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For the first time, we report on projective synchronization between two time delay chaotic systems with single time delays. It overcomes some limitations of the previous work, where projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve projective synchronization in infinitedimensional chaotic systems. We give a general method with which we can achieve projective synchronization in time-delayed chaotic systems. The method is illustrated using the famous delay-differential equations related to optical bistability. Numerical simulations fully support the analytical approach. 相似文献
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In this Letter, the synchronization problem is investigated for a class of stochastic complex networks with time delays. By utilizing a new Lyapunov functional form based on the idea of ‘delay fractioning’, we employ the stochastic analysis techniques and the properties of Kronecker product to establish delay-dependent synchronization criteria that guarantee the globally asymptotically mean-square synchronization of the addressed delayed networks with stochastic disturbances. These sufficient conditions, which are formulated in terms of linear matrix inequalities (LMIs), can be solved efficiently by the LMI toolbox in Matlab. The main results are proved to be much less conservative and the conservatism could be reduced further as the number of delay fractioning gets bigger. A simulation example is exploited to demonstrate the advantage and applicability of the proposed result. 相似文献
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In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction. 相似文献
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《Physica A》2005,351(1):133-141
It is shown that the nonlinear dynamics of chaotic time-delay systems can be reconstructed using a new type of neural network with two modules: one for nonfeedback part with input data delayed by the embedding time, and a second one for the feedback part with input data delayed by the feedback time. The method is applied to both simulated and experimental data from an electronic analog circuit of the Mackey–Glass system. Better results are obtained for the modular than for feedforward neural networks for the same number of parameters. It is found that the complexity of the neural network model required to reconstruct nonlinear dynamics does not increase with the delay time. Synchronization between the data and the model with diffusive coupling is also achieved. We have also shown by iterating the model from the present point that the dynamics can be predicted with a forecast horizon larger than the feedback delay time. 相似文献
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Using a recently proposed algorithmic scheme for correlation dimension analysis of hyperchaotic attractors, we study two well-known hyperchaotic flows and two standard time delayed hyperchaotic systems in detail numerically. We show that at the transition to hyperchaos, the nature of the scaling region changes suddenly and the attractor displays two scaling regions for embedding dimension M ≥ 4. We argue that it is an indication of a strong clustering tendency of the underlying attractor in the hyperchaotic phase. Because of this sudden qualitative change in the scaling region, the transition to hyperchaos can be easily identified using the discontinuous changes in the dimension (D 2) at the transition point. We show this explicitely for the two time delayed systems. Further support for our results is provided by computing the spectrum of Lyapunov Exponents (LE) of the hyperchaotic attractor in all cases. Our numerical results imply that the structure of a hyperchaotic attractor is topologically different from that of a chaotic attractor with inherent dual scales, at least for the two general classes of hyperchaotic systems we have analysed here. 相似文献
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A whole impulsive control scheme of nonlinear systems with time-varying delays, which is an extension for impulsive control of nonlinear systems without time delay, is presented in this paper. Utilizing the Lyapunov functions and the impulsive-type comparison principles, we establish a series of different conditions under which impulsively controlled nonlinear systems with time-varying delays are asymptotically stable. Then we estimate upper bounds of impulse interval and time-varying delays for asymptotically stable control. Finally a numerical example is given to illustrate the effectiveness of the method. 相似文献
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M.?Wolfrum S.?Yanchuk P.?H?vel E.?Sch?ll 《The European physical journal. Special topics》2010,191(1):91-103
We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion
of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in
such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic
approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including
the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity
modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback. 相似文献
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M.J. Bünner M. Ciofini A. Giaquinta R. Hegger H. Kantz R. Meucci A. Politi 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2000,10(2):165-176
High-dimensional chaos displayed by multi-component systems with a single time-delayed feedback is shown to be accessible
to time series analysis of a scalar variable only. The mapping of the original dynamics onto scalar time-delay systems defined
on sufficiently high dimensional spaces is thoroughly discussed. The dimension of the “embedding” space turns out to be independent
of the delay time and thus of the dimensionality of the attractor dynamics. As a consequence, the procedure described in the
present paper turns out to be definitely advantageous with respect to the standard embedding technique in the case of high-dimensional
chaos, when the latter is practically unapplicable. The mapping is not exact when delayed maps are used to reproduce the dynamics
of time-continuous systems, but the errors can be kept under control. In this context, the approximation of delay-differential
equations is discussed with reference to different classes of maps. Appropriate tools to estimate the a priori unknown delay time and the number of hidden components are introduced. The generalized Mackey-Glass system is investigated
in detail as a testing ground for the theoretical considerations.
Received 14 June 1999 and Received in final form 4 November 1999 相似文献
10.
In this paper,we propose a method for the projective synchronization between two different chaotic systems with variable time delays.Using active control approach,the suitable controller is constructed to make the states of two different diverse time delayed systems asymptotically synchronize up to the desired scaling factor.Based on the Lyapunov stability theory,the sufficient condition for the projective synchronization is calculated theoretically.Numerical simulations of the projective synchronization between Mackey-Glass system and Ikeda system with variable time delays are shown to validate the effectiveness of the proposed algorithm. 相似文献
11.
