Localized chaotic patterns in weakly dissipative systems - Springer Link

Eur. Phys. J. Special Topics 223, 141–154 (2014) © EDP Sciences, Springer-Verlag 2014 DOI: 10.1140/epjst/e2014-02089-x

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Localized chaotic patterns in weakly dissipative systems D. Urzagasti1 , D. Laroze1,2 , and H. Pleiner2,a 1 2

Instituto de Alta Investigaci´ on, Universidad de Tarapac´ a, Casilla 7D, Arica, Chile Max-Planck-Institute for Polymer Research, 55021 Mainz, Germany Received 11 September 2013 / Received in final form 25 November 2013 Published online 23 January 2014 Abstract. A generalized parametrically driven damped nonlinear Schr¨ odinger equation is used to describe, close to the resonance, the dynamics of weakly dissipative systems, like a harmonically coupled pendula chain or an easy-plane magnetic wire. The combined effects of parametric forcing, spatial coupling, and dissipation allows for the existence of stable non-trivial uniform states as well as homogeneous pattern states. The latter can be regular or chaotic. A new family of localized states that connect asymptotically a non-trivial uniform state with a spatio-temporal chaotic pattern is numerically found. We discuss the parameter range, where these localized structures exist. This article is dedicated to Prof. Helmut R. Brand on the occasion of his 60th birthday.

1 Introduction Localized structures can be found in many different driven systems, like chiral bubbles in liquid crystals, current filaments in gas discharge, spots in chemical reactions, localized states in fluid surface waves, oscillons in granular media, isolated states in thermal convection, solitary waves in nonlinear optics, solitons in magnetic materials and Bose-Einstein condensates, localized states in generic subcritical instabilities, to mention a few. Recent reviews of the state of the art can be found in Refs. [1–4]. In one-dimensional spatial systems, localized states can be described as spatial trajectories that connect one steady state with itself, which means, they are homoclinic orbits from the dynamical system point of view [5]. On the other hand, domain walls or front solutions can be seen as spatial trajectories connecting two different steady states and are, thus, heteroclinic curves of the corresponding dynamical system [6]. A particular class of localized states that has been in the focus of many efforts in the past decade, are localized patterns. These are structures that vary in space over only a small part of the system [7, 8]. Hence, localized patterns are homoclinic trajectories that link a uniform state with itself, the latter being able to coexist with the patterned state [5, 6]. The existence of localized patterns based on front interaction was developed in Ref. [9], and subsequently in Ref. [10]. A geometrical interpretation of the a

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existence, the stability properties, and the bifurcation diagram of localized patterns in one-dimensional extended systems has been proposed [11,12]. We remark that there are many different types of various complexity, such as oscillatory, propagative, and chaotic localized patterns [13–25]. Here, we deal with chaotic ones. When the driving force acting on a system depends on the system’s variables, the system is called parametrically driven. The generation of standing surface (Faraday) waves of a vertically vibrated Newtonian fluid, is one of the classical parametrically driven hydrodynamic instabilities [26], where the injection of energy is made through vertical oscillations. These surface waves are subharmonic, that is, the system responds strongest at half the forcing frequency (2:1 resonance) [27]. This type of parametric resonance is commonly termed strong resonance [28]. Parametric instabilities close to the 2:1 resonance for the particular case of weakly dissipative systems [29], i.e. genuine time reversible systems that are only slightly perturbed by the injection and dissipation of energy, are modeled by the parametrically driven damped nonlinear Schr¨ odinger (PDDNLS) equation [30]. This model has been derived and studied to describe pattern formation in several physical frameworks, such as nonlinear lattices [31], optical fibers [32], Kerr-type optical parametric oscillators [33], magnetization dynamics of easy-plane ferromagnets subject to an oscillatory magnetic field [34–36], parametrically driven damped coupled pendula chains [37], and in general form [38–42]. However, it has recently been shown that this conventional approach is not able to describe the case of stable localized states that link two zones of asymptotically uniform oscillations with opposite phases, [43–46], nor, more recently, localized states that connect asymptotically a uniform oscillatory state with an extended wave in a magnetic nanowire [47]. To get such stable solutions, one has to rely on a generalization of the classical PDDNLS equation derived in Ref. [43]. Viewed as an amplitude equation obtained by the systematic expansion procedure of the weakly nonlinear stability analysis, this generalization of the PDDNLS is obtained by taking into account contributions of the next higher expansion step. Another generalization that leads to stable homogeneous states is the complexification of the parameters of the PDDNLS equation producing a parametrically driven complex Ginzburg Landau model [48– 50]. Furthermore, other recent generalizations applied to intrinsic localized modes in parametrically driven arrays of nonlinear resonators can be found in Ref. [51]. The purpose of the present work is to discuss the existence and the properties of chaotic localized structures of parametrically driven systems close to the 2:1 resonance. To this aim a generalized PDDNLS equation is used that can be viewed as the amplitude equation for the different states (Sect. 2). We find analytically an implicit expression for the amplitudes of the finite-amplitude, homogeneous solutions (Sect. 2.1) and derive a cubic-quintic Ginzburg-Landau equation that describes the amplitude of extended spatial patterns (Sect. 2.2). We then numerically show that in a certain region of the parameters the pattern solution coexists with the finite-amplitude, homogeneous one, thus, allowing for localized patterns (Sect. 3). In particular, we discuss localized chaotic patterns. A summary is presented in Sect. 4.

