RESONANT GLUING BIFURCATIONS

2000 ◽  
Vol 10 (09) ◽  
pp. 2141-2160 ◽  
Author(s):  
ROBERT W. GHRIST
Keyword(s):  
Saddle Point ◽  
Parameter Space ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Centre Manifold ◽  
Principal Eigenvalues ◽  
Homoclinic Connections ◽  
Homoclinic Bifurcations ◽  
Zero Entropy

We consider the codimension-three phenomenon of homoclinic bifurcations of flows containing a pair of orbits homoclinic to a saddle point whose principal eigenvalues are in resonance. We concentrate upon the simplest possible configuration, the so-called "figure-of-eight," and reduce the dynamics near the homoclinic connections to those on a two-dimensional locally invariant centre manifold. The ensuing resonant gluing bifurcations exhibit features of both gluing bifurcations and resonant homoclinic bifurcations. Under certain twist conditions, the bifurcation structure is extremely rich, although describing zero-entropy flows. The analysis carefully exploits the topology of the orbits, the centre manifold and the parameter space.

COMMON DYNAMICAL FEATURES OF PERIODICALLY DRIVEN STRICTLY DISSIPATIVE OSCILLATORS

1993 ◽  
Vol 03 (03) ◽  
pp. 703-715 ◽  
Author(s):  
ULRICH PARLITZ
Keyword(s):  
Periodic Orbits ◽  
Parameter Space ◽  
Nonlinear Oscillators ◽  
Two Dimensional ◽  
Topological Properties ◽  
Basic Pattern ◽  
Bifurcation Structure ◽  
Periodically Driven ◽  
Dynamical Features ◽  
The Way

Periodically driven strictly dissipative nonlinear oscillators in general possess a recurring bifurcation structure in parameter space. It consists of slightly modified versions of a basic pattern of bifurcation curves that was found to be essentially the same for many different oscillators. The periodic orbits involved in these bifurcation scenarios also possess common topological properties characterized in terms of their torsion numbers and the way they are connected when parameters are varied. In this paper, this typical bifurcation structure of periodically driven strictly dissipative oscillators will be presented and discussed in terms of examples from Duffing’s equation. Furthermore a family of two-dimensional maps is given that models (strictly) dissipative oscillators and shows essential features of the bifurcation pattern found.

scholarly journals Homoclinic and heteroclinic bifurcations in a two-dimensional endomorphism

2003 ◽  
Vol 16 (2) ◽  
pp. 273-283
Author(s):  
Ilham Djellit ◽  
Mohamed Ferchichi
Keyword(s):  
Saddle Point ◽  
Chaotic Attractor ◽  
Invariant Manifolds ◽  
Basin Of Attraction ◽  
Homoclinic Orbits ◽  
Specific Information ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Homoclinic Bifurcations ◽  
Noninvertible Maps

Our study concerns global bifurcations occurring in noninvertible maps, it consists to show that there exists a link between contact bifurcations of a chaotic attractor and homoclinic bifurcations of a saddle point or a saddle cycle being on the boundary of the chaotic attractor. We provide specific information about the intricate dynamics near such points. We study particularly a two-dimensional endomorphism of (Z\ - Z$ - Z\) type. We will show that points of contact, between boundary of the attractor and its basin of attraction, converge toward the saddle point or the saddle cycle. These points of contact are also points of intersection between the stable and unstable invariant manifolds. This gives rise to the birth of homoclinic orbits (homoclinic bifurcations).

A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

1993 ◽  
Vol 03 (02) ◽  
pp. 293-321 ◽  
Author(s):  
JÜRGEN WEITKÄMPER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Parallel Computers ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Logistic Mapping ◽  
Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 239-244
Author(s):  
SERGE GALAM
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Parameter Space ◽  
Landau Theory ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Theory Analysis ◽  
Continuous Transition ◽  
Dimensional Parameter ◽  
First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

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.

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.

BI-SPIRALING HOMOCLINIC CURVES AROUND A T-POINT IN CHUA'S EQUATION

2004 ◽  
Vol 14 (05) ◽  
pp. 1789-1793 ◽  
Author(s):  
FERNANDO FERNÁNDEZ-SÁNCHEZ ◽  
EMILIO FREIRE ◽  
ALEJANDRO J. RODRÍGUEZ-LUIS
Keyword(s):  
Two Dimensional ◽  
Homoclinic Connections ◽  
Parameter Bifurcation

In this work, the existence of curves of homoclinic connections that bi-spiral around a T-point between two saddle-focus equilibria is detected in Chua's equation. That is, the homoclinic curve emerges spiraling from a T-point in a parameter bifurcation plane and ends, by a different spiral, at the same T-point. This new phenomenon is related to the existence of more than one intersection between the two-dimensional manifolds of the involved equilibria at the T-point.

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