We study the dynamics of a pair of parametrically-driven coupled nonlinear mechanical resonators of the kind that is typically encountered in applications involving microelectromechanical and nanoelectromechanical systems (MEMS & NEMS). We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using a version of the high-dimensional Melnikov approach, developed by Kovacic and Wiggins [Physica D, 57, 185 (1992)], we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Shilnikov orbits, indicating a loss of integrability and the existence of chaos. Our analytical calculations of Shilnikov orbits are confirmed numerically.
We examine the Melnikov criterion for transition to chaos in case of a single degree of freedom nonlinear oscillator with the Ueda well potential and an external periodic excitation term. Using effective Hamiltonian we have examined homoclinic orbits and cross-sections of stable and unstable manifolds which gave the condition of transition to chaos through a homoclinic bifurcation.
A review on the application of Melnikov's method to control homoclinic and heteroclinic chaos in low-dimensional, non-autonomous and dissipative, oscillator systems by weak harmonic excitations is presented, including diverse applications such as chaotic escape from a potential well, chaotic solitons in Frenkel-Kontorova chains, and chaotic charged particles in the field of an electrostatic wave packet.
We study the origin of homoclinic chaos in the classical 3D model proposed by O. Rössler in 1976. Of our particular interest are the convoluted bifurcations of the Shilnikov saddle-foci and how their synergy determines the global unfolding of the model, along with transformations of its chaotic attractors. We apply two computational methods proposed, 1D return maps and a symbolic approach specifically tailored to this model, to scrutinize homoclinic bifurcations, as well as to detect the regions of structurally stable and chaotic dynamics in the parameter space of the Rössler model.
A method of controlling Shil'nikov's type chaos using windows that appear in the 1 dimensional bifurcation diagram when perturbations are applied, and using existence of stable homoclinic orbits near the unstable one is presented and applied to the electronic Chua's circuit. A demonstration of the chaos control in the electronic circuit experiments and their simulations and bifurcation analyses are given.
The chaotic spike train of a homoclinic dynamical system is self-synchronized by re-inserting a small fraction of the delayed output. Due to the sensitive nature of the homoclinic chaos to external perturbations, stabilization of very long periodic orbits is possible. On these orbits, the dynamics appears chaotic over a finite time, but then it repeats with a recurrence time that is slightly longer than the delay time. The effect, called delayed self-synchronization (DSS), displays analogies with neurodynamic events which occur in the build-up of long term memories.
The dynamical bifurcations of a laser with a saturable absorber were calculated, with the 3-2 level model, as function of the gain parameter. The average power of the laser is shown to have specific behavior at bifurcations. The succession of periodic-chaotic windows, known to occur in the homoclinic chaos, was studied numerically. A critical exponent of 1/2 is found on the tangent bifurcations from chaotic into periodic pulsations.
In an early work, Bernoulli shift dynamics of submanifolds was established in a neighborhood of a homoclinic tube. In this article, we will present a concrete example: sine-Gordon equation under a quasi-periodic perturbation.
We developed a powerful computational approach to elaborate on onset mechanisms of deterministic chaos due to complex homoclinic bifurcations in diverse systems. Its core is the reduction of phase space dynamics to symbolic binary representations that lets one detect regions of simple and complex dynamics as well as fine organization structures of the latter in parameter space. Massively parallel simulations shorten the computational time to disclose highly detailed bifurcation diagrams to a few seconds.
Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulae expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well…
We prove the presence of chaos near a homoclinic orbit in the modified Li-Yorke sense [10] by implementing chaotic perturbations. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Ott-Grebogi-Yorke and Pyragas control methods are used to stabilize almost periodic motions.