2 Theoretical model To describe the phenomena close to the 2:1 parametric resonance of weakly dissipative systems, we consider the dimensionless generalized parametrically driven damped nonlinear Schr¨ odinger equation ∂2A ∂A = −iνA − i|A|2 A − i 2 − μA + γ A¯ + NA , ∂T ∂Z

(1)

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with ν the detuning parameter, μ the damping coefficient, and γ the driving parameter, while A¯ stands for the complex conjugate of A, and NA will be discussed in detail below. For NA = 0, Eq. (1) is the standard PDDNLS equation [34]. This equation has often been studied to understand Faraday waves [46, 52], soliton like solutions [34, 41], two-soliton states [36, 42], and spatio-temporal chaos [39]. In the special case odinger equation, which serves NA = μ = γ = 0, Eq. (1) reduces to the nonlinear Schr¨ as a Hamiltonian, time reversible model [53]. The PDDNLS equation has different homogeneous solutions (A = const.). ones are The trivial one is A = 0, and nontrivial A±,± = ±x 0 (1 ± iy0 ), where x0 = (γ − μ)(φ − ν)/2γ and y0 = (μ − γ)/(μ + γ) with φ ≡ γ 2 − μ2 . All these solutions merge by a saddle-node bifurcation at γ = μ. Generally, the nontrivial states are unstable everywhere as a consequence of the spatial coupling [43], only A±,+ is marginally stable for zero detuning (ν = 0). Hence, one can expect that any (small) correction of the PDDNLS equation will render this state linearly stable or unstable. Hence, NA can lead to the stability of the non-trivial homogeneous solution. The generalization of the PDDNLS equation described by NA can be derived systematically from the underlying physical model [43, 46] using the weakly nonlinear analysis. In order γ 5/2 of this expansion one gets the general expression NA = γ

∂2A bA|A|4 + δA3 + βA3 |A|2 + αA|A|2 + iaA|A|4 + κ 2 − cA|A|2 . ∂Z

(2)

which describes additional features not present in the ordinary PDDNLS equation, in particular, diffusion (∼ κ), nonlinear dissipation (∼ c), and nonlinear parametric forcing (∼ b, δ, β, α), while the term ∼ a describes a higher (quintic) nonlinear saturation in addition to the cubic one already present in PDDNLS. Not all of these extra terms are necessary to recover the stability of the homogeneous states, e.g. in Ref. [48] the PDDNLS equation is amended by the (extra) diffusion term (∼ κ), allowing for localized solutions that connect the zero state with a non-trivial homogeneous one. In Ref. [49, 50], the authors study localized states in a forced complex Ginzburg Landau equation using terms of the form ∼ a, κ, and c. In the following, we will analyze the particular case of κ = 0 and c = 0, since the effect of these coefficients have already been studied in the afore mentioned references, and since they do not appear in the physical example which is provided in the Appendix. Finally, we remark that the generalization NA preserves the reflection symmetry A → −A of the PDDNLS equation. In the next two subsections we describe stable homogeneous solutions of the generalized system.

