scholarly journals Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Chengwei Dong ◽  
Lian Jia ◽  
Qi Jie ◽  
Hantao Li
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Unstable Periodic Orbits ◽  
Nonlinear Dynamical ◽  
System A ◽  
Return Maps ◽  
Phase Portrait Analysis ◽  
Encoding Method ◽  
First Return Maps ◽  
Parameter Values

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

Bifurcation Control of Ro¨ssler System

2003 ◽  
Author(s):  
Z. Q. Wu ◽  
P. Yu
Keyword(s):  
Feedback Control ◽  
Nonlinear Dynamical Systems ◽  
Equilibrium Points ◽  
New Method ◽  
Control Law ◽  
Equilibrium Solutions ◽  
Nonlinear Dynamical ◽  
The Stability ◽  
Parameter Values ◽  
Feasible System

In this paper, a new method is proposed for controlling bifurcations of nonlinear dynamical systems. This approach employs the idea used in deriving the transition variety sets of bifurcations with constraints to find the stability region of equilibrium points in parameter space. With this method, one can design, via a feedback control, appropriate parameter values to delay either static, or dynamic or both bifurcations as one wishes. The approach is applied to consider controlling bifurcations of the Ro¨ssler system. The uncontrolled Ro¨ssler has two equilibrium solutions, one of which exhibits static bifurcation while the other has Hopf bifurcation. When a feedback control based on the new method is applied, one can delay the bifurcations and even change the type of bifurcations. An optimal control law is obtained to stabilize the Ro¨ssler system using all feasible system parameter values. Numerical simulations are used to verify the analytical results.

scholarly journals Control of Chaos in the Rössler System

1997 ◽  
Vol 50 (2) ◽  
pp. 263 ◽  
Author(s):  
Stuart Corney
Keyword(s):  
Periodic Orbits ◽  
Time Series Data ◽  
Control Method ◽  
Three Dimensional ◽  
Dynamical Variable ◽  
Series Data ◽  
External Parameter ◽  
Unstable Periodic Orbits ◽  
System A ◽  
Linear Differential

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.

Experimental verification of horseshoes from electronic circuits

1995 ◽  
Vol 353 (1701) ◽  
pp. 59-64 ◽  
Keyword(s):  
Dynamical Systems ◽  
Real Time ◽  
Experimental Verification ◽  
Electronic Circuit ◽  
Nonlinear Dynamical Systems ◽  
Electronic Circuits ◽  
Poincaré Maps ◽  
Nonlinear Dynamical ◽  
Poincare Maps ◽  
Return Maps

Poincaré maps are an important tool in analysing the behaviour of nonlinear dynamical systems. If the system to be investigated is an electronic circuit or can be modelled by an electronic circuit, these maps can be visualized on an oscilloscope thereby facilitating real-time investigations. In this paper, sequences of return maps eventually leading to horseshoes are described. These maps are experimentally taken both from non-autonomous and autonomous circuits.

scholarly journals Template analysis of a nonlinear dynamo

2011 ◽  
Vol 468 (2137) ◽  
pp. 288-302 ◽  
Author(s):  
Irene M. Moroz
Keyword(s):  
Symmetry Breaking ◽  
Periodic Orbits ◽  
Linear Series ◽  
Lorenz Attractor ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Template Analysis ◽  
Parameter Values

In this paper, we extend our previous template analysis of a self-exciting Faraday disc dynamo with a linear series motor to the case of a nonlinear series motor. This introduces two additional nonlinear symmetry-breaking terms into the governing dynamo equations. We investigate the consequences for the identification of a possible template on which the unstable periodic orbits (UPOs) lie. By computing Gauss linking numbers between pairs of UPOs, we show that their values are not incompatible with those for a template for the Lorenz attractor for its classic parameter values.

