Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems

Kocarev, Ljupčo; Parlitz, Ulrich

doi:10.1103/physrevlett.76.1816

Cited by 974 publications

(573 citation statements)

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“…Their characteristics and ability to generate complex dynamics are demonstrated using a theoretical example and a recording from a biological system (the human heart). We also show how the properties of these systems allow the deterministic dynamics to be extracted.The new class of systems is a subclass of nonautonomous systems whose definition is provided by a skewproduct flow [10,24,25] generated by unidirectionally coupled differential equations (also known as a masterslave configuration [7], or as drive and response systems [26])where p 2 R n , x 2 R m . The nonautonomous system x can be considered as driven by the system p in the sense that _ x ¼ gðx; pðtÞÞ for any given solution pðtÞ.…”

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confidence: 99%

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Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

2013

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Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks. DOI: 10.1103/PhysRevLett.111.024101 PACS numbers: 05.45.Xt, 05.65.+b, 05.90.+m, 89.75.Fb Dynamical systems are generally seen as being either deterministic or stochastic. The advent of dynamical chaos several decades ago attracted much attention and illustrated that even complex dynamics can be deterministic. Dynamical systems can also be classified as autonomous (self-contained) or nonautonomous (subject to external influences). In reality, nonautonomous systems are the commoner, but they are far harder to treat. Until now they were mostly treated as stochastic or, alternatively, attempts were made to reformulate them as autonomous. Neither approach captures the characteristic properties of these systems.In this Letter we propose a new class of nonautonomous systems and name them chronotaxic to characterize oscillatory systems with time-varying, but stable, amplitudes and frequencies. Nonautonomous oscillatory systems appear in various fields of research including neuroscience [1,2], cardiovascular dynamics [3], climate [4,5] and evolutionary science [6], as well as in complex systems and networks [7][8][9]. Although we are witnessing a rapid development of the theory of nonautonomous [10,11] and random dynamical systems [12,13], nonautonomous oscillatory systems with stable but time-varying characteristic frequencies have to date not been addressed. When treated in an inverse approach such systems are usually considered as stochastic. In an attempt to cope with the problem, several methods for the inverse approach were introduced, including wavelet-based decomposition [14] Common to all these systems is that they are oscillatory, have stable amplitudes, and frequencies that are resistant to external perturbations. The variety of systems with these characteristics suggests that their dynamics are generated from a universal basis. To date, the description of stable oscillatory dynamics has been based on the model of autonomous self-sustained limit cycle oscillators [23]. While this model provides stable amplitude dynamics, frequencies of oscillations within this model can be easily changed even by weakest external perturbations.The new class of nonautonomous oscillatory dynamical systems that we now propose account for such dynamics. The novelty of these systems is that not only are the amplitude dynamics stable but also the frequencies of the oscillations are time dependent and stable-i.e., their timedependent values cannot be easily altered by external perturbations. Their c...

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confidence: 99%

“…The new class of systems is a subclass of nonautonomous systems whose definition is provided by a skewproduct flow [10,24,25] generated by unidirectionally coupled differential equations (also known as a masterslave configuration [7], or as drive and response systems [26])…”

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confidence: 99%

Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

2013

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“…Consider a three-dimensional dynamical system of ordinary differential equations (ODEs) in the form (1) with where i ∈ {x, y, z} and monomials are of a polynomial form up to the second-order. It was shown in [7,13,14] that when a small subset of coefficients a i,j is nonzero, system (1) can be transformed into a standard form whose structure is (2) where the first variable X 1 = h(x, y, z) = s is some function of the state variables, h being the so-called measurement function.…”

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confidence: 99%

Discovering independent parameters in complex dynamical systems

Lainscsek

Weyhenmeyer

Sejnowski

et al. 2015

Chaos, Solitons & Fractals

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The transformation of a nonlinear dynamical system into a standard form by using one of its variables and its successive derivatives can be used to identify the relationships that may exist between the parameters of the original system such as the subset of the parameter space over which the dynamics is left invariant. We show how the size of the attractor or the time scale (the pseudo-period) can be varied without affecting the underlying dynamics. This is demonstrated for the Rössler and the Lorenz systems. We also consider the case when two Rössler systems are unidirectionally coupled and when a Lorenz system is driven by a Rössler system. In both cases, the dynamics of the coupled system is affected.