2.1 Homogeneous solutions The generalized PDDNLS equation still has the trivial solution A = 0 and other nontrivial homogeneous ones. In particular cases, analytical expressions can be given. For example, for κ = 0 and c = 0 an implicit equation for the amplitudes is found γ2 =

μ2 (ν + |A|2 − a|A|4 ))2 + · 2 4 2 (1 + [δ + α]|A| + [β + b]|A| ) (1 + [α − δ]|A|2 + [b − β]|A|4 )2

(3)

These states bifurcate from A = 0 at γ 2 = μ2 + ν 2 ; this latter relation defines the first Arnold tongue commonly used in γ vs. ν diagrams. For the modulus of the amplitude, |A|2 , as a function of γ, Fig. 1 shows the comparison between the

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Fig. 1. |A|2 as a function of γ for ν = −0.05, μ = 0.35, b = 1/12, δ = 4/15, β = −1/24, α = −0.65, c = 0 and κ = 0. The dots are numerical solutions of Eq. (1), while the continuous line represents the analytical solution given by Eq. (3).

analytical expression given by Eq. (3) and numerical solutions of Eq. (1). We can observe a perfect agreement between both. We numerically found these states to be stable in a wide range of parameters. Their stability is the crucial requirement for generating localized states that connect a homogeneous state with itself. Different types of connections are possible which produce different types of localized states. For example, localized states that connect the zero state with a non-trivial one [48], or localized states that connect a nontrivial state with itself [43]. Due to the symmetry of the system, also heteroclinic connections are possible. We would like to remark that these non-zero homogeneous solutions in the amplitude equation description represent uniform oscillatory states with a constant maximum amplitude in the appropriate physical system. Examples are the parametric pendulum [27] and the spin precession in a time dependent magnetic field [54–56], both amended by spatial interactions. 2.2 Spatial instability in the generalized PDDNLS equation In order to understand the mechanism of pattern formation in the model (1), we study the stability of the trivial solution A = 0. Without the spatial coupling the trivial state is stable outside the Arnold tongue. The spatial coupling, expressed by the Laplacian term in the PDDNLS equation, modifies this scenario and the zero state exhibits a spatial instability at γ = μ for positive detuning, ν > 0, giving rise to the occurrence of pattern states. Close to the bifurcation, the ansatz [52, 57] 1 3 2ikc Z ikc Z B e e + c.c. + O(5) Re A = B + 8ν Im A = − with kc

3 B|B|2 eikc Z + c.c. + O(5) 2γ

(4)

√ ν, results in an amplitude equation for the pattern amplitude B ∂2B ∂B = ΓB + χ1 |B|2 B − χ2 |B|4 B − i 2 ∂ξ ∂Z

(5)


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with Γ = γ − μ the bifurcation parameter and χ1 = 3γ(α + δ), χ2 = 9/(2γ) and ξ = Γ T . This equation is a cubic-quintic Ginzburg-Landau equation for the complex amplitude B. Note that without the generalization of Eq. (2) there is χ1 = 0, and previous results presented in Ref. [52] are recovered. Hence, in the generalized case the instability can be either super- or sub-critical, which is not possible in the standard approach. We remark that close to the bifurcation the pattern amplitude B is constant and increases with Γ according to the law, χ1 ± 4Γχ2 + χ21 , |B0 | = 2χ2 which reduces for χ1 = 0 to the well-known power law [52], |B0 | ∼ (γ −μ)1/4 . The pattern structure described by Eq. (5) not only exists, and is stable, for ν > 0, for which it has been derived, but also for ν < 0, if γ 2 ≥ μ2 + ν 2 (i.e. within the Arnold tongue). Reducing γ (at negative ν) the pattern structure vanishes at the Arnold tongue by a saddle-node bifurcation, before the γ = μ line is reached. If γ is increased to high values within the Arnold tongue, the pattern suffers multiple bifurcations and becomes a chaotic state. Thus, inside the Arnold tongue stable homogeneous states coexist with patterns. Generally, the coexistence of two stationary states leads to the possibility of obtaining localized patterns [47, 49, 50], where one of the possible states exists in a small portion of the system, squeezed between larger areas, where the other possible state exists. In the next section we will present a novel type of localized pattern that connects a non-zero, asymptotically uniform solution with a spatio-temporal chaotic pattern (and back).

