Periodic orbits,basins of attraction and chaotic beats in two coupled Kerr oscillators

Kerr oscillators are model systems which have practical applications in nonlinear optics. Optical Kerr effect, i.e., interaction of optical waves with nonlinear medium with polarizability χ ⁽³⁾ is the basic phenomenon needed to explain, for example, the process of light transmission in fibers and optical couplers. In this paper, we analyze the two Kerr oscillators coupler and we show that there is a possibility to control the dynamics of this system, especially by switching its dynamics from periodic to chaotic motion and vice versa. Moreover, the switching between two different stable periodic states is investigated. The stability of the system is described by the so-called maps of Lyapunov exponents in parametric spaces. Comparison of basins of attractions between two Kerr couplers and a single Kerr system is also presented.     

Stable amplitude chimera states and chimera death in repulsively coupled chaotic oscillators

Amplitude chimera states, representing a spontaneous symmetry breaking of a population of coupled identical oscillators into two distinct clusters with one oscillating in spatial coherent amplitude, while the other displaying oscillations in a spatially incoherent manner, have been observed as a kind of transient dynamics in the process of transition to the in-phase synchronization in coupled limit-cycle oscillators. Here, we obtain a kind of stable amplitude chimera state in the chaotic regime of a system of repulsively coupled Lorenz oscillators. With the increment of the coupling strength, the coupled oscillators transit from spatiotemporal chaos to amplitude chimera states then to coherent oscillation death or chimera death states. Moreover, the number of clusters in amplitude chimera patterns has a power-law dependence on the number of coupled neighbors. The amplitude chimera and the chimera death states coexist at certain coupling strength. Moreover, the amplitude chimera and the amplitude death patterns are related to the initial condition for given coupling strength. Our findings of amplitude chimera states and chimera death states in coupled chaotic system may enrich the knowledge of the symmetry-breaking-induced pattern formation.     

Asymptotically stable periodic orbits of a coupled electromechanical system

In this paper an electromechanical system is analyzed. The existence and asymptotic stability of a periodic orbit are obtained in a mathematically rigorous way as well as an expansion of the period by using an adequate small parameter. For the analytical results the main tool used is the regular perturbation theory. Some results, such as the growing of the period according to some powers of the parameters and the relation 2:1 between the period of the cart, which is a part of the electromechanical system, and the period of the current, are compatible with earlier numerical findings.     

Stabilizing torsion-free periodic orbits using method of harmonic oscillators

Nonlinear Dynamics - An odd number of real Floquet multipliers greater than unity prevents the classical time delayed feedback control from stabilizing torsion-free orbits of nonautonomous systems....     

A pair of van der Pol oscillators coupled by fractional derivatives

We consider the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators: 1 2 where D ^?? x is the order ?? derivative of x(t), and 0<??<1. We use a two-variable perturbation method on the system??s corresponding variational equations to derive expressions for the transition curves separating regions of stability from instability in the ??, ?? parameter plane. The perturbation results are validated with numerics and through direct comparison with known results in the limiting cases of ??=0 and ??=1, where the fractional coupling reduces to position coupling and velocity coupling, respectively.     

Modeling a synthetic biological chaotic system: relaxation oscillators coupled by quorum sensing

Chaos exists in biological systems. Through investigating synthetic genetic relaxation oscillators coupled by quorum sensing, this paper reports a chaotic system. The detailed dynamical behaviors of this chaotic biological system are investigated, including Lyapunov exponents spectrum, bifurcation, and Poincaré mapping.     

Simplified model and analysis of a pair of coupled thermo-optical MEMS oscillators

Nonlinear Dynamics - Motivated by the dynamics of microscale oscillators with thermo-optical feedback, a simplified third-order model capturing the key features of these oscillators is developed,...     

Bifurcations which destroy the out-of-phase mode in a pair of linearly coupled relaxation oscillators

This paper presents two mechanisms, the Hopf bifurcation and the infinite period bifurcation, which lead to non-existence of the out-of-phase mode of oscillation in a pair of identical van der Pol relaxation oscillators with diffusive coupling in the displacements and velocities. We show that the out-of-phase mode is associated with a periodic solution in a two dimensional invariant out-of-phase manifold. Phase plane techniques are used to find the region of parameter space where this periodic solution exists and how it is destroyed at the boundaries of this region.     

