scholarly journals No Chaos in Dixon’s System

2021 ◽  
Vol 31 (03) ◽  
pp. 2150044
Author(s):  
Werner M. Seiler ◽  
Matthias Seiß
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Homoclinic Orbits ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Continuous Dynamical System ◽  
Two Parameters

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

BIFURCATION ANALYSIS IN A PWM CURRENT-CONTROLLED H-BRIDGE INVERTER

2011 ◽  
Vol 21 (03) ◽  
pp. 985-996 ◽  
Author(s):  
HIROYUKI ASAHARA ◽  
TAKUJI KOUSAKA
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Bifurcation Analysis ◽  
Specific Property ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Rigorous Analysis ◽  
Bifurcation Phenomena ◽  
The One ◽  
Switched Dynamical System

This paper introduces the complete bifurcation analysis in a PWM current-controlled H-Bridge inverter in a wide parameter space. First, we briefly explain the behavior of the waveform in the circuit in terms of the switched dynamical system. Then, the consecutive waveform during the duration of the clock interval is exactly discretized, and the return map is defined for the rigorous analysis. Using the map, we derive the one- and two-dimensional bifurcation diagrams, and discuss the specific property of each bifurcation phenomena in the circuit.

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (02) ◽  
pp. 399-404 ◽  
Author(s):  
T. SÜNNER ◽  
H. SAUERMANN
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Dynamical Behaviour ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

scholarly journals Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950111 ◽  
Author(s):  
Mohammed-Salah Abdelouahab ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Search Algorithm ◽  
Two Dimensional ◽  
Mixed Mode Oscillations ◽  
The Stability ◽  
Two Parameters ◽  
Complex Phenomena ◽  
Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

Effects of DC electric fields on neuronal excitability: A bifurcation analysis

2014 ◽  
Vol 28 (18) ◽  
pp. 1450114 ◽  
Author(s):  
Yanqiu Che ◽  
Huiyan Li ◽  
Chunxiao Han ◽  
Xile Wei ◽  
Bin Deng ◽  
...  
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Parameter Space ◽  
Bifurcation Analysis ◽  
Neural Control ◽  
Neuronal Excitability ◽  
Modulation Method ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Dc Electric Fields

In this paper, the effects of external DC electric fields on the neuro-computational properties are investigated in the context of Morris–Lecar (ML) model with bifurcation analysis. We obtain the detailed bifurcation diagram in two-dimensional parameter space of externally applied DC current and trans-membrane potential induced by external DC electric field. The bifurcation sets partition the two-dimensional parameter space in terms of the qualitatively different behaviors of the ML model. Thus the neuron's information encodes the stimulus information, and vice versa, which is significant in neural control. Furthermore, we identify the electric field as a key parameter to control the transitions among four different excitability and spiking properties, which facilitates the design of electric fields based neuronal modulation method.

HORSESHOE CHAOS IN A HYBRID PLANAR DYNAMICAL SYSTEM

2012 ◽  
Vol 22 (08) ◽  
pp. 1250202 ◽  
Author(s):  
QING-JU FAN
Keyword(s):  
Dynamical System ◽  
Cross Section ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Buck Converter ◽  
Two Dimensional ◽  
Topological Horseshoe ◽  
Piecewise Linear System ◽  
Shift Map

In this paper, we study the chaotic dynamics of a voltage-mode controlled buck converter, which is typically a switched piecewise linear system. For the two-dimensional hybrid system, we consider a properly chosen cross-section and the corresponding Poincaré map, and show that the dynamics of the system is semi-conjugate to a 2-shift map, which implies the chaotic behavior of this system. The essential tool is a topological horseshoe theory and numerical method.

BIFURCATIONS AND CHAOS IN CELLULAR NEURAL NETWORKS

2003 ◽  
Vol 12 (04) ◽  
pp. 417-433 ◽  
Author(s):  
M. BIEY ◽  
P. CHECCO ◽  
M. GILLI
Keyword(s):  
Neural Networks ◽  
Bifurcation Analysis ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
First Order ◽  
Stable Periodic ◽  
Torus Breakdown

The dynamic behavior of first-order autonomous space invariant cellular neural networks (CNNs) is investigated. It is shown that complex dynamics may occur in very simple CNN structures, described by two-dimensional templates that present only vertical and horizontal couplings. The bifurcation processes are analyzed through the computation of the limit cycle Floquet's multipliers, the evaluation of the Lyapunov exponents and of the signal spectra. As a main result a detailed and accurate two-dimensional bifurcation diagram is reported. The diagram allows one to distinguish several regions in the parameter space of a single CNN. They correspond to stable, periodic, quasi-periodic, and chaotic behavior, respectively. In particular it is shown that chaotic regions can be reached through two different routes: period doubling and torus breakdown. We remark that most practical CNN implementations exploit first order cells and space-invariant templates: so far only a few examples of complex dynamics and no complete bifurcation analysis have been presented for such networks.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

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