3 Simulations This section is divided in two parts. In the first subsection we briefly discuss the quantities, which we use to characterize spatio-temporal regimes, and in the second one some numerical results and their analysis are presented.

3.1 Dynamical indicators We will essentially characterize the different types of dynamical behavior of the system by the energy function, Q, and the largest Lyapunov exponent, λmax . The first one is the norm, • , of the amplitude defined by 1 Q(T ) = 2L

L

−L

|A(T, Z)|2 dZ,

(6)

which is frequently used to characterize non-regular dynamics in optics [18–21], localized patterns in fluids, and other physical systems [58–60]. The one-dimensional system is assumed to be of length 2L. We remark that Q is generally a function of time, which reflects the temporal information of the patterns, i.e. in a stationary regime, Q is constant, while in a (quasi-) periodic one, Q is a (quasi-) periodic function of time. Consequently, if the system is in a chaotic regime, the time series of Q will be chaotic, too. To understand the time series of Q, we first take the Fast Fourier Transform (FFT) which gives us a complex discrete signal, S ( ), in frequency space

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Fig. 2. Spatio-temporal diagram of |A| (color or gray-scale coded) for an extended pattern (left) and a localized one (right) at ν = −0.05 and γ = 0.47. The two states are obtained at the same parameters, but using different initial conditions.

Fig. 3. Real (left) and imaginary (right) part of the amplitude A of the localized pattern as a function of space at ν = −0.05 and γ = 0.47.

= ( 1 , ..., n ), producing a set of pairs { k , S ( k )}. For this signal we calculate its power spectrum |S ( )|. In general, when |S ( )| has a finite number of discrete peaks, the time series is regular, while if there is a continuum of peaks, the series can be chaotic. To complement the information (in particular to discriminate between quasiperiodic and chaotic dynamics) and to provide a more quantitative aspect of the dynamics, we calculate the largest Lyapunov exponent [61], defined by λmax = lim

T →∞

1 T

ln

δA(T, Z) δA0

,

(7)

where δA satisfies the differential equation ∂δA ¯ · δA, =J ∂T

(8)


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Fig. 4. Spatio-temporal diagram of |A| (color or gray-scale coded) for an extended pattern state (left) and a localized one (right) at ν = −0.05 and γ = 1.20.

Fig. 5. Localized pattern energy, Q, as a function of time (left) and its corresponding Fourier power spectrum (right) at ν = −0.05 and γ = 1.20.

¯ the Jacobian matrix of Eq. (1). This number quantifies how fast the distance with J between two initially close trajectories δA of the vector field A either vanishes exponentially (λmax < 0) or diverges (λmax > 0). The latter is the hallmark of chaotic behavior. This method has been extensively used for many different dynamical systems to quantify chaos [13–17,54–56, 61–69]. 3.2 Numerical results To solve numerically the amplitude equation, Eq. (1), we use a variable-step fifth-order Runge-Kutta (RK) scheme for the time evolution and a second-order central finitedifference method to approximate the spatial derivatives over the system’s length 2L. Here we consider L = 100 and use N = 1500 lattice points, which implies ΔZ = 200/1500 ≈ 0.13. We use a double precision RK method provided in Ref. [70], which admits the error tolerance to be chosen and which we take as 10−7 implying that errors occur in the seventh significant digit at the most. We use Neumann boundary conditions. In addition, we have used two different types of initial conditions, an extended pattern over the whole box, and a localized domain with two bumps in a