Let $F$ be a $C^3$ diffeomorphism on a Banach space $B$. $F$ has a homoclinic tube asymptotic to an invariant manifold. Around the homoclinic tube, Bernoulli shift dynamics of submanifolds is established through shadowing lemma. This work removes an uncheckable condition of Silnikov [Equation (11), page 625 of L. P. Silnikov, Soviet Math. Dokl., vol.9, no.3, (1968), 624-628]. Also, the result of Silnikov does not imply Bernoulli shift dynamics of a single map, rather only provides a labeling of all invariant tubes around the homoclinic tube. The work of Silnikov was done in ${\mathbb R}^n$, an…
Homoclinic and heteroclinic orbits provide a skeleton of the full dynamics of a chaotic dynamical system and are the foundation of semiclassical sums for quantum wave packet, coherent state, and transport quantities. Here, the homoclinic orbits are organized according to the complexity of their phase-space excursions, and exact relations are derived expressing the relative classical actions of complicated orbits as linear combinations of those with simpler excursions plus phase-space cell areas bounded by stable and unstable manifolds. The total number of homoclinic orbits increases exponentia…
We analyze, by means of Melnikov method, the possibility of modifying the threshold of homoclinic chaos in general 1-dimensional problems, by introducing small periodic resonant modulations. We indicate in particular a prescription in order to increase the threshold (i.e. to prevent chaos), and consider then its application to the bistable Duffing-Holmes potential. All results are confirmed both by numerical and by analog simulations, showing that small modulations can in fact sensibly influence the onset of chaos.
We describe and characterize rigorously the chaotic behavior of the sine-Gordon equation. The existence of invariant manifolds and the persistence of homoclinic orbits for a perturbed sine--Gordon equation are established. We apply a geometric method based on Mel'nikov's analysis to derive conditions for the transversal intersection of invariant manifolds of a hyperbolic point of the perturbed Poincare map.
Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wavefunctions localized along periodic orbits we reveal the existence of an oscillatory behavior, that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit.
Tunnelling from a chaotic potential well is explained in terms of a set of complex periodic orbits which contain information about the real dynamics inside the well as well as the complex dynamics under the confining barrier. These orbits are associated with trajectories which are homoclinic to a real trajectory emerging from the optimal tunnelling path. The theory is verified by considering a model double-well problem.
Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. We consider here, following a (suitably adjusted) classical analytic method, the case of non-hyperbolic points and show that, under a Melnikov-type condition plus an additional assumption, the negatively and positively asymptotic sets persist under periodic perturbations, together with their infinitely many intersections on the Poincaré section. We also examine, by means of essentially the same procedure, the case of (heteroclinic) orbits tending to the infinity; this case includes in particular th…
In addition to the well known scarring effect of periodic orbits, we show here that homoclinic and heteroclinic orbits, which are cornerstones in the theory of classical chaos, also scar eigenfunctions of classically chaotic systems when associated closed circuits in phase space are properly quantized, thus introducing strong quantum correlations. The corresponding quantization rules are also established. This opens the door for developing computationally tractable methods to calculate eigenstates of chaotic systems.
The Melnikov method is applied to periodically perturbed open systems modeled by an inverse--square--law attraction center plus a quadrupolelike term. A compactification approach that regularizes periodic orbits at infinity is introduced. The (modified) Smale-Birkhoff homoclinic theorem is used to study transversal homoclinic intersections. A larger class of open systems with degenerated (nonhyperbolic) unstable periodic orbits after regularization is also briefly considered.
We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers $(λe^{i\vphi}, λe^{-i\vphi}, γ)$, where $0<λ<1<|γ|$ and $|λ^2γ|=1$. We show that in a three-parameter family, $g_{\eps}$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\eps =0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional Hénon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attra…
There is a long tradition of studying chaotic trajectories in systems whose integrability is broken by means of an external perturbation. Here we explore a different route to chaos, in the dynamics of extended bodies, which arises due to finite-size corrections to the otherwise integrable motion of a test particle. We find that cyclic changes in the overall shape of the body may lead to the onset of chaos. This is applied to the Duffing and Yukawa potentials. For Kepler's potential, periodic deviations from spherical symmetry give rise to chaotic regions around the unperturbed parabolic orbit.