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“…In the first case, a replica subsystem driven by chaotic signals of the chaotic system can synchronize identically with the drive system [1][2][3][4][5], if the largest conditional Lyapunov is negative. This is referred to as identical synchronization.Secondly, a driven system, which is not a replica of the drive system, however, may not achieve identical synchronization, but generalized synchronization [6][7][8], if the largest conditional Lyapunov exponent is negative. Two identical systems, driven by the same signal, thus may come to the same final state due to the negative largest conditional Lyapunov exponent.…”

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confidence: 99%

“…Secondly, a driven system, which is not a replica of the drive system, however, may not achieve identical synchronization, but generalized synchronization [6][7][8], if the largest conditional Lyapunov exponent is negative. Two identical systems, driven by the same signal, thus may come to the same final state due to the negative largest conditional Lyapunov exponent.…”

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Synchronization with positive conditional Lyapunov exponents

Zhou

Lai

1998

Phys. Rev. E

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Synchronization of chaotic system may occur only when the largest conditional Lyapunov exponent of the driven system is negative. The synchronization with positive conditional Lyapunov reported in a recent paper (Phys. Rev. E, 56, 2272Rev. E, 56, (1997) is a combined result of the contracting region of the system and the finite precision in computer simulations. PACS number(s): 05.45.+b; 1 Sensitivity to initial conditions is a generic feature of chaotic dynamical systems. Two chaotic orbits, starting from slightly different initial points in the state space, separate exponentially with time, and become totally uncorrelated. As a result, independent identical chaotic systems cannot synchronize with each other. The sensitivity is quantitatively described by positive Lyapunov exponent(s) in the Lyapunov exponent spectrum of the chaotic system.However, chaotic systems linked by common signal can synchronize with each other. Several cases could be distinguished. In the first case, a replica subsystem driven by chaotic signals of the chaotic system can synchronize identically with the drive system [1][2][3][4][5], if the largest conditional Lyapunov is negative. This is referred to as identical synchronization.Secondly, a driven system, which is not a replica of the drive system, however, may not achieve identical synchronization, but generalized synchronization [6][7][8], if the largest conditional Lyapunov exponent is negative. Two identical systems, driven by the same signal, thus may come to the same final state due to the negative largest conditional Lyapunov exponent.Lyapunov exponents are also employed to characterize behavior of random dynamical systems [9]: the system is chaotic (non-chaotic) when the largest Lyapunov exponent is positive (negative). The sensitivity of a chaotic system may also be suppressed by noise, and identical chaotic systems subjected to common noise can synchronize with each other. Maritan and Banavar[10] studied the behavior of the noise-driven logistic maps and reported synchronization phenomenon. It turned out that the observed synchronization was an outcome of finite precision in numerical simulations [11,12], while the Lyapunov exponent of the noisy logistic map is positive [11].Very recently, Shuai et al [13] claimed that synchronization can be achieved with positive conditional Lyapunov exponents. In a one-way coupled map lattice, they observed, through computer simulations, synchronization of spatiotemporal chaos with many positive components in the conditional Lyapunov exponent spectrum. Based on these results, they drew the conclusion that the conditional Lyapunov exponents cannot be used as a criterion for synchronous chaotic systems.Whether such a claim is true is of great importance for our understanding of synchronization. In this paper, we reexamined such synchronization phenomenon, revealing that it is yet another example of round-off induced phenomenon.In [13], Shuai et al studied a driven one-way coupled map latticewhere x 0 (t) is a hyperchaotic signal f...

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Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems

Cited by 974 publications

References 18 publications

Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

Discovering independent parameters in complex dynamical systems

Synchronization with positive conditional Lyapunov exponents

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