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Fig. 6. Bifurcation diagram of Qmax (left) and the largest Lyapunov exponent, λmax , (right) as a function of γ at ν = −0.05. In the inset (left) Qmax is blown up by a factor of 105 .

central part of the box. We have also checked smaller and larger box sizes to guarantee that there are no finite size effects for L = 100. After any transients have faded away we have continued the calculations for at least twice the full transient time with a maximum integration time T = 4.8 × 103 . Moreover, we have changed N and ΔZ to verify that none of the presented results depends sensitively on the discretization used. The main results are given in Figs. 2–9. In particular, we concentrate the discussion on the influence of the driving force. Therefore, we use the driving force coefficient, γ, as the bifurcation parameter. In the following the fixed parameter values are μ = 0.35, b = 1/12, δ = 4/15, β = −1/24, α = −0.65, and a = 1/6. The new parameters, which appear in NA , are chosen very close, but not identical, to the parameters obtained in the pendulum’s derivation (see Appendix). This choice ensures realistic parameter values, but also avoids probable artefacts of a specific system and, therefore, provides a general description. All figures are taken for a detuning of ν = −0.05, but other values of ν are discussed briefly at the end of this section. We remark that in all spatio-temporal diagrams presented the time scale starts after a long lapse after the transients. Figure 2 shows spatio-temporal diagrams of the absolute value of the amplitude |A| (color coded) in the stationary regime at γ = 0.470 for both, the extended and the localized pattern state. The latter is a consequence of the coexistence between stable homogeneous and extended pattern states. Since this is a stationary regime, the energy function Q is also constant, i.e. Q = 0.257 for the particular parameter values. The largest Lyapunov exponent is negative, λmax ≈ −1.42×10−2 , as expected. Figure 3 shows the spatial dependence of the real and the imaginary part of A for the localized state. Figure 4 shows spatio-temporal diagrams of the absolute value of the amplitude |A| in the chaotic regime at γ = 1.20 for both, the extended and the localized pattern state. The largest Lyapunov exponent is positive, λmax ≈ 0.1879, indicating chaotic dynamics. In addition, Q is a chaotic function of time and shown on the left side of Fig. 5. The corresponding Fourier power spectrum of Q, given on right side of Fig. 5, is continuous and, thus, confirms the chaotic behavior of the time series in this regime. In order to investigate how the system changes its dynamical behavior as a function of the control parameter γ, we determine the bifurcation diagram and calculate the largest Lyapunov exponent, λmax (γ). The bifurcation diagram (left frame of Fig. 6) is obtained by taking repeatedly the maximum value of the energy function Qmax in


149

Fig. 7. Spatio-temporal diagram of the localized pattern state (left) and the corresponding Fourier power spectrum of Q (right) at ν = −0.05 and γ = 0.7289 in a quasi-periodic regime just below the chaotic island.

Fig. 8. Real (left) and imaginary (right) part of the amplitude A (color or gray-scale coded) as a function of space and γ for a snapshot at T = 4.8 × 103 for ν = −0.05.

a given time interval at different times (well after the transients); this is done for very many different values of the control parameter γ. If there is a unique Qmax , then the system is stationary or periodic, while for a finite continuous distribution of Qmax values, the behavior is either quasi-periodic or chaotic. To discriminate between the latter possibilities, the Lyapunov exponent λmax is shown on the right frame of Fig. 6 and when it has positive values the system is in a chaotic regime. In both frames we have calculated 3.6 × 103 points for the γ dependence. For each value of γ the system has been evolved with the same initial conditions. In Fig. 6 we can observe several transitions between regular and chaotic states. In particular, there is a chaotic island in the range γ ∈ (0.729, 0.773). Above γ = 1.028 the system becomes chaotic again through a non-smooth transition, and continues to be chaotic for the values of γ considered here. On the other hand, the first transition from a regular to a chaotic localized pattern seems to be rather smooth. In particular, the fluctuations of Qmax and the positive Lyapunov exponent are much smaller in this chaotic island than in the upper chaotic regime. Just below this transition to the chaotic island, at γ = 0.7289, the Lyapunov exponent is zero within the numerical