Grégory Faye 《Physica D: Nonlinear Phenomena》2010,239(9):561-578
In this paper we study neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integro-differential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we show the existence and uniqueness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable. We also present a fairly detailed study of the numerical computation of these solutions. This is, to our knowledge, the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed. Another original contribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our numerical schemes on a variety of examples that are relevant to modeling in neuroscience. 相似文献
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We conjecture that in one-dimensional spatially extended systems the propagation velocity of correlations coincides with a zero of the convective Lyapunov spectrum. This conjecture is successfully tested in three different contexts: (i) a Hamiltonian system (a Fermi-Pasta-Ulam chain of oscillators); (ii) a general model for spatiotemporal chaos (the complex Ginzburg-Landau equation); (iii) experimental data taken from a CO2 laser with delayed feedback. In the last case, the convective Lyapunov exponent is determined directly from the experimental data. 相似文献
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In this paper, synchronisation of fractional-order time delayed chaotic systems in ring networks is investigated. Based on Lyapunov stability theory, a new generic synchronisation criterion for N-coupled chaotic systems with time delay is proposed. The synchronisation scheme is applied to N-coupled fractional-order time delayed simplified Lorenz systems, and the Adomian decomposition method (ADM) is developed for solving these chaotic systems. Performance analysis of the synchronisation network is carried out. Numerical experiments demonstrate that synchronisation realises in both state variables and intermediate variables, which verifies the effectiveness of the proposed method. 相似文献
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J. Doyne Farmer 《Physica D: Nonlinear Phenomena》1982,4(3):366-393
We study the chaotic attractors of a delay differential equation. The dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of Lyapunov exponents according to a conjecture of Kaplan and Yorke. Assuming this conjecture to be valid, as the delay parameter is varied, from computations of the spectrum of Lyapunov exponents, we observe a roughly linear increase from two to twenty in the dimension, while the metric entropy remains roughly constant. These results are compared to a linear analysis, and the asymptotic behavior of the Lyapunov exponents is derived. 相似文献
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In this paper, we study the effect of parameter mismatch on the synchronization of a class of coupled chaotic systems with time delays. In the presence of parameter mismatch, the delayed coupled chaotic systems are investigated in terms of the quasisynchronization. A simple and yet easily applicable criterion for quasisynchronization of a large class of coupled chaotic systems with delays is derived based on the Lyapunov stability theory. As an example, the Ikeda oscillator is simulated, thereby validating the theoretical result in this paper. 相似文献
16.
Simultaneous identification of unknown time delays and model parameters in uncertain dynamical systems with linear or nonlinear parameterization by autosynchronization
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In this paper, we propose a general method to simultaneously identify both unknown time delays and unknown model parameters in delayed dynamical systems based on the autosynchronization technique. The design procedure is presented in detail by constructing a specific Lyapunov function and linearizing the model function with nonlinear parameterization. The obtained result can be directly extended to the identification problem of linearly parameterized dynamical systems. Two Wpical numerical examples confirming the effectiveness of the identification method are given. 相似文献
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K. P. Harikrishnan G. Ambika R. Misra 《The European physical journal. Special topics》2013,222(3-4):839-846
Transition to hyperchaos is uaually studied by computing the spectrum of Lyapunov Exponents (LE). But such a procedure can be employed mainly when the equations governing the dynamical system are known. However, if the information available on the system is only through time series, the method becomes difficult to implement. We show that the transition to hyperchaos is followed by a sudden change in the topological structure of the underlying attractor. Our numerical results indicate that the transition to hyperchaos can be characterized accurately through the computation of correlation dimension (D 2) from time series. We use two standard time delayed hyperchaotic systems as examples since, for such systems, D 2 varies smoothly as a function of the time delay τ which can be used as the control parameter. 相似文献
18.
《Physics letters. A》1999,263(3):167-174
One of the most important results of dynamical systems theory is the possibility to determine dynamical invariants by virtue of a long-term integration. In particular, this applies to the set of Lyapunov exponents of systems with chaotic solutions. However, we demonstrate that the structure of a Lyapunov spectrum, i.e., the signs of the (nonzero) exponents, is accessible already if the local flow is known within some small (in principle infinitesimal) time interval. We present various examples, including one in an embedding space, and discuss possible applications. 相似文献
19.
V. PyragasK. Pyragas 《Physics letters. A》2011,375(44):3866-3871
We propose a simple adaptive delayed feedback control algorithm for stabilization of unstable periodic orbits with unknown periods. The state dependent time delay is varied continuously towards the period of controlled orbit according to a gradient-descent method realized through three simple ordinary differential equations. We demonstrate the efficiency of the algorithm with the Rössler and Mackey-Glass chaotic systems. The stability of the controlled orbits is proven by computation of the Lyapunov exponents of linearized equations. 相似文献