Chaotic transient and the improvement of system flexibility

2007 ◽  
Vol 365 (4) ◽  
pp. 328-334
Author(s):  
Luiz Felipe R. Turci ◽  
Elbert E.N. Macau ◽  
Takashi Yoneyama
Keyword(s):  
Chaotic Transient

PARTIAL CONTROL OF TRANSIENT CHAOS IN ELECTRONIC CIRCUITS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250032 ◽  
Author(s):  
ALEXANDRE WAGEMAKERS ◽  
SAMUEL ZAMBRANO ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Analog Circuit ◽  
Control Method ◽  
Initial Conditions ◽  
Simple Extension ◽  
The Novel ◽  
Partial Control ◽  
Time Dynamics ◽  
The One ◽  
Chaotic Transient ◽  
Chaotic Saddle

We present an analog circuit implementation of the novel partial control method, that is able to sustain chaotic transient dynamics. The electronic circuit simulates the dynamics of the one-dimensional slope-three tent map, for which the trajectories diverge to infinity for nearly all the initial conditions after behaving chaotically for a while. This is due to the existence of a nonattractive chaotic set: a chaotic saddle. The partial control allows one to keep the trajectories close to the chaotic saddle, even if the control applied is smaller than the effect of the applied noise, introduced into the system. Furthermore, we also show here that similar results can be implemented on a circuit that simulates a horseshoe-like map, which is a simple extension of the previous one. This encouraging result validates the theory and opens new perspectives for the application of this technique to systems with higher dimensions and continuous time dynamics.

Evidence of a chaotic transient in a closed unstirred cerium catalyzed Belousov-Zhabotinsky system

1996 ◽  
Vol 263 (3-4) ◽  
pp. 429-434 ◽  
Author(s):  
Mauro Rustici ◽  
Mario Branca ◽  
Carlo Caravati ◽  
Nadia Marchettini
Keyword(s):  
Chaotic Transient

SUPER PERSISTENT CHAOTIC TRANSIENTS IN PHYSICAL SYSTEMS: EFFECT OF NOISE ON PHASE SYNCHRONIZATION OF COUPLED CHAOTIC OSCILLATORS

2001 ◽  
Vol 11 (10) ◽  
pp. 2607-2619 ◽  
Author(s):  
VICTOR ANDRADE ◽  
YING-CHENG LAI
Keyword(s):  
Phase Synchronization ◽  
Scaling Law ◽  
Unstable Periodic Orbits ◽  
Chaotic Oscillators ◽  
Physical Systems ◽  
Temporal Phase ◽  
Additive White Noise ◽  
Chaotic Transients ◽  
Phase Slips ◽  
Chaotic Transient

A super persistent chaotic transient is typically induced by an unstable–unstable pair bifurcation in which two unstable periodic orbits of the same period coalesce and disappear as a system parameter is changed through a critical value. So far examples illustrating this type of transient chaos utilize discrete-time maps. We present a class of continuous-time dynamical systems that exhibit super persistent chaotic transients in parameter regimes of positive measure. In particular, we examine the effect of noise on phase synchronization of coupled chaotic oscillators. It is found that additive white noise can induce phase slips in integer multiples of 2π's in parameter regimes where phase synchronization is expected in the absence of noise. The average time durations of the temporal phase synchronization are in fact characteristic of those of super persistent chaotic transients. We provide heuristic arguments for the scaling law of the average transient lifetime and verify it using numerical examples from both the system of coupled Chua's circuits and that of coupled Rössler oscillators. Our work suggests a way to observe super persistent chaotic transients in physically realizable systems.

Periodic Windows and Chaotic Transient in Coupled Map Lattices

1994 ◽  
Vol 3 (5) ◽  
pp. 353-359 ◽  
Author(s):  
Qu Zhi-lin ◽  
Xu Chang-ye ◽  
Ma Ben-kun ◽  
Hu Gang
Keyword(s):  
Coupled Map Lattices ◽  
Chaotic Transient

scholarly journals Super persistent chaotic transients

1985 ◽  
Vol 5 (3) ◽  
pp. 341-372 ◽  
Author(s):  
Celso Grebogi ◽  
Edward Ott ◽  
James A. Yorke
Keyword(s):  
Bifurcation Point ◽  
Average Length ◽  
Asymptotic State ◽  
Average Lifetime ◽  
Bounded Noise ◽  
Small Random Perturbations ◽  
Image Position ◽  
Chaotic Transients ◽  
Parameter Values ◽  
Chaotic Transient

AbstractThe unstable-unstable pair bifurcation is a bifurcation in which two unstable fixed points or periodic orbits of the same period coalesce and disappear as a system paremeter is raised. For parameter values just above that at which unstable orbits are destroyed there can be chaotic transients. Then, as the bifurcation is approached from above, the average length of a chaotic transient diverges, and, below the bifurcation point, the chaotic transient may be regarded as having been converted into a chaotic attractor. It is argued that unstable-unstable pair bifurcations should be expected to occur commonly in dynamical systems. This bifurcation is an example of the crisis route to chaos. The most striking fact about unstable-unstable pair bifurcation crises is that long chaotic transients persist even for parameter values relatively far from the bifurcation point. These long-lived chaotic transients may prevent the time asymptotic state from being reached during experiments. An expression giving a lower bound for the average lifetime of a chaotic transient is derived and shown to agree well with numerical experiments. In particular, this bound on the average lifetime, (τ), satisfiesfor α near α*, where k1 and k2 are constants and α* is the value of the parameter a at which the crisis occurs. Thus, as a approaches α* from above, (τ) increases more rapidly than any power of (α − α*)−1. Finally, we discuss the effect of adding bounded noise (small random perturbations) on these phenomena and argue that the chaotic transients should be lengthened by noise.

scholarly journals Improving the Complexity of the Lorenz Dynamics

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
María Pilar Mareca ◽  
Borja Bordel
Keyword(s):  
Cross Sections ◽  
Secure Communications ◽  
The Novel ◽  
Return Maps ◽  
Transient Solutions ◽  
The Lorenz System ◽  
The Stability ◽  
First Return Maps ◽  
Chaotic Transient ◽  
New System

A new four-dimensional, hyperchaotic dynamic system, based on Lorenz dynamics, is presented. Besides, the most representative dynamics which may be found in this new system are located in the phase space and are analyzed here. The new system is especially designed to improve the complexity of Lorenz dynamics, which, despite being a paradigm to understand the chaotic dissipative flows, is a very simple example and shows great vulnerability when used in secure communications. Here, we demonstrate the vulnerability of the Lorenz system in a general way. The proposed 4D system increases the complexity of the Lorenz dynamics. The trajectories of the novel system include structures going from chaos to hyperchaos and chaotic-transient solutions. The symmetry and the stability of the proposed system are also studied. First return maps, Poincaré sections, and bifurcation diagrams allow characterizing the global system behavior and locating some coexisting structures. Numerical results about the first return maps, Poincaré cross sections, Lyapunov spectrum, and Kaplan-Yorke dimension demonstrate the complexity of the proposed equations.

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