150


Fig. 9. Δ (color or gray-scale coded) as a function of time and γ (left) and its corresponding time average ΔT as a function of γ (right).

error (λmax ≈ −3.5 × 10−4 ). The left frame of Fig. 7 shows the spatio-temporal diagram of |A| for this particular point in the parameter space. The right frame displays the Fourier power spectrum of Q revealing that the temporal motion is fourperiodic. In addition, from Fig. 6 one can observe that in the upper chaotic regime γ > 1.0279 the bifurcation diagram consists of two distinguishable parts. First, the set of Qmax values is rather compact, while for larger γ values it becomes diffuse. In addition, the function λmax (γ) seems to change its mean slope at the same point. This issue is consistent with the fact that the width of the localized pattern starts to increases for larger γ, as can be seen in Fig. 8. There, the real and imaginary part of A are shown as a function of space and γ for a snapshot at a very large time T = 4.8 × 103 . We can observe that the localized chaotic state start to penetrate the homogenous state at that γ, where also the changes in Qmax and λmax (γ) are observed. In order to quantify this phenomenon we calculate a statistical width of the local2 ized pattern, Δ(T ), describing deviations from the uniform state. Using |A| as the statistical weight distribution, Δ = 2 |δΣ|, where

L δΣ(T ) =

−L

|A(T, Z)|2 (Z − Z0 (T ))2 dZ L2 , −

L 3 |A(T, Z)|2 dZ −L

and

L Z0 (T ) = −L L

|A(T, Z)|2 Z dZ

−L

|A(T, Z)|2 dZ

·

The left frame of Fig. 9 shows Δ (color coded) as a function of time and γ. One can observe that it is constant for the stationary localized pattern, but has a time dependence for the chaotic windows. For the second chaotic window (γ > 1.0X) the time dependence reveals two branches of different behavior. This is even more pronounced, when the time average of Δ, denoted by Δ T , is considered in the right frame of Fig. 9. Above that γ, where the chaotic pattern penetrates the homogeneous solution, Δ T (γ) increases and has a quite different functional form compared to that in the other chaotic and regular regimes.


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Fig. 10. Spatio-temporal diagrams of the (color or gray-scale coded) real part of the amplitude A of localized states that initially are separated by a distance d. The top-left, top-right, bottom-left and bottom-right frames are for d = 11, d = 29, d = 100 and d = 150, respectively. The fixed parameters are those of Fig. 5.

In addition, we have determined the existence range for localized chaotic states as a function of the detuning ν for two different values of the driving parameter. At γ = 1.03 there are localized chaotic states in the range ν ∈ {−0.05, 0.07} while for γ = 1.2 two windows exist, ν ∈ {−0.17, −0.12} ∪ {−0.09, 0.07}. Above these ranges other spatio-temporal solutions appear, such as nonlinear waves. Notice that for γ = 1.2 the island between the two chaotic ranges features regular localized patterns. The complete γ vs. ν phase diagram is still in progress and will be published elsewhere. Finally, let us comment on the interaction of these localized chaotic structures. Starting, as an initial condition, with two localized states of the type of Figure 5, separated by a distance d, rather complex final states evolve that require a very long integration time, about 240 times longer than the integration time used for a single chaotic localized pattern. We numerically find that for small distances (0 < d 28) the final state is a single localized chaotic object, albeit with a distinct internal structure (Fig. 10 top-left). For intermediate distances (29 d 150) the two distinct localized structures remain (Fig. 10 bottom-left), however with an interaction between them that fades away for large initial distances, d 150 (Fig. 10 bottom-right). At the transition between the two regimes (d ≈ 29) no stationary state is found within the integration time used (Fig. 10 top-right).

