Geometry of a chaotic layer

An analysis is made of the dependence of the geometric shape of the chaotic layer near the separatrix of a nonlinear resonance of a Hamiltonian system on the parameters of this system. A separatrix algorithmic mapping, which describes the motion near the separatrix in the presence of an asymmetric perturbation having an arbitrary degree of asymmetry. The separatrix algorithmic mapping is an algorithm containing conditional transfer instructions, is considered. An analytic procedure is derived to reduce the separatrix algorithmic mapping to the unified surface of the cross section of the initial Hamiltonian system (mapping synchronization procedure). It is observed that in the case of the high-frequency perturbation λ → +∞ (where λ is the ratio of the perturbation frequency to the frequency of small phase oscillations at resonance), the chaotic layer is subjected to strong bending in the sense that during motion near the separatrix theamplitude of the energy deviations relative to the unperturbed separatrix value is much larger than the layer width. However, the synchronized separatrix algorithmic mapping ensures an accurate representation of the phase portrait of the layer for both low and high values of the parameter λ provided that the amplitude of the perturbation is fairly small. This is demonstrated by comparing the phase portraits obtained using the synchronized separatrix algorithmic mapping with the results of direct numerical integrations of the initial Hamiltonian system.     

Drastic facilitation of the onset of global chaos

We show that the onset of global chaos in a time periodically perturbed Hamiltonian system may occur at unusually small magnitudes of perturbation if the unperturbed system possesses more than one separatrix. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with separatrices of the unperturbed system. We develop the asymptotic theory and verify it in simulations.     

Chaos healing by separatrix disappearance and quasisingle helicity states of the reversed field pinch

The resilience to chaotic perturbations of one-parameter one-degree-of-freedom Hamiltonian dynamics is shown to increase when its corresponding separatrix vanishes due to a saddle-node bifurcation. This is first highlighted for the magnetic chaos related to quasisingle helicity (QSH) states of the reversed field pinch. It provides a rationale for the confinement improvement of helical structures experimentally found for QSH plasmas; such a feature would not be expected from the classical resonance overlap picture as the separatrix disappearance occurs when the amplitude of the dominant mode increases.     

Dynamical chaos for a limited power supply for fluid oscillations in cylindrical tanks

Forced oscillations of the fluid surface in a cylindrical tank due to interaction with the excitation mechanism of a limited power supply (so-called “limited excitation” phenomena) are investigated in detail. On the basis of analysis of the largest Lyapunov exponents for a complex system—a tank with fluid and an excitation arrangement—the three types of steady-state regimes are found: equilibrium positions, periodic and chaotic regimes. Phase portraits, Poincaré sections and maps, distributions of spectral densities and invariant measures are constructed and thoroughly studied. Attention is concentrated mainly on the properties of chaotic attractors and schemes of transition from “order” to chaos. It is established that different scenarios of transition to chaos and various structures of chaotic attractors are possible in the same physical system. The new scenario transition to chaos which generalizes scenario of Pomeau-Manneville is revealed. It is shown that chaotic regimes with the single-mode fluid free surface oscillations can originate only due to interaction with the excitation mechanism.     

Chaotic layer of a pendulum under low-and medium-frequency perturbations

The amplitude of the separatrix map and the size of a pendulum chaotic layer are studied numerically and analytically as functions of the adiabaticity parameter at low and medium perturbation frequencies. Good agreement between the theory and numerical experiment is found at low frequencies. In the medium-frequency range, the efficiency of using resonance invariants of separatrix mapping is high. Taken together with the known high-frequency asymptotics, the results obtained in this work reconstruct the chaotic layer pattern throughout the perturbation frequency range.     

Chaotic transport and damping from θ-ruffled separatrices

Variations in magnetic or electrostatic confinement fields give rise to trapping separatrices, and neoclassical transport theory analyzes effects from collision-induced separatrix crossings. Experiments on pure electron plasmas now quantitatively characterize a broad range of transport and wave damping effects due to "chaotic" separatrix crossings, which occur due to equilibrium plasma rotation across θ-ruffled separatrices, and due to wave-induced separatrix fluctuations.     

From flow to map in an experimental high-dimensional electro-optic nonlinear delay oscillator

An optoelectronic nonlinear delay oscillator seeded by a pulsed laser source is used to experimentally demonstrate a new transition scenario for the general class of delay differential dynamics, from continuous to discrete time behavior. This transition scenario differs from the singular limit map, or adiabatic approximation model that is usually considered. The transition from the map to the flow is observed when increasing the pulse repetition rate. The mechanism of this transition opens the way to new interpretations of the general properties of delay differential dynamics, which are universal features of many other scientific domains. We anticipate that the nonlinear delay oscillator architecture presented here will have significant applications in chaotic communication systems.     

Chaotically spiking canards in an excitable system with 2D inertial fast manifolds

We introduce a new class of excitable systems with two-dimensional fast dynamics that includes inertia. A novel transition from excitability to relaxation oscillations is discovered where the usual Hopf bifurcation is followed by a cascade of period doubled and chaotic small excitable attractors and, as they grow, by a new type of canard explosion where a small chaotic background erratically but deterministically triggers excitable spikes. This scenario is also found in a model for a nonlinear Fabry-Perot cavity with one pendular mirror.     

The chaotic layer of a nonlinear resonance under low-frequency perturbation

Conditions whereby the chaotic layer of a nonlinear resonance is described in terms of low-frequency separatrix mapping are discussed. In this case, the accurate estimation of the size of the layer requires the arrangement of resonances at its edge to be known. The resonance picture is constructed using the separatrix mapping invariants of the first three orders. The variation of the layer size with the mapping amplitude is traced with the criterion for resonance overlapping. Results obtained by direct calculation and by invariants analysis are compared. Issues that remain to be solved are noted.     