Special subsets of orbits in chaotic systems, e.g. periodic orbits, heteroclinic orbits, closed orbits, can be considered as skeletons or scaffolds upon which the full dynamics of the system is built. In particular, as demonstrated in previous publications [Phys. Rev. E 95, 062224 (2017), Phys. Rev. E 97, 022216 (2018)], the determination of homoclinic orbits is sufficient for the exact calculation of classical action functions of unstable periodic orbits, which have potential applications in semiclassical trace formulas. Here this previous work is generalized to the calculation of classical a…
We present a case study elaborating on the multiplicity and self-similarity of homoclinic and heteroclinic bifurcation structures in the 2D and 3D parameter spaces of a nonlinear laser model with a Lorenz-like chaotic attractor. In a symbiotic approach combining the traditional parameter continuation methods using MatCont and a newly developed technique called the Deterministic Chaos Prospector (DCP) utilizing symbolic dynamics on fast parallel computing hardware with graphics processing units (GPUs), we exhibit how specific codimension-two bifurcations originate and pattern regions of chaotic…
We numerically investigate the dynamics of a closed chain of unidirectionally coupled oscillators in a regime of homoclinic chaos. The emerging synchronization regimes show analogies with the experimental behavior of a single chaotic laser subjected to a delayed feedback
We predict theoretically and verify experimentally the suppression of chaos in the Lorenz system driven by a high-frequency periodic or stochastic parametric force. We derive the theoretical criteria for chaos suppression and verify that they are in a good agreement with the results of numerical simulations and the experimental data obtained for an analog electronic circuit.
Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for $N>4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also …
We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward…
We study a third-order nonlinear ordinary differential equation whose solutions, under certain specific conditions, are individual pulses. These correspond to homoclinic orbits in the phase space of the equation and we study the possible pulse types in some detail. Sufficiently close to the conditions under which a homoclinic orbit exists, the solutions take the form of trains of well-separated pulses. A measure of closeness to homoclinic conditions provides a small parameter for the development of an asymptotic solution consisting of superposed, isolated pulses. The solvability condition in t…
Extreme-Mass-Ratio Inspirals Embedded in Dark Matter Halo: Existence of Homoclinic Orbit and Horizon-Induced Chaos
University of Science and Technology of China（中国科学技术大学）；CAS Key Laboratory for Researches in Galaxies and Cosmology, School of Astronomy and Space Science, University of Science and Technology of China（中国科学院星系与宇宙学重点实验室，中国科学技术大学天文与空间科学学院）；Shanghai University（上海大学）；Sogang University（西江大学）；Center for Quantum Spacetime, Sogang University（西江大学量子时空中心）
In pipes and channels, the onset of turbulence is initially dominated by localized transients, which lead to sustained turbulence through their collective dynamics. In the present work, we study the localized turbulence in pipe flow numerically and elucidate a state space structure that gives rise to transient chaos. Starting from the basin boundary separating laminar and turbulent flow, we identify transverse homoclinic orbits, the presence of which necessitates a homoclinic tangle and chaos. A direct consequence of the homoclinic tangle is the fractal nature of the laminar-turbulent boundary…
In a 2D conservative Hamiltonian system there is a formal integral $Φ$ besides the energy H. This is not convergent near a stable periodic orbit, but it is convergent near an unstable periodic orbit. We explain this difference and we find the convergence radius along the asymptotic curves. In simple mappings this radius is infinite. This allows the theoretical calculation of the asymptotic curves and their intersections at homoclinic points. However in more complex mappings and in Hamiltonian systems the radius of convergence is in general finite and does not allow the theoretical calculation …
Small-sized systems exhibit a finite number of routes to chaos. However, in extended systems, not all routes to complex spatiotemporal behavior have been fully explored. Starting from the sine-Gordon model of parametrically driven chain of damped nonlinear oscillators, we investigate a route to spatiotemporal chaos emerging from standing waves. The route from the stationary to the chaotic state proceeds through quasiperiodic dynamics. The standing wave undergoes the onset of oscillatory instability, which subsequently exhibits a different critical frequency, from which the complexity originate…
We study bifurcations associated with stability of the lowest stationary point (SP) of a damped parametrically forced pendulum by varying $ω_0$ (the natural frequency of the pendulum) and $A$ (the amplitude of the external driving force). As $A$ is increased, the SP will restabilize after its instability, destabilize again, and so {\it ad infinitum} for any given $ω_0$. Its destabilizations (restabilizations) occur via alternating supercritical (subcritical) period-doubling bifurcations (PDB's) and pitchfork bifurcations, except the first destabilization at which a supercritical or subcritical…
In this paper a Lorenz-like system, describing the process of rotating fluid convection, is considered. The present work demonstrates numerically that this system, also like the classical Lorenz system, possesses a homoclinic trajectory and a chaotic self-excited attractor. However, for considered system, unlike the classical Lorenz one, along with self-excited attractor a hidden attractor can be localized. Analytical-numerical localization of hidden attractor is presented.