A GEOMETRIC MODEL FOR THE DUFFING OSCILLATOR

1993 ◽  
Vol 03 (03) ◽  
pp. 685-691 ◽  
Author(s):  
J.W.L. McCALLUM ◽  
R. GILMORE
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Attractor ◽  
Geometric Model ◽  
Duffing Oscillator ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Full Period ◽  
Parameter Values

A geometric model for the Duffing oscillator is constructed by analyzing the unstable periodic orbits underlying the chaotic attractors present at particular parameter values. A template is constructed from observations of the motion of the chaotic attractor in a Poincaré section as the section is swept for one full period. The periodic orbits underlying the chaotic attractor are found and their linking numbers are computed. These are compared with the linking numbers from the template and the symbolic dynamics of the orbits are identified. This comparison is used to validate the template identification and label the orbits by their symbolic dynamics.

scholarly journals STABILITY AND STABILIZATION OF NONLINEAR DYNAMICAL SYSTEMS

2017 ◽  
Vol 20 (1) ◽  
pp. 61-70
Author(s):  
P. Sattayatham ◽  
R. Saelim ◽  
S. Sujitjorn
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Feedback Controllers ◽  
Nonlinear Dynamical ◽  
System A ◽  
Continuous Feedback ◽  
Stability And Stabilization ◽  
The Stability

Exponential and asymptotic stability for a class of nonlinear dynamical systems with uncertainties is investigated.  Based on the stability of the nominal system, a class of bounded continuous feedback controllers is constructed.  By such a class of controllers, the results guarantee exponential and asymptotic stability of uncertain nonlinear dynamical system.  A numerical example is also given to demonstrate the use of the main result.

A Comparison of Analytical Methods for Studying a Single-DOF Nonlinear System

1997 ◽  
Author(s):  
Ramesh S. Guttalu ◽  
Henryk Flashner
Keyword(s):  
Stability Analysis ◽  
Nonlinear Dynamical Systems ◽  
Point Mapping ◽  
Nonlinear Dynamical ◽  
Averaging Methods ◽  
Direct Numerical Solution ◽  
Parameter Values ◽  
Support Motion ◽  
Comparison Of Analytical Methods

Abstract Qualitative analysis of periodic nonlinear dynamical systems is often carried out using perturbation and averaging methods. Important topics of study for this class of systems are determination of periodic solutions, their stability analysis and their bifurcation characteristics. Since perturbation and averaging methods rely on expanding the solution in terms of a small parameter, important questions such as convergence and accuracy of the solution arise which are often difficult to answer. This paper presents a preliminary study of the comparison of the method of averaging, Liapunov-Floquet transformation, and point mapping. The methods are applied for the analysis of a damped pendulum subjected to periodic support motion. To evaluate the accuracy of various methods, the results of stability analysis are compared with direct numerical solution for different parameter values of the system.

scholarly journals REACTION PATHS AND ELEMENTARY BIFURCATIONS TRACKS: THE DIABATIC 1B2-STATE OF OZONE

2006 ◽  
Vol 16 (07) ◽  
pp. 1913-1928 ◽  
Author(s):  
S. C. FARANTOS ◽  
ZHENG-WANG QU ◽  
H. ZHU ◽  
R. SCHINKE
Keyword(s):  
Periodic Orbits ◽  
Chemical Reactions ◽  
Nonlinear Dynamical Systems ◽  
Equilibrium Points ◽  
Excited Electronic State ◽  
Quantum Mechanical ◽  
Energy Localization ◽  
Nonlinear Dynamical ◽  
Bifurcation Phenomena ◽  
Families Of Periodic Orbits

Bifurcations of equilibrium points and periodic orbits are common in nonlinear dynamical systems when some parameters change. The vibrational motions of a molecule are nonlinear, and the bifurcation phenomena are seen in spectroscopy and chemical reactions. Bifurcations may lead to energy localization in specific bonds, and thus, they have important consequences for elementary chemical reactions, such as isomerization and dissociation/association. In this article we investigate how elementary bifurcations, such as saddle-node and pitchfork bifurcations, appear in small molecules and show their manifestations in the quantum mechanical frequencies and in the topology of wave functions. We present the results of classical and quantum mechanical calculations on a new (diabatic) potential energy surface of ozone for the 1 B 2 state.This excited electronic state of ozone is pertinent for the absorption of the harmful UV radiation from the sun. We demonstrate that regular localized overtone states, which extend from the bottom of the well up to the dissociation or isomerization barrier, are associated with families of periodic orbits emanated from elementary bifurcations.

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