DETERMINISTIC CHAOS SEEN IN TERMS OF FEEDBACK CIRCUITS: ANALYSIS, SYNTHESIS, "LABYRINTH CHAOS"

This paper aims to show how complex nonlinear dynamic systems can be classified, analyzed and synthesized in terms of feedback circuits. The Rössler equations for deterministic chaos are revisited and generalized in this perspective. It is shown that once a proper set of feedback circuits is present in the Jacobian matrix of the system, the chaotic character of trajectories is remarkably robust versus changes in the nature of the nonlinearities. "Labyrinth chaos", whereby simple differential systems generate large lattices of many unstable steady states embedded in a chaotic attractor, is constructed using this technique. In the limit case of a single three-element circuit without diagonal elements, one finds systems possessing an infinite lattice of unstable steady states between which trajectories percolate in a deterministic chaotic way.

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18th Annual Spring Meeting: Division of Psychoanalysis (39). Psychoanalysis and Sexuality: Reflections of an Old Love Affair. Summaries: Theoretical. Nonlinear Dynamic Systems and Deterministic Chaos: Conceptual Advances or Aesthetically Pleasing Distractions?

PsycEXTRA Dataset ◽

10.1037/e404582005-010 ◽

1998 ◽

Author(s):

Craig Piers

Keyword(s):

Dynamic Systems ◽

Nonlinear Dynamic ◽

Deterministic Chaos ◽

Nonlinear Dynamic Systems ◽

Love Affair ◽

Spring Meeting

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THE RÖSSLER EQUATIONS REVISITED IN TERMS OF FEEDBACK CIRCUITS

Journal of Biological System ◽

10.1142/s0218339099000152 ◽

1999 ◽

Vol 07 (02) ◽

pp. 225-237 ◽

Cited By ~ 7

Author(s):

R. THOMAS

Keyword(s):

Steady State ◽

Chaotic Dynamics ◽

Jacobian Matrix ◽

Complex Dynamics ◽

Deterministic Chaos ◽

Logical Structure ◽

Steady States ◽

Global Behavior ◽

Feedback Circuits ◽

By Products

We would like to show that complex dynamics can be deciphered and at least partly understood in terms of its underlying logical structure, more specifically in terms of the feedback circuits built in the differential equations. This approach has permitted to build a number of new three- and four-variable systems displaying chaotic dynamics. Starting from the well-known Rössler equations for deterministic chaos, one asks how systems with the same types of steady states can be synthesized ab initio from appropriate feedback circuits (= appropriate logical structure). It is found that, granted an appropriate logical structure, the existence of a domain of chaotic dynamics is remarkably robust towards changes in the nature of the nonlinearity used, and towards those sign changes which respect the nature (positive vs negative) of the feedback circuits. Using logical arguments, it was also easy to find related systems with a single steady state. A variety of 3- and 4-d systems based on other combinations of feedback circuits and generating chaotic dynamics are described. The aim of this work is to contribute to a better understanding of the respective roles of feedback circuits and nonlinearity — both essential — in so-called "non trivial behavior", including deterministic chaos. Special emphasis is put on the interest of using the Jacobian matrix and its by-products (characteristic equation, eigenvalues, …) not only close to steady states (where linear stability analysis can be performed) but also elsewhere in phase space, where precious indications about the global behavior can be collected.

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GENERALIZED SYNCHRONIZATION IN AN ARRAY OF NONLINEAR DYNAMIC SYSTEMS WITH APPLICATIONS TO CHAOTIC CNN

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413500168 ◽

2013 ◽

Vol 23 (01) ◽

pp. 1350016 ◽

Cited By ~ 13

Author(s):

LEQUAN MIN ◽

GUANRONG CHEN

Keyword(s):

Neural Network ◽

Numerical Simulations ◽

Dynamic Systems ◽

Nonlinear Dynamic ◽

Cellular Neural Network ◽

Nonlinear Dynamic Systems ◽

Generalized Synchronization ◽

Complex Dynamic ◽

Differential Systems ◽

Difference Systems

This paper establishes some generalized synchronization (GS) theorems for a coupled discrete array of difference systems (CDADS) and a coupled continuous array of differential systems (CCADS). These constructive theorems provide general representations of GS in CDADS and CCADS. Based on these theorems, one can design GS-driven CDADS and CCADS via appropriate (invertible) transformations. As applications, the results are applied to autonomous and nonautonomous coupled Chen cellular neural network (CNN) CDADS and CCADS, discrete bidirectional Lorenz CNN CDADS, nonautonomous bidirectional Chua CNN CCADS, and nonautonomously bidirectional Chen CNN CDADS and CCADS, respectively. Extensive numerical simulations show their complex dynamic behaviors. These theorems provide new means for understanding the GS phenomena of complex discrete and continuously differentiable networks.

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