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4 Summary We have presented a novel type of localized states that link an asymptotically homogeneous state with a sub-harmonic pattern in parametrically driven systems close to the parametric resonance. To describe this phenomenon we have employed a generalized parametrically driven damped nonlinear Schr¨ odinger equation, which contains higher order nonlinear terms, since the standard model lacks stable nontrivial homogeneous solutions which play a crucial role to obtain localized states. In particular, we have found the localized pattern to be stationary, q-periodic or chaotic, depending on the parameters. We have concentrated on the chaotic localized patterns, and different tools to characterize the dynamical behavior, such as bifurcation diagrams, Fourier power spectra and Lyapunov exponents, have been used. Localized chaotic states exist within the Arnold tongue and compete with regular ones. Varying the driving strength at constant detuning (or vise versa) results in various transitions between these states. The chaotic states come in two rather different phenomenological forms, either with small chaotic indicators (“energy” and positive Lyapunov exponent) or large ones; there are localized chaotic regimes, where the spatial width of the chaotic part is rather well defined and others, where the chaotic area penetrates the homogeneous one as a function of the driving force. Finally, let us remark that due to the universal nature of the considered model we expect to observe localized spatio-temporal chaos in several driven physical systems, such as a vertically oscillating fluid layers, magnetic systems, forced nonlinear lattices, and optical fibers. D.U. and D.L. would like to thank M.G. Clerc (Univ. of Chile) for valuable discussions on the generalization of the PDDNLS equation. D. L. acknowledges partial financial support from FONDECYT 1120764, Millennium Scientific Initiative, P 10−061−F , Basal Program Center for Development of Nanoscience and Nanotechnology (CEDENNA), UTA-project 8750 − 12. D. U. acknowledges the PhD program fellowship through the Performance Agreement Project’s UTA/Mineduc.

Appendix In this appendix we derive the coefficients that appear in NA for a vertically driven, damped, elastically coupled linear pendula chain, which is described in the continuum limit by [52] ∂2θ ∂θ ∂2θ = k 2, + [Ω20 + γ0 sin ((2Ω0 + ν0 /2)t)]sin(θ) + μ0 2 ∂t ∂t ∂z

(A.1)

where θ(t, z) is the angle between the pendulum and the vertical (gravity) axis at time t and position z along the chain. Ω0 is the pendulum’s natural frequency, γ0 the amplitude of the parametric forcing, μ0 is the damping coefficient, k the elastic coupling coefficient, and ν0 the detuning frequency, which measures how far off the frequency of the driving force is from the parametric resonance. This system has a zero amplitude solution θ = 0. We focus on the case ∼ γ0 ∼ μ0 ∼ ν0 , where is an arbitrary small parameter 1. Close to the parametric instability, we introduce the ansatz (A.2) θ = A(T, Z)ei(Ω0 +ν0 /2)t + c.c. + h.o.t., where A(T, Z) is the envelope of a uniform oscillation, T , Z are slow variables, while c.c. and h.o.t. mean complex conjugate and higher order terms, respectively. Introducing this ansatz in Eq. (A.1) and expanding the nonlinear terms in a Taylor series


153

up to quintic-order, one gets and after some algebra the amplitude equation ∂A ∂2A ∂2A = −iνA − i|A|2 A − i 2 − μA + γ A¯ + κ 2 − cA|A|2 ∂T ∂Z ∂Z +γ bA|A|4 + δA3 + βA3 |A|2 + αA|A|2 + iaA|A|4 ,

(A.3)

where T =

Ω0 t, 4

Ω0 Z = √ z, 2k

μ=

2 μ0 , Ω0

ν=

2 ν0 , Ω0

1 , 6

b=

γ=

γ0 , Ω02

with the coefficients κ = 0,

c = 0,

1 α=− , 2

δ=

1 , 6

a=

1 , 12

β=−

1 · 24

Note that Eq. (A.3) is of the form of Eq. (1) with a specific choice of parameters. More details can be found in Refs. [43–46].

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