CHAOTIC SYNCHRONIZATION OF A ONE-DIMENSIONAL ARRAY OF NONLINEAR ACTIVE SYSTEMS

1993 ◽  
Vol 03 (04) ◽  
pp. 1067-1074 ◽  
Author(s):  
V. PÉREZ-VILLAR ◽  
A. P. MUÑUZURI ◽  
V. PÉREZ-MUÑUZURI ◽  
L. O. CHUA
Keyword(s):  
Stability Analysis ◽  
Phase Space ◽  
Dynamical Systems ◽  
Chaotic Synchronization ◽  
Active Systems ◽  
One Dimensional ◽  
Chaotic Dynamical Systems ◽  
First Case ◽  
Theoretical Predictions

Linear stability analysis is used to study the synchronization of N coupled chaotic dynamical systems. It is found that the role of the coupling is always to stabilize the system, and then synchronize it. Computer simulations and experimental results of an array of Chua's circuits are carried out. Arrays of identical and slightly different oscillators are considered. In the first case, the oscillators synchronize and sync-phase, i.e., each one repeats exactly the same behavior as the rest of them. When the oscillators are not identical, they can also synchronize but not in phase with each other. The last situation is shown to form structures in the phase space of the dynamical variables. Due to the inevitable component tolerances (±5%), our experiments have so far confirmed our theoretical predictions only for an array of slightly different oscillators.

scholarly journals Modelling of Chaotic Processes with Caputo Fractional Order Derivative

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1027 ◽  
Author(s):  
Kolade M. Owolabi ◽  
José Francisco Gómez-Aguilar ◽  
G. Fernández-Anaya ◽  
J. E. Lavín-Delgado ◽  
E. Hernández-Castillo
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Spectral Method ◽  
Fractional Order ◽  
Linear Stability Analysis ◽  
Numerical Approximation ◽  
Fractional Power ◽  
Chaotic Dynamical Systems ◽  
Chebyshev Spectral Method ◽  
Caputo Fractional Order Derivative

Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for the numerical approximation. In each of the systems considered, linear stability analysis is established. A range of chaotic behaviours are obtained at the instances of fractional power which show the evolution of the species in time and space.

An Algorithm for the Computation of Preimages in Noninvertible Mappings

1998 ◽  
Vol 08 (02) ◽  
pp. 415-422 ◽  
Author(s):  
Chia-Hsing Nien ◽  
Frederick J. Wicklin
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Computational Algorithm ◽  
Singularity Theory ◽  
Periodic Points ◽  
Discrete Dynamical Systems ◽  
Global Knowledge ◽  
Stable Manifolds ◽  
One Dimensional ◽  
Accurate Computations

For discrete dynamical systems generated by iterating a diffeomorphism, every point in the phase space has a unique preimage and it is straightforward to compute geometric structures such as inverse orbits and one-dimensional stable manifolds of periodic points. For noninvertible mappings, however, some points have multiple preimages; others may have no preimages. This makes the computation of inverse orbits difficult, because accurate computations require global knowledge about the way the mapping folds and pleats phase space. In this article we use ideas from singularity theory to examine the geometry of noninvertible mappings. We use the geometry to derive a computational algorithm for efficiently computing preimages in noninvertible mappings.

DERIVING CHAOTIC DYNAMICAL SYSTEMS FROM ENERGY FUNCTIONALS

2001 ◽  
Vol 01 (03) ◽  
pp. 377-388 ◽  
Author(s):  
PAUL BRACKEN ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Order Differential Equation ◽  
Lagrange Equation ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Energy Functionals ◽  
Chaotic Dynamical Systems ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Simple one-dimensional chaotic dynamical systems are derived by optimizing energy functionals. The Euler–Lagrange equation yields a nonlinear second-order differential equation whose solution yields a 2–1 map which admits an absolutely continuous invariant measure. The solutions of the differential equation are studied.

ABRUPT DIMENSION CHANGES AT BASIN BOUNDARY METAMORPHOSES

1992 ◽  
Vol 02 (03) ◽  
pp. 533-541 ◽  
Author(s):  
BAE-SIG PARK ◽  
CELSO GREBOGI ◽  
YING-CHENG LAI
Keyword(s):  
Dynamical Systems ◽  
Numerical Experiments ◽  
System Parameter ◽  
Analytic Calculation ◽  
Two Dimensional ◽  
Qualitative Model ◽  
One Dimensional ◽  
Basin Boundary ◽  
Chaotic Dynamical Systems ◽  
Basin Boundaries

Basin boundaries in chaotic dynamical systems can be either smooth or fractal. As a system parameter changes, the structure of the basin boundary also changes. In particular, the dimension of the basin boundary changes continuously except when a basin boundary metamorphosis occurs, at which it can change abruptly. We present numerical experiments to demonstrate such sudden dimension changes. We have also used a one-dimensional analytic calculation and a two-dimensional qualitative model to explain such changes.

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