We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Henon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable…
We present results of numerical simulations of coupled Ginzburg-Landau equations that describe parametrically excited waves. In one dimension we focus on a new regime in which the Eckhaus sideband instability does not lead to an overall change in the wavelength via the occurrence of a single phase slip but instead leads to ``double phase slips''. They are characterized by the phase slips occurring in sequential pairs, with the second phase slip quickly following and negating the first. The resulting dynamics range from transient excursions from a fixed point resembling those seen in excitable …
An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films has allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via gluing bifurcations in optically excited nematic liquid crystals previously proposed by G. Demeter and L. Kramer. The model of these last authors, proposed about twenty years before, does not have the central symmetry which one would expect for minimal dimensional models for…
We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of the Duffing system with nonlinear fractional damping and external excitation. Using perturbation methods we have found a critical forcing amplitude above which the system may behave chaotically. The results have been verified by numerical simulations using standard nonlinear tools as Poincare maps and a Lyapunov exponent. Above the critical Melnikov amplitude $μ_c$, which is the sufficient condition of a global homoclinic bifurcation, we have observed the region with a transie…
The study of nonlinear dynamics or chaos theory has emerged in the last three decades or so as an important interdisciplinary area of research encompassing a wide range of fields like: fluids, plasmas, biomedical sciences, finance, turbulence, astronomy, material sciences, etc. In plasma chaos was first experimentally observed by Boswell. Different other nonlinear dynamics related phenomena like, the intermittency route to a chaos, Homoclinic chaos, Period adding route to chaos and period subtracting, mode locking, period pulling, etc., had been observed by several researchers. In this thesis,…
It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Ha…
Spatio-temporal chaos in parametrically driven waves is investigated in one and two dimensions using numerical simulations of Ginzburg-Landau equations. A regime is identified in which in one dimension the dynamics are due to double phase slips. In very small systems they are found to arise through a Hopf bifurcation off a mixed mode. In large systems they can lead to a state of localized spatio-temporal chaos, which can be understood within the framework of phase dynamics. In two dimensions the double phase slips are replaced by bound defect pairs. Our simulations indicate the possibility of …
A manifestly relativistically covariant form of the van der Pol oscillator in 1+1 dimensions is studied. We show that the driven relativistic equations, for which $x$ and $t$ are coupled, relax very quickly to a pair of identical decoupled equations, due to a rapid vanishing of the ``angular momentum'' (the boost in 1+1 dimensions). A similar effect occurs in the damped driven covariant Duffing oscillator previously treated. This effect is an example of entrainment, or synchronization (phase locking), of coupled chaotic systems. The Lyapunov exponents are calculated using the very efficient me…
Chaotic dynamics can be effectively studied by continuation from an anti-integrable limit. We use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations. We find a bound on the parameter range for which the Henon map exhibits a complete binary horseshoe as well as a subshift of finite type. We classify homoclinic bifurcations, and study those for the area preserving case in detail. Simple forcing relations between homoclinic orbits are established. We show that a symmetry of the map gives rise to constraints on certain sequences of homocli…
We study in this paper the behavior of a periodically driven nonlinear mechanical system. Bifurcation diagrams are found which locate regions of quasiperiodic, periodic and chaotic behavior within the parameter space of the system. We also conduct a symbolic analysis of the model, which demonstrates that the symbolic dynamics of two-dimensional maps can be applied effectively to the study of ordinary differential equations in order to gain global knowledge about them.
