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Register a new userDiscontinuous Attractor Dimension at the Synchronization Transition of Time-Delayed Chaotic Systems
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The attractor dimension at the transition to complete synchronization in a network of chaotic units with time-delayed couplings is investigated. In particular, we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents for iterated maps and for two coupled semiconductor lasers. We argue that the Kaplan-Yorke dimension must be discontinuous at the transition and compare it to the correlation dimension. For a system of Bernoulli maps we indeed find a jump in the correlation dimension. The magnitude of the discontinuity in the Kaplan-Yorke dimension is calculated for networks of Bernoulli units as a function of the network size. Furthermore the scaling of the Kaplan-Yorke dimension as well as of the Kolmogorov entropy with system size and time delay is investigated.
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Networks of nonlinear units with time-delayed couplings can synchronize to a common chaotic trajectory. Although the delay time may be very large, the units can synchronize completely without time shift. For networks of coupled Bernoulli maps, analytic results are derived for the stability of the chaotic synchronization manifold. For a single delay time, chaos synchronization is related to the spectral gap of the coupling matrix. For networks with multiple delay times, analytic results are obtained from the theory of polynomials. Finally, the analytic results are compared with networks of iterated tent maps and Lang-Kobayashi equations which imitate the behaviour of networks of semiconductor lasers.
Synchronization of chaotic units coupled by their time delayed variables are investigated analytically. A new type of cooperative behavior is found: sublattice synchronization. Although the units of one sublattice are not directly coupled to each other, they completely synchronize without time delay. The chaotic trajectories of different sublattices are only weakly correlated but not related by generalized synchronization. Nevertheless, the trajectory of one sublattice is predictable from the complete trajectory of the other one. The spectra of Lyapunov exponents are calculated analytically in the limit of infinite delay times, and phase diagrams are derived for different topologies.
A new method of virtual unknown parameter is proposed to synchronize two different systems with unknown parameters and disturbance in finite time. Virtual unknown parameters are introduced in order to avoid the unknown parameters from appearing in the controllers and parameters update laws when the adaptive control method is applied. A single virtual unknown parameter is used in the design of adaptive controllers and parameters update laws if the Lipschitz constant on the nonlinear function can be found, while multiple virtual unknown parameters are adopted if the Lipschitz constant cannot be determined. Numerical simulations show that the present method does make the two different chaotic systems synchronize in finite time.
Time-delay systems are, in many ways, a natural set of dynamical systems for natural scientists to study because they form an interface between abstract mathematics and data. However, they are complicated because past states must be sensibly incorporated into the dynamical system. The primary goal of this paper is to begin to isolate and understand the effects of adding time-delay coordinates to a dynamical system. The key results include (i) an analytical understanding regarding extreme points of a time-delay dynamical system framework including an invariance of entropy and the variance of the Kaplan-Yorke formula with simple time re-scalings; (ii) computational results from a time-delay mapping that forms a path between dynamical systems dependent upon the most distant and the most recent past; (iii) the observation that non-trivial mixing of past states can lead to high-dimensional, high-entropy dynamics that are not easily reduced to low-dimensional dynamical systems; (iv) the observed phase transition (bifurcation) between low-dimensional, reducible dynamics and high or infinite-dimensional dynamics; and (v) a convergent scaling of the distribution of Lyapunov exponents, suggesting that the infinite limit of delay coordinates in systems such are the ones we study will result in a continuous or (dense) point spectrum.
Certain nonlinear systems can switch between dynamical attractors occupying different regions of phase space, under variation of parameters or initial states. In this work we exploit this feature to obtain reliable logic operations. With logic output 0 or 1 mapped to dynamical attractors bounded in distinct regions of phase space, and logic inputs encoded by a very small bias parameter, we explicitly demonstrate that the system hops consistently in response to an external input stream, operating effectively as a reliable logic gate. This system offers the advantage that very low-amplitude inputs yield highly amplified outputs. Additionally, different dynamical variables in the system yield complementary logic operations in parallel. Further, we show that in certain parameter regions noise aids the reliability of logic operations, and is actually necessary for obtaining consistent outputs. This leads us to a generalization of the concept of Logical Stochastic Resonance to attractors more complex than fixed point states, such as periodic or chaotic attractors. Lastly, the results are verified in electronic circuit experiments, demonstrating the robustness of the phenomena. So we have combined the research directions of Chaos Computing and Logical Stochastic Resonance here, and this approach has the potential to be realized in wide-ranging systems.
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