The Effects of Symmetry Breaking Perturbation on the Dynamics of a Novel Chaotic System with Cyclic Symmetry: Theoretical Analysis and Circuit Realization

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Jacques Kengne ◽  
Sandrine Zoulewa Dountsop ◽  
Jean Chamberlain Chedjou ◽  
Khabibullo Nosirov
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Period Doubling ◽  
Cyclic Symmetry ◽  
Multiple Attractors ◽  
Circuit Realization ◽  
Analog Device ◽  
Nonlinear Dynamical ◽  
The One ◽  
Parameter Ranges ◽  
New System

Symmetry is an important property shared by a large number of nonlinear dynamical systems. Although the study of nonlinear systems with a symmetry property is very well documented, the literature has no sufficient investigation on the important issues concerning the behavior of such systems when their symmetry is broken or altered. In this work, we introduce a novel autonomous 3D system with cyclic symmetry having a piecewise quadratic nonlinearity [Formula: see text] where parameter [Formula: see text] is fixed and parameter [Formula: see text] controls the symmetry and the nonlinearity of the model. Obviously, for [Formula: see text] the system presents both cyclic and inversion symmetries while the inversion symmetry is explicitly broken for [Formula: see text]. We consider in detail the dynamics of the new system for both two regimes of operation by using classical nonlinear analysis tools (e.g. bifurcation diagrams, plots of largest Lyapunov exponents, phase space trajectory plots, etc.). Several nonlinear patterns are reported such as period doubling, periodic windows, parallel bifurcation branches, hysteresis, transient chaos, and the coexistence of multiple attractors of different topologies as well. One of the most gratifying features of the new system introduced in this work is the existence of several parameter ranges for which up to twelve disconnected periodic and chaotic attractors coexist. This latter feature is rarely reported, at least for a simple system like the one discussed in this work. An electronic analog device of the new cyclic system is designed and implemented in PSpice. A very good agreement is observed between PSpice simulation and the theoretical results.

NOISE-INDUCED BACKWARD BIFURCATIONS OF STOCHASTIC 3D-CYCLES

2010 ◽  
Vol 09 (01) ◽  
pp. 89-106 ◽  
Author(s):  
I. BASHKIRTSEVA ◽  
L. RYASHKO ◽  
P. STIKHIN
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Nonlinear Dynamical Systems ◽  
Period Doubling ◽  
Sensitivity Function ◽  
Function Technique ◽  
Stochastic Bifurcation ◽  
Nonlinear Dynamical ◽  
General Method

We study stochastically forced multiple limit cycles of nonlinear dynamical systems in a period-doubling bifurcation zone. Noise-induced transitions between separate parts of the cycle are considered. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation (BSB). In this paper, for the BSB analysis we suggest a stochastic sensitivity function technique. As a result, a method for the estimation of critical values of noise intensity corresponding to BSB is proposed. The constructive possibilities of this general method for the detailed BSB analysis of the multiple stochastic cycles of the forced Roessler system are demonstrated.

scholarly journals Fundamental Constants in Time from the Big-Bang

2020 ◽  
Author(s):  
Heikki Sipilä ◽  
Ari Lehto
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Structure Constant ◽  
Period Doubling ◽  
Big Bang ◽  
Atomic Spectroscopy ◽  
Fine Structure Constant ◽  
Theoretical Interpretation ◽  
Fundamental Constants ◽  
Spatial Direction ◽  
Nonlinear Dynamical

Our understanding and theoretical interpretation of observations in astrophysics and cosmology depends on our knowledge of the fundamental constants and their possible dependence on time and space. Atomic spectroscopy and radio astronomy give important information on the validity and stability of the fundamental constants. The possible dependence of the fine structure constant alpha on time and spatial direction is an active topic of research.Period doubling is a universal property of nonlinear dynamical systems, and the doubling is exact in principle. The value of the elementary charge squared can be calculated by the period doubling process from the Planck charge and thereby the value of alpha.If ‘old’ and ‘new’ electrons are identical, then the Planck charge, i.e. a set of natural constants, has remained constant over time. In this article we show that the value of alpha calculated from the Planck charge is 0.007 % larger than the current accepted value of alpha.

Head Motion Stabilization During Quadruped Robot Locomotion

2011 ◽  
Vol 2 (1) ◽  
pp. 39-62 ◽  
Author(s):  
Miguel Oliveira ◽  
Cristina P. Santos ◽  
Lino Costa ◽  
Ana Rocha ◽  
Manuel Ferreira
Keyword(s):  
Global Optimization ◽  
Head Movement ◽  
Nonlinear Dynamical Systems ◽  
Fitness Function ◽  
Central Pattern Generators ◽  
Quadruped Robot ◽  
Biologically Inspired ◽  
Nonlinear Dynamical ◽  
Pattern Generators ◽  
The One

In this work, the authors propose a combined approach based on a controller architecture that is able to generate locomotion for a quadruped robot and a global optimization algorithm to generate head movement stabilization. The movement controllers are biologically inspired in the concept of Central Pattern Generators (CPGs) that are modelled based on nonlinear dynamical systems, coupled Hopf oscillators. This approach allows for explicitly specified parameters such as amplitude, offset and frequency of movement and to smoothly modulate the generated oscillations according to changes in these parameters. The overall idea is to generate head movement opposed to the one induced by locomotion, such that the head remains stabilized. Thus, in order to achieve this desired head movement, it is necessary to appropriately tune the CPG parameters. Three different global optimization algorithms search for this best set of parameters. In order to evaluate the resulting head movement, a fitness function based on the Euclidean norm is investigated. Moreover, a constraint-handling technique based on tournament selection was implemented.