A Dynamical Phase Transition of Binary Species BECs Mixtures in a Double Well Potential

We investigate the critical behavior in tunneling dynamics of a binary mixture of Bose-Einstein condensates (BECs) trapped in a symmetric double well potential. By gradually increasing the interspecies interaction, we characterize a continuous dynamical phase transition behavior which involves power law scaling. This dynamical phase transition is a consequence of separatrix crossing as we revealed by poincaré section analysis.     

Dynamics of domain walls in pattern formation with traveling-wave forcing

We study dynamics of domain walls in pattern forming systems that are externally forced by a moving space-periodic modulation close to 2:1 spatial resonance. The motion of the forcing induces nongradient dynamics, while the wave number mismatch breaks explicitly the chiral symmetry of the domain walls. The combination of both effects yields an imperfect nonequilibrium Ising-Bloch bifurcation, where all kinks (including the Ising-like one) drift. Kink velocities and interactions are studied within the generic amplitude equation. For nonzero mismatch, a transition to traveling bound kink-antikink pairs and chaotic wave trains occurs.     

Chaos quasisynchronization induced by impulses with parameter mismatches

This paper studies the effect of parameter mismatch on the impulsive synchronization of a class of coupled chaotic systems. A new definition for global quasisynchronization is introduced and used to analyze the synchronous behavior of coupled chaotic systems in the presence of parameter mismatch. Using the linear decomposition and comparison-system methods, a global synchronization error bound together with a sufficient condition is derived. Numerical simulations on the chaotic Chua's circuit are presented to verify the theoretical results.     

Dynamics of ensemble of inhibitory coupled Rulkov maps

The motif of three inhibitory coupled Rulkov elements is studied. Possible dynamical regimes, including different types of sequential activity, winner-take-all activity and chaotic activity, are in the focus of this paper. In particular, a new transition scenario from sequential activity to winner-take-all activity through chaos is uncovered. This study can be used in high performance computation of large neuron-like ensembles for the modeling of neuron-like activity.     

Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.     

Contribution from the secondary harmonics of a disturbance to the separatrix map of the Hamiltonian system

The special role of low-frequency secondary harmonics with frequencies that are sums of and differences between primary frequencies entering into the Hamiltonian in explicit form has been already discussed in the literature. These harmonics are of the second order of smallness and constitute a minor fraction of the disturbance. Nevertheless, under certain conditions, their contribution to the amplitude of the separatrix map of the system may be several orders of magnitude higher than the contributions from primary harmonics and, thereby, govern the formation of dynamic chaos. This work generalizes currently available theoretical and numerical data on this issue. The role of secondary harmonics is demonstrated with a pendulum the disturbance of which in the Hamiltonian is represented by two asymmetric closely spaced high-frequency harmonics. An analytical expression for the contribution of the secondary harmonics to the separatrix map amplitude for this system is derived, and the range of very low secondary frequencies not studied earlier is considered using this equation. The domains where the separatrix map amplitude linearly grows with frequency and the chaotic layer size is frequency-independent are indicated. Theoretical predictions are compared with numerical data.     

Time delay induced partial death patterns with conjugate coupling in relay oscillators

We investigate the dynamics of delay-coupled relay oscillators with conjugate (or dissimilar) coupling and find the partial death with the phase-flip transition. This phenomenon is quite general and occurs for the limit cycle as well as chaotic relay oscillators. In the regime of partial death, parts of the system oscillate with large amplitude, while other element stays at rest. Using the Stuart-Landau and Rössler oscillators, we demonstrate that partial amplitude death is a robust dynamical state in coupled oscillators. We also studied the mismatch delay and find different types of dynamical pattern with partial death.     

分子高激发振动态的动力学特性研究

运用经典哈密顿代数方法,结合单摆的运动特点表示两个化学键之间的振动耦合.对水分子高激发态下两个氧氢键(O—H)伸缩振动动力学的研究结果表明,靠近分界线的中间能级的相空间中较易出现混沌轨道,而较高或较低能级的相空间中则具有比较规则的周期运动 关键词： 高激发振动 共振 混沌     

Limiting phase trajectories and dynamic transitions in nonlinear periodic systems

The intensity of energy exchange between parts of periodic nonlinear Frenkel-Kontorova and Klein-Gordon lattices is analyzed based on a concept of limiting phase trajectories introduced earlier. It is demonstrated that, with increasing nonlinearity parameter in these lattices, two dynamic transitions take place successively. The first transition is due to the bifurcation of the lower (with respect to frequency) normal mode because of its instability. It is accompanied by the occurrence of two additional normal modes and the separatrix between them. In this case, after this transition and before it, complete energy exchange between parts of the system is possible. The second transition takes place as a result of merging of the limiting phase trajectory with the separatrix, after which complete energy exchange between parts of the system is impossible. Analytical results are proven by numerical data.     

The Feigenbaum—Sharkovskii—Magnitskii scenario of transition to chaos in a circuit with a tunnel diode

The scenarios of transition to a chaotic state in a circuit with a tunnel diode are experimentally studied. It is shown that there are regions on the plane of control parameters where the Feigenbaum scenario and the Sharkovskii order alternate and chaos is reached through intermittency.     

Suppression of chaos in the vicinity of a separatrix

The standard Melnikov method for analyzing the onset of chaos in the vicinity of a separatrix is used to explore the possibility of suppressing chaos of dynamical systems of a certain class. Analytical expressions are obtained for external perturbations that eliminate chaotic behavior. These results are supplemented with a numerical analysis of the Duffing-Holmes-oscillator and pendulum equations.