We examine the Mielnikov criterion for transition to chaos in case of one degree of freedom nonlinear oscillator with non symmetric potential. This system subjected to an external periodic force shows homoclinic transition from regular vibrations to chaos just before escape from the potential well. Especially we study the effect of a second resonant excitation with different phase on the system transition to chaos and propose a way of its control.
We examine the Melnikov criterion for a global homoclinic bifurcation and a possible transition to chaos in case of a single degree of freedom nonlinear oscillator with a symmetric double well nonlinear potential. The system was subjected simultaneously to parametric periodic forcing and self excitation via negative damping term. Detailed numerical studies confirm the analytical predictions and show that transitions from regular to chaotic types of motion are often associated with increasing the energy of an oscillator and its escape from a single well.
In this work, inspired in the symbolic dynamic of chaotic systems and using machine learning techniques, a control strategy for complex systems is designed. Unlike the usual methodologies based on modeling, where the control signal is obtained from an approximation of the dynamic rule, here the strategy rest upon an approach of a function, that from the current state of the system, give the necessary perturbation to bring the system closer to a homoclinic orbit that naturally goes to the target. The proposed methodology is data-driven or can be developed in a based-model context and is illustr…
A long wavelength optical lattice is generated in a two-level medium by low-frequency contrapropagating beams. Then a short wave length gap soliton generated by evanescent boundary instability (supratransmission) undergoes a dynamics shown to obey the Newton equation of the parametrically driven pendulum, hence presenting extremely rich, possibly chaotic, dynamical behavior. The theory is sustained by numerical simulations and provides an efficient tool to study soliton trajectories.
We study period doublings in $N$ $(N=2,3,4, \dots)$ coupled parametrically forced damped pendulums by varying $A$ (the amplitude of the external driving force) and $c$ (the strength of coupling). With increasing $A$, the stationary point undergoes multiple period-doubling transitions to chaos. We first investigate the two-coupled case with $N=2$. For each period-doubling transition to chaos, the critical set consists of an infinity of critical line segments and the zero-coupling critical point lying on the line $A=A^*_i$ in the $A-c$ plane, where $A^*_i$ is the $i$th transition point for the u…
Intrinsically coupled nonlinear systems present different oscillating components that exchange energy among themselves. A paradigmatic example is the spring pendulum, which displays spring, pendulum, and coupled oscillations. We analyze the energy exchanges among the oscillations, and obtain that it is enhanced for chaotic orbits. Moreover, the highest rates of energy exchange for the coupling occur along the homoclinic tangle of the primary hyperbolic point embedded in a chaotic sea. The results show a clear relation between internal energy exchanges and the dynamics of a coupled system.
Chains of parametrically driven, damped pendula are known to support soliton-like clusters of in-phase motion which become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the pinning of the soliton on a "long" impurity (a longer pendulum) expands dramatically its stability region whereas "short" defects simply repel solitons producing effective partition of the chain. We also show that defects may spontaneously nucleate solitons.
We investigate the semiclassical mechanism of tunneling process in non-integrable systems. The significant role of complex-phase-space chaos in the description of the tunneling process is elucidated by studying a simple scattering map model. Behaviors of tunneling orbits are encoded into symbolic sequences based on the structure of complex homoclinic tanglement. By means of the symbolic coding, the phase space itineraries of tunneling orbits are related with the amounts of imaginary parts of actions gained by the orbits, so that the systematic search of significant tunneling orbits becomes pos…
We show that the classical dynamics of independent particles can determine the quantum properties of interacting electrons in the ballistic regime. This connection is established using diagrammatic perturbation theory and semiclassical finite-temperature Green functions. Specifically, the orbital magnetism is greatly enhanced over the Landau susceptibility by the combined effects of interactions and finite size. The presence of families of periodic orbits in regular systems makes their susceptibility parametrically larger than that of chaotic systems, a difference which emerges from correlatio…
Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all …
Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and Maslov indices. For chaotic systems, the relative actions of such orbits can be expressed in terms of phase space areas bounded by segments of stable and unstable manifolds, and Moser invariant curves. This also generates direct relations between periodic orbits and homoclinic (heteroclinic) orbit actions. Simpler, explicit approximate expressions following f…