Growth Phenomena — Algebraic and Geometric Descriptions

2001 ◽  
Vol 08 (01) ◽  
pp. 63-71
Author(s):  
Andrzej Jamiołkowski
Keyword(s):  
Vector Fields ◽  
Nonlinear Dynamical Systems ◽  
Polynomial Systems ◽  
Polynomial Vector ◽  
Finsler Spaces ◽  
Polynomial Vector Fields ◽  
Associative Algebras ◽  
Nonlinear Dynamical ◽  
Volterra Systems ◽  
The One

An enormous variety of nonlinear dynamical systems can be — by suitable introduction of new coordinates — represented in the form of polynomial systems and then can be reduced to Volterra systems, where the nonlinearities are at most quadratic. In this paper, we discuss a link between systems of differential equations with homogeneous quadratic polynomial vector fields and non-associative algebras on the one hand and the question of representation of such systems as geodesics in some Finsler spaces on the other hand.

scholarly journals Constrained attractor selection using deep reinforcement learning

2020 ◽  
pp. 107754632093014
Author(s):  
Xue-She Wang ◽  
James D Turner ◽  
Brian P Mann
Keyword(s):  
Reinforcement Learning ◽  
Gradient Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Learning Approaches ◽  
Multiple Attractors ◽  
Nonlinear Dynamical ◽  
Cross Entropy Method ◽  
Policy Gradient ◽  
Attractor Selection

This study describes an approach for attractor selection (or multistability control) in nonlinear dynamical systems with constrained actuation. Attractor selection is obtained using two different deep reinforcement learning methods: (1) the cross-entropy method and (2) the deep deterministic policy gradient method. The framework and algorithms for applying these control methods are presented. Experiments were performed on a Duffing oscillator, as it is a classic nonlinear dynamical system with multiple attractors. Both methods achieve attractor selection under various control constraints. Although these methods have nearly identical success rates, the deep deterministic policy gradient method has the advantages of a high learning rate, low performance variance, and a smooth control approach. This study demonstrates the ability of two reinforcement learning approaches to achieve constrained attractor selection.

scholarly journals A New Chaotic System with Only Nonhyperbolic Equilibrium Points: Dynamics and Its Engineering Application

Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-16
Author(s):  
Maryam Zolfaghari-Nejad ◽  
Mostafa Charmi ◽  
Hossein Hassanpoor
Keyword(s):  
Chaotic System ◽  
Infinite Number ◽  
Equilibrium Points ◽  
Engineering Application ◽  
Period Doubling ◽  
Spectral Entropy ◽  
Multiple Attractors ◽  
Initial Values ◽  
New Chaotic System ◽  
New System

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system.

CANDYS/QA—A SOFTWARE SYSTEM FOR QUALITATIVE ANALYSIS OF NONLINEAR DYNAMICAL SYSTEMS

1992 ◽  
Vol 02 (04) ◽  
pp. 773-794 ◽  
Author(s):  
ULRIKE FEUDEL ◽  
WOLFGANG JANSEN
Keyword(s):  
Dynamical Systems ◽  
Qualitative Analysis ◽  
Nonlinear Dynamical Systems ◽  
Period Doubling ◽  
Path Following ◽  
Invariant Sets ◽  
Software System ◽  
Nonlinear Dynamical ◽  
Bifurcation Phenomena ◽  
Two Parameter

Numerical methods are often needed if bifurcation phenomena in nonlinear dynamical systems are studied. In this paper the software system CANDYS/QA for numerical qualitative analysis is presented. A wide class of problems is treated: computation of invariant sets (e.g., steady-states and periodic orbits), path-following (continuation) of such sets, and the related bifurcation phenomena. The following bifurcation situations are detected and the corresponding critical points are calculated during path-following: turning, bifurcation, Hopf bifurcation, period-doubling, torus bifurcation points (one-parameter problems) as well as cusp and Takens-Bogdanov points (two-parameter problems). A number of newly developed methods (e.g., for computation of the Poincaré map) as well as algorithms from the literature are described to demonstrate the whole procedure of a qualitative analysis by numerical means. An illustrative example analyzed by CANDYS/QA is included.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250093 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
JIANZHE HUANG
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Dynamical

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.

ON STRONG AND WEAK CHAOTIC PARTIAL SYNCHRONIZATION

2000 ◽  
Vol 10 (01) ◽  
pp. 179-203 ◽  
Author(s):  
Yu. MAISTRENKO ◽  
O. POPOVYCH ◽  
M. HASLER
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Chaotic Synchronization ◽  
State Variables ◽  
Large Parameter ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Rich Variety ◽  
Parameter Ranges ◽  
One Dimensional Maps

We study coupled nonlinear dynamical systems with chaotic behavior in the case when two or more (but not all) state variables synchronize, i.e. converge to each other asymptotically in time. It is shown that for symmetrical systems, such partial chaotic synchronization is usually only weak, whereas with nonsymmetrical coupling it can be strong in large parameter ranges. These facts are illustrated with systems of three coupled one-dimensional maps, for which a rich variety of different "partial chaotic synchronizing" phenomena takes place.

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