Partial synchronization in coupled chemical chaotic oscillators

In this paper we investigate the problem of partial synchronization in diffusively coupled chemical chaotic oscillators with zero-flux boundary conditions. The dynamical properties of the chemical system which oscillates with Uniform Phase evolution, yet has Chaotic Amplitudes (UPCA) are first discussed. By combining numerical and analytical methods, the impossibility of full global synchronization in a network of two or three coupled chemical oscillators is discovered. Mathematically, stable partial synchronization corresponds to convergence to a linear invariant manifold of the global state space. The sufficient conditions for exponential stability of the invariant manifold in a network of three coupled chemical oscillators are obtained via the nonlinear contraction principle.     

Synchronization of two coupled multimode oscillators with time-delayed feedback

Effects of synchronization in a system of two coupled oscillators with time-delayed feedback are investigated. Phase space of a system with time delay is infinite-dimensional. Thus, the picture of synchronization in such systems acquires many new features not inherent to finite-dimensional ones. A picture of oscillation modes in cases of identical and non-identical coupled oscillators is studied in detail. Periodical structure of amplitude death and “broadband synchronization” zones is investigated. Such a behavior occurs due to the resonances between different modes of the infinite-dimensional system with time delay.     

Predictability portraits for chaotic motions

The reliability of forecasts for chaotic motions varies with the state of the dynamical system. We define quantities that measure predictability and investigate their dependence on the initial state for different forecasting times. Two model systems are investigated, a driven damped pendulum and the Lorenz system. We use two different numerical methods to analyse the effect of finite resolution in determining the initial conditions on the reliability of forecasts. For the pendulum we also compare numerical forecasts with experimental data.     

Synchronization of time-delay coupled pulse oscillators

We present a detailed study of the dynamics of pulse oscillators with time-delayed coupling. We get the return maps, obtain strict solutions and analyze their stability. For the case of two oscillators, a periodical structure of synchronization regions is found in parameter space, and the regions corresponding to in-phase and antiphase regimes alternate with growth of time delay. Two types of switching between in-phase and antiphase regimes are studied. We also show that for different parameters coupling delay may have synchronizing or desynchronizing effect. Another novel result is that phase locked regimes exist for arbitrary large values. The specificity of system dynamics with large delay is studied.     

Ragged oscillation death in coupled nonidentical oscillators

In this paper, the effect of spatial frequencies distributions on the oscillation death in a ring of coupled nonidentical oscillators is studied. We find that the rearrangement of the spatial frequencies may deform the domain of oscillation death and give rise to a ragged oscillation death in some parameter spaces. The usual critical curves with shape V in the parameter space of frequency-mismatch vs coupling-strength may become the shape W (or even shape WV). This phenomenon has been not only numerically observed in coupled nonidentical nonlinear systems, but also well supported by our theoretical analysis.     

Weakly coupled oscillators and topological degree

By using the topological degree of Brouwer for mappings along with averaging method, we study the existence of forced periodic solutions for certain weakly coupled periodically perturbed ordinary differential equations.     

EEG simulation by 2D interconnected chaotic oscillators

An artificial neuronal network composed by 2D interconnected chaotic oscillators is explored for brain waves (EEG) simulation. For the inverse problem solution a parallel real-coded genetic algorithm (PRCGA) is proposed. In order to conduct thorough comparison between the simulated and target signal characteristics, a spectrum analysis of the signals is undertaken. A good matching between the theoretical and experimental EEG signals has been achieved. Numerical results of calculations are presented and discussed.     

Spatial and temporal patterns in coupled Belousov-Zhabotinsky oscillators

The coupling of two identical B-Z oscillators is examined. The coupling, unlike previous work, is only via the cerium component. Formation of spatial and temporal inhomogeneities is found. Transitions from the spatially homogeneous steady state to the inhomogeneous one, and then to the spatially homogeneous one and inhomogeneous oscillations are reported as a function of the coupling rate and the stoichiometric factor. The feasibility of experimental verification of these results and the biological and ecological implications are discussed.     

