Robust Estimation-Based Control of Chaotic Behavior in an Oscillator with Inertial and Elastic Symmetric Nonlinearities

2003 ◽  
Vol 9 (6) ◽  
pp. 665-684 ◽  
Author(s):  
A. A. Al-Qaisia ◽  
A. M. Harb ◽  
A. A. Zaher ◽  
M. A. Zohdy
Keyword(s):  
Robust Estimation ◽  
Nonlinear Oscillator ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Physical Parameters ◽  
Bifurcation And Chaos ◽  
Transition To Chaos ◽  
Simulation Results ◽  
Forced Nonlinear Oscillator ◽  
Period Doubling Bifurcations

In this paper, we study the dynamics of a forced nonlinear oscillator with inertial and elastic symmetric nonlinearities using modern nonlinear, bifurcation and chaos theories. The results for the response have shown that, for a certain combination of physical parameters, this oscillator exhibits a chaotic behavior or a transition to chaos through a sequence of period doubling bifurcations. The main objective of this paper is to control the chaotic behavior for this type of oscillator. A nonlinear estimation-based controller is proposed and the transient performance is investigated. The design of the parameter update mechanism is analyzed while discussing ways to extend its performance to further account for other types of uncertainties. We examine robustness problems as well as ways to tune the controller parameters. Simulation results are presented for the uncontrolled and controlled cases, verifying the effectiveness and the capability of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.

HIERARCHY OF INSTABILITIES AND SOLITONS IN AN ACOUSTIC DISTRIBUTED SYSTEM WITH FEEDBACK

1994 ◽  
Vol 04 (06) ◽  
pp. 1525-1534
Author(s):  
S.G. DOLINCHUK ◽  
V.I. ZADOROZHNII ◽  
A.M. FEDORCHENKO
Keyword(s):  
Steady State ◽  
Distributed System ◽  
Negative Feedback ◽  
Positive Feedback ◽  
Period Doubling ◽  
Physical Parameters ◽  
Transition To Chaos ◽  
Hard Excitation ◽  
Additional Feedback ◽  
Period Doubling Bifurcations

We have considered the hierarchy of instabilities, the transition to chaos, and the periodic generation of three-wave solitons in an acoustic distributed system with additional feedback and pumping. At positive feedback only steady-state conditions are shown to be stable. Self-oscillations, period-doubling bifurcations, and transition from quasiperiodicity to chaos of parametrically coupled waves have been found at negative feedback. “Hard” excitation of the generator leads to the formation of solitons. Established effects have been studied on dependence with physical parameters of a TeO 2 crystal.

CHAOTIC BEHAVIOR INDUCED BY THERMAL MODULATION IN A MODEL OF THE CONVECTIVE FLOW OF A FLUID MIXTURE IN A POROUS MEDIUM

1999 ◽  
Vol 09 (02) ◽  
pp. 383-396 ◽  
Author(s):  
J.-M. MALASOMA ◽  
P. WERNY ◽  
C.-H. LAMARQUE
Keyword(s):  
Porous Medium ◽  
Convective Flow ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Type I ◽  
Fluid Mixture ◽  
Numerical Investigations ◽  
Period Doubling Bifurcations

Numerical investigations of the global behavior of a model of the convective flow of a binary mixture in a porous medium are reported. We find a complex behavior characterized by the presence of coexisting periodic, quasiperiodic and chaotic attractors. Bifurcations of periodic solutions and routes to chaos via type-I intermittency and period-doubling bifurcations are described. Boundary crises and band merging crises have also been observed.

EFFECT OF DISSIPATION ON THE HAMILTONIAN CHAOS IN COUPLED OSCILLATOR SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 859-867 ◽  
Author(s):  
V. SHEEJA ◽  
M. SABIR
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Dissipation Function ◽  
Hamiltonian Chaos ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Route To Chaos ◽  
Period Doubling Bifurcations ◽  
External Driving

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

Multiple Scenarios of Transition to Chaos in the Alternative Splicing Model

2017 ◽  
Vol 27 (02) ◽  
pp. 1730006 ◽  
Author(s):  
Vladislav V. Kogai ◽  
Vitaly A. Likhoshvai ◽  
Stanislav I. Fadeev ◽  
Tamara M. Khlebodarova
Keyword(s):  
Alternative Splicing ◽  
Period Doubling ◽  
Genetic System ◽  
Protein Isoforms ◽  
Type I ◽  
Transition To Chaos ◽  
Delayed Arguments ◽  
The Mathematical Model ◽  
Period Doubling Bifurcations ◽  
Type Of Transition

We have investigated the scenarios of transition to chaos in the mathematical model of a genetic system constituted by a single transcription factor-encoding gene, the expression of which is self-regulated by a feedback loop that involves protein isoforms. Alternative splicing results in the synthesis of protein isoforms providing opposite regulatory outcomes — activation or repression. The model is represented by a differential equation with two delayed arguments. The possibility of transition to chaos dynamics via all classical scenarios: a cascade of period-doubling bifurcations, quasiperiodicity and type-I, type-II and type-III intermittencies, has been numerically demonstrated. The parametric features of each type of transition to chaos have been described.

On the Practical Relevance of Bifurcation and Chaos in Chemical and Biochemical Reaction Engineering

Fractals ◽  
1997 ◽  
Vol 05 (03) ◽  
pp. 549-560
Author(s):  
S. S. E. H. Elnashaie ◽  
A. E. Abasaeed ◽  
G. Ibrahim
Keyword(s):  
Catalytic Cracking ◽  
Chaotic Behavior ◽  
Research Work ◽  
Period Doubling ◽  
Reaction Engineering ◽  
Fluid Catalytic Cracking ◽  
Biochemical Reaction ◽  
Enzymatic System ◽  
Engineering Systems ◽  
Bifurcation And Chaos

This paper presents a condensed exposition of some of the research work on the bifurcation and chaotic behavior of industrially and fundamentally important chemical and biochemical reaction engineering systems. Two cases are presented: a forced industrial fluid catalytic cracking unit and an enzymatic system. The Poincare bifurcation diagrams obtained for both cases reveal their richness in dynamic behavior. Various routes to chaotic behavior have been identified including period doubling, intermittency, crisis.

BIFURCATION PHENOMENA IN THE 1:1 RESONANT HORN FOR THE FORCED VAN DER POL—DUFFING EQUATION

1992 ◽  
Vol 02 (01) ◽  
pp. 93-100 ◽  
Author(s):  
A.S. DMITRIEV ◽  
U.A. KOMLEV ◽  
D.V. TURAEV
Keyword(s):  
Period Doubling ◽  
Duffing Equation ◽  
Van Der Pol ◽  
Parallel Processes ◽  
Transition To Chaos ◽  
Bifurcation Phenomena ◽  
Small Dissipation ◽  
Period Doubling Bifurcations

This paper presents the 1:1 resonant horn bifurcation phenomena for the forced van der Pol—Duffing equation. It is shown that the transition to chaos in the case of small dissipation evolves in two parallel processes: a sequence of period-doubling bifurcations and the birth, growth and merging of homoclinic structures.

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