Collective dynamics induced by diversity taken from two-point distribution in globally coupled chaotic oscillators

The effect of parameter mismatch (diversity) taken from two-point distribution is studied numerically and theoretically in globally coupled Rössler chaotic systems. Two cases including mixed populations consisting of elements with different timescales and attractors are considered. In these two cases, the probability p of two-point distribution, which acts as an asymmetrical coupling on the system, plays a crucial role in determining the evolution of systems, and the rich dynamical phenomena are observed, especially for amplitude death (AD). The relationships between various dynamics are also discussed.     

Periodic,almost periodic and chaotic motions of forced self-sustained oscillators

Steady motions of the Van der Pol oscillator and an oscillator with hysteresis are studied numerically in this paper. Some features of periodic, almost periodic and chaotic motions of forced self-sustained oscillators are investigated. This paper has been presented at the ICTAM XVI Lyngby.     

Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller

In this paper we deal with the control of chaotic systems. Knowing that a chaotic attractor contains a myriad of unstable periodic orbits (UPO’s), the aim of our work is to stabilize some of the UPO’s embedded in the chaotic attractor and which have interesting characteristics. First, using the input-to-state linearization method in conjunction with a time-delayed state feedback, we design a control signal that can achieve stabilization. Next, an adaptive time-delayed state feedback is proposed which shows at once efficiency and simplicity and circumvents the construction complexity of the first controller. Finally, we propose a reduced order sliding mode observer to estimate the necessary states for the design of an adaptive time delayed state feedback controller. This last controller has one main advantage, it in fact achieves UPO stabilization without using the system model. The efficacy of the proposed methods is illustrated by numerical simulations onto Chua’s system.     

Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits

We study the nature of motion in a 3D potential composed of perturbed elliptic oscillators. Our technique is to use the results obtained from the 2D potential in order to find the initial conditions generating regular or chaotic orbits in the 3D potential. Both 2D and 3D potentials display exact periodic orbits together with extended chaotic regions. Numerical experiments suggest that the degree of chaos increases rapidly as the energy of the test particle increases. About 97?% of the phase plane of the 2D system is covered by chaotic orbits for large energies. The regular or chaotic character of the 2D orbits is checked using the S(c) dynamical spectrum, while for the 3D potential we use the S(c) spectrum, along with the P(f) spectral method. Comparison with other dynamical indicators shows that the S(c) spectrum gives fast and reliable information about the character of motion.     

Homoclinic orbits and an invariant chaotic set in a new 4D piecewise affine systems

Nonlinear Dynamics - It is a challenging task to prove mathematically the existence of homoclinic orbits in a high-dimensional dynamical system. Here we first introduce a new four-dimensional (4D)...     

Transition to escape in a system of coupled oscillators

We study a Hamiltonian system of coupled oscillators derived from two forced pendulums, connected with a torsional spring. The uncoupled limit is described by two identical oscillators, each possessing a homoclinic orbit separating bounded from unbounded motion. We focus on intermediate energy levels which lead to detained motions, defined as trajectories that, though unbounded as t → ∞, oscillate within the region defined by the homoclinic orbit of the unperturbed system for a long but finite time. We analyze the existence and behavior of these motions in terms of equipotential surfaces. These curves provide bounds on the motion of the system and are shown to be closed for low energies. However, above some critical energy level the equipotential curves become open. The detained trajectories are shown to arise from the region of phase space that was, for appropriate energies, stochastic. These motions remain within this region for long times before finally “leaking out” of the opening in the equipotential curves and proceeding to infinity.     

Homoclinic orbits in a shallow arch subjected to periodic excitation

Homoclinic orbits in a shallow arch subjected to periodic excitation are investigated in the presence of 1:1 internal resonance and external resonance. The method of multiple scales is used to obtain a set of near-integrable systems. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect equilibria in a resonance band of the slow manifold. These orbits arise from singular homoclinic orbits and are composed of alternating slow and fast pieces. The result obtained imply the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the class of shallow arch systems.     

A semi-numerical method for periodic orbits in a bisymmetrical potential

We use a semi-numerical method to find the position and period of periodic orbits in a bisymmetrical potential, made up of a two dimensional harmonic oscillator, with an additional term of a Plummer potential, in a number of resonant cases. The results are compared with the outcomes obtained by the numerical integration of the equations of motion and the agreement is good. This indicates that the semi-numerical method gives general and reliable results. Comparison with other methods of locating periodic orbits is also made.