Asymmetric type II periodic motions for nonlinear impact oscillators

In this paper a general class of nonlinear impact oscillators is considered for Type II periodic motions. This system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation. The unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law. By approximating the Poincaré map to O(ε)$O\left(\epsilon \right)$ directly, a general method of Melnikov type for detecting the existence of asymmetric Type II subharmonic orbits outside the homoclinic cycles is presented.     

Geometrical properties of coupled oscillators at synchronization

We study the synchronization of N nearest neighbors coupled oscillators in a ring. We derive an analytic form for the phase difference among neighboring oscillators which shows the dependency on the periodic boundary conditions. At synchronization, we find two distinct quantities which characterize four of the oscillators, two pairs of nearest neighbors, which are at the border of the clusters before total synchronization occurs. These oscillators are responsible for the saddle node bifurcation, of which only two of them have a phase-lock of phase difference equals ± π/2. Using these properties we build a technique based on geometric properties and numerical observations to arrive to an exact analytic expression for the coupling strength at full synchronization and determine the two oscillators that have a phase-lock condition of ± π/2.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

Synchronization of coupled chaotic FitzHugh-Nagumo systems

This paper addresses dynamic synchronization of two FitzHugh-Nagumo (FHN) systems coupled with gap junctions. All the states of the coupled chaotic system, treating either as single-input or two-input control system, are synchronized by stabilizing their error dynamics, using simplest and locally robust control laws. The local asymptotic stability, chosen by utilizing the local Lipschitz nonlinear property of the model to address additionally the non-failure of the achieved synchronization, is ensured by formulating the matrix inequalities on the basis of Lyapunov stability theory. In the presence of disturbances, it ensures the local uniform ultimate boundedness. Furthermore, the robustness of the proposed methods is ensured against bounded disturbances besides providing the upper bound on disturbances. To the best of our knowledge, this is the computationally simplest solution for synchronization of coupled FHN modeled systems along with unique advantages of less conservative local asymptotic stability of synchronization errors with robustness. Numerical simulations are carried out to successfully validate the proposed control strategies.     

Autowave processes in continual chains of unidirectionally coupled oscillators

We introduce a mathematical model of a continual circular chain of unidirectionally coupled oscillators. It is a nonlinear hyperbolic boundary value problem obtained from a circular chain of unidirectionally coupled ordinary differential equations in the limit as the number of equations indefinitely increases. We study the attractors of this boundary value problem. Combining analytic and numerical methods, we establish that one of the following two alternatives takes place in this problem: either the buffer phenomenon (unbounded accumulation of stable periodic motions) or chaotic attractors of arbitrarily high Lyapunov dimensions.     

On resonances in systems of two weakly coupled oscillators

Assuming that two weakly coupled oscillators are essentially nonlinear we construct the most suitable form of a shortened 3-dimensional system which describes behavior of solutions inside non-degenerate resonance zones. We analyze a model system of that kind and establish the existence of limit cycles of different types and also the existence of nonregular attractors which are explained by the existence of saddle-focus loops.      

Symmetry and phaselocking in chains of weakly coupled oscillators

Weakly coupled chains of oscillators with nearest-neighbor interactions are analyzed for phaselocked solutions. It is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound. It is also shown that qualitative behavior of these solutions depends on whether the coupling is “diffusive” or “I synaptic”. terms defined in the paper. The methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two-point (continuum) boundary value problem that is easier to analyze; the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law. An application to the neurophysiology of motor behavior is discussed briefly.     

Universal occurrence of mixed-synchronization in counter-rotating nonlinear coupled oscillators

By coupling counter-rotating coupled nonlinear oscillators, we observe a “mixed” synchronization between the different dynamical variables of the same system. The phenomenon of amplitude death is also observed. Results for coupled systems with co-rotating coupled oscillators are also presented for a detailed comparison. Results for Landau–Stuart and Rössler oscillators are presented.     

The dynamics ofn weakly coupled identical oscillators

Summary We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.     

A variational approach to chaotic dynamics in periodically forced nonlinear oscillators

We prove that a class of problems containing the classical periodically forced pendulum equation displays the main features of chaotic dynamics. The approach is based on the construction of multibump type heteroclinic solutions to periodic orbits by the use of global variational methods.