

###  1.2 Landform detection on Mars using HiRISE images

((a)) Type 1a

((b)) Type 1b

((c)) Type 2a

((d)) Type 2b

((e)) Type 3

((f)) Type 4

Figure 2: Principal types of pits and skylights found on Mars terrain: (a)
Skylight with possible cave entrance (Type 1a). (b) Pit with possible relation
to cave entrance (Type 1b). (c) “Bowl” pit with a possible connection to lava
tubes (Type 2a). (d) Pit with uncertain connection to lava tubes or dikes
(Type 2b). (e) Coalescent pits (Type 3). (f) Pit with a possible connection to
lava tubes (Type 4) [20].

Mapping planetary landforms plays a crucial role in various tasks such as
surveying, environmental monitoring, resource management, and planning. On
Earth, for example, the presence of water triggers several geological and
geomorphological processes [1]. Conversely, on Mars, researchers have found
correlations between the presence of certain landforms such as pits,
sinkholes, and landslides and the possible presence of water [2, 3]. However,
identifying, classifying, and drawing regions of interest manually is a
complex and time-consuming process [20]– one that would greatly benefit from
automation.

In this regard, the identification and segmentation of various Martian
landforms have gained increasing attention in recent years [20, 16, 14, 21].
Figure 2 shows an overview of some of the pits and skylights that can be
identified on the Martian terrain. In this study, we focus only on these
landforms, utilising a dataset prepared exclusively for it (cf. Section 2.1).
Automatic detection and segmentation of these landforms have the potential to
accelerate the identification of potential landing sites for future missions,
study the geological history of Mars, and contribute to a better understanding
of the planet’s potential habitability. Therefore, this endeavour is of
significant importance in planetary science.

##  2 Method

###  2.1 Dataset

The data used in this work are images acquired by image sensors operating in
the visible (VIS) and Near InfraRed (NIR) spectrums on board probes orbiting
Mars. This data set is composed of images by HiRISE instrument and downloaded
both as Reduced Data Record (RDR) and Experiment Data Record (EDR) format from
public space archives such as PDS Geosciences Node Orbital Data Explorer (ODE)
333https://ode.rsl.wustl.edu/. With this, Nodjoumi et al. [20] released a
processed dataset with 486 samples. This dataset is split into 405 images for
training, 25 for validation and the rest for testing. In their work, they
train a Mask-RCNN using all images annotated manually. In order to explore the
applicability of knowledge distillation, we incrementally select train samples
for annotation and subsequently train the domain decoder with these. This, in
effect, is analogous to learning correct prompts for the task, with the least
amount of annotated samples.

((a)) Image

((b)) Automatic

((c)) Point prompt

((d)) Box prompt

Figure 3: An overview of generation of segmentation masks with the three
different prompt settings in SAM. The box prompt delineates the land mass from
the adjacent shadow in comparison to the point prompt.

###  2.2 Prompt-mode selection for annotation

We conducted an evaluation of the SAM model using the three different prompt
settings:  (a) In the automatic prompt setting, SAM generates single-point
input prompts in a grid pattern across the input image and selects high-
quality masks using non-maximal suppression. All parameters were set to their
default values. In the case of multiple masks being obtained, we selected the
mask with the highest returned IoU score.  (b) In the point prompt setting, we
used the centre of the ground truth regions of interest as the point prompts
for SAM.  (c) In the box prompt setting, we computed the bounding box for SAM
around the ground-truth mask. Figure 3 illustrates the mask generation on an
exemplary sample for the three modes. Clearly, the automatic prompt simply
segments all regions in a semantic agnostic way. Point and box prompts
generate high-quality segmentation masks, with an average image level IoU
above 90 %. Although in our case, point and box prompt performed relatively
comparably on simpler cases, we empirically found box prompt to be most
reliable in occluded and shadowy scenes and thus chose that to be used for
final annotations. In practice, the expert would need only a few seconds to
draw boxes around all relevant regions of interest on a sample.

###  2.3 Domain Decoder

####  2.3.1 Why not directly fine-tune the SAM decoder?

A recent work [8] from the medical domain corroborates our observation that
the model underperforms significantly in comparison to state-of-the-art
without training and with just the use of prompts. So fine-tuning the model
would be necessary. However, we also observe that the decoder in SAM has
learnt patterns other than that specific to the task and is prone to detecting
other regions not relevant to our task. In our case, we empirically observed
fine-tuned model 444The SAM decoder is fine-tuned via training with a set of
25 annotated images for 100 epochs. to give spurious results. Figure 4
illustrates an exemplary fail-case of fine-tuning the SAM decoder with the
labels. SAM decoder even when fine-tuned is optimal only when prompts are
available, and thus is hard to be used without human-in-the-loop or additional
information from the ground truth. All of the recently developed works [9, 11,
8] use prompts derived from the ground truth labels for the problem-specific
task. This is not a realistic scenario in our application. We, therefore,
choose to train a separate decoder to learn the problem-specific semantics.

