Numerical test for hyperbolicity of chaotic dynamics in time-delay systems

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.Comment: 7 pages, 5 figure

Dynamics of a Limit Cycle Oscillator under Time Delayed Linear and Nonlinear Feedbacks

We study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single Hopf bifurcation oscillator. Our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips, frequency suppression, multiple periodic states and chaos. Such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators. Our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator.We construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms. We find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like birhythmicity, phase reversals, radial trapping, phase jumps and spiraling patterns in the amplitude space. Our results may find useful applications in physical, chemical or biological systems.Comment: VERSION 4: Fig. 10(d) added, an uncited reference removed; (To appear in Physica D) (17 pages, 21 figures, two column, aps RevTeX); VERSION 3: Revised. In Section 2, small tau approximation added; Section 3 is divided into subsections; periodic solution discussed in detail; Figs. 7 and 11 discarded; Figs. 12 and 14 altered; three new figures (now Figs. 10, 11 and 21) added. VERSION 2: Figs. 1 and 2 replace

Synchronization of Coupled Boolean Phase Oscillators

We design, characterize, and couple Boolean phase oscillators that include state-dependent feedback delay. The state-dependent delay allows us to realize an adjustable coupling strength, even though only Boolean signals are exchanged. Specifically, increasing the coupling strength via the range of state-dependent delay leads to larger locking ranges in uni- and bi-directional coupling of oscillators in both experiment and numerical simulation with a piecewise switching model. In the unidirectional coupling scheme, we unveil asymmetric triangular-shaped locking regions (Arnold tongues) that appear at multiples of the natural frequency of the oscillators. This extends observations of a single locking region reported in previous studies. In the bidirectional coupling scheme, we map out a symmetric locking region in the parameter space of frequency detuning and coupling strength. Because of large scalability of our setup, our observations constitute a first step towards realizing large-scale networks of coupled oscillators to address fundamental questions on the dynamical properties of networks in a new experimental setting.Comment: 8 pages, 8 figure

Autonomous Learning by Simple Dynamical Systems with Delayed Feedbacks

A general scheme for construction of dynamical systems able to learn generation of the desired kinds of dynamics through adjustment of their internal structure is proposed. The scheme involves intrinsic time-delayed feedback to steer the dynamics towards the target performance. As an example, a system of coupled phase oscillators, which can by changing the weights of connections between its elements evolve to a dynamical state with the prescribed (low or high) synchronization level, is considered and investigated

Dynamical systems with time-dependent coupling: Clustering and critical behaviour

We study the collective behaviour of an ensemble of coupled motile elements whose interactions depend on time and are alternatively attractive or repulsive. The evolution of interactions is driven by individual internal variables with autonomous dynamics. The system exhibits different dynamical regimes, with various forms of collective organization, controlled by the range of interactions and the dispersion of time scales in the evolution of the internal variables. In the limit of large interaction ranges, it reduces to an ensemble of coupled identical phase oscillators and, to some extent, admits to be treated analytically. We find and characterize a transition between ordered and disordered states, mediated by a regime of dynamical clustering.Comment: to appear in Physica

Synchronization problems for unidirectional feedback coupled nonlinear systems

In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.Comment: To appear in Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Ana