Figure 4: Landforms of interest are harder to detect without prompts while
using the SAM decoder. While the untuned model will segment all surrounding
regions, the fine-tuned model still struggles with ignoring the regions of
non-interest.

We employ a lightweight decoder (cf. Figure 1) comprised of only three
upsampling layers using deconvolutions that maps the bottleneck $z$ to an
image of the desired size, in our case $(3\times 900\times 900)$. The
bottleneck is obtained by passing the image through SAMs encoder. During
training, only the weights of the decoder are updated. We use a sigmoid
activation to map the logits in the range $[0,1]$ for binary segmentation. In
this manner, we train the decoder with incremental sets of SAM-annotated
images. The incremental function $\mathbf{\Delta}(\mathbb{N})$ is used in step
sizes with $\mathbb{N}\in\\{5,10,15,20,25,50\\}$. All models are trained for a
total of 100 training epochs, without additional control. We compare the
performance using mean Intersection over Union (mIoU), micro F1, accuracy, and
micro-precision and recall. Micro metrics are chosen to better represent the
performance under data imbalance. Figure 5 shows the evolution of the metrics.
We observe that the performance improvement with additional training samples
after a handful is non-significant, with any differences being representative
of stochasticity in evaluation rather than true information gain. By observing
the metrics above and with qualitative evaluation it can be inferred that
depending on the complexity of the domain-specific task, a very small number
of annotations can suffice for a representative performance (cf. Figure 6).

Figure 5: Development of the evaluation metrics with increasing sizes
$\mathbf{\Delta}(\mathbb{N})$ of annotated training samples. Increasing
training size beyond a handful of samples yields trivial overall improvement.

Further, we compare the performance of this approach with $\mathbf{\Delta}(5)$
against the Mask-RCNN model proposed in existing literature (cf. Table 1) for
the same task, which serves as the benchmark for our comparison. This model is
trained with the full training size of 405 manually annotated images. The
authors [20] in this work only reported macro metrics and noted that about
1000 positive labels were required for satisfactory performance. We clearly
see that knowledge distillation through SAM by utilising relatively minuscule
labels surpasses the benchmark on most reported metrics. In spite of the
precision being slightly lower, the recall is substantially higher. It is to
be noted that in tasks like these, recall should be given a higher importance
to precision, since missing a region of interest is more critical than falsely
identifying one.

Table 1: Comparison of the state of the art vs our proposed approach trained only with 5 labelled samples. The authors in [20] train their model with 405 samples and report macro metrics. model | macro F1 | accuracy | macro precision | macro recall
---|---|---|---|---
Mask-RCNN [20] | 0.811 | 0.774 | 0.952 | 0.706
ours ($\mathbf{\Delta}(5)$) | 0.86 | 0.96 | 0.89 | 0.93

Figure 6:  Example predictions on the test set. The domain decoder identifies
all regions of interest reasonably well.

##  3 Conclusion

In this work, we extended the SAM framework and applied it to the segmentation
of landforms like pits and skylights on the surface of Mars using HiRISE
images. We observed that SAM has a high accuracy in separating various
semantic regions, however, it cannot be directly applied to domain-specific
tasks due to a lack of problem-specific bias. To this end, we developed and
applied a domain-specific decoder that takes the image embedding generated by
SAMs image encoder and learns the problem-specific semantics with
substantially fewer labels. By training the domain decoder with only 5
labelled images sampled randomly, we demonstrated an equivalent if not
superior performance to the existing Mask-RCNN method for the same task that
was trained with over 400 labelled images.

We also explored the applicability of SAMs decoder for annotation using the
various out-of-box prompts. We observed that the fully automatic mode is prone
to marking irrelevant regions, and further can also miss some regions of
interest if it doesn’t know where to look. The point-based mode can be
ambiguous at times. In contrast, the bounding box-based mode can clearly
specify the ROI and obtain reasonable segmentation results without multiple
trials and errors. We can therefore conclude that the bounding box-based
segmentation mode can be a useful setting for rapid annotation by the domain
expert.

In conclusion, our study reveals that SAM can effectively be exploited to
accelerate domain-specific segmentation tasks. This work presents the first
attempt to adapt SAM to geological mapping by fine-tuning through knowledge
distillation. As part of future work, it might be worthwhile to investigate
how the process of annotation can be automated, further lowering the load of
human-in-the-loop. We hope this work will motivate more studies to build
segmentation foundation models in the planetary science domain.

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