Local theory of the slanted homoclinic snaking bifurcation diagram

2008 ◽  
Vol 78 (3) ◽  
Author(s):  
U. Bortolozzo ◽  
M. G. Clerc ◽  
S. Residori
Keyword(s):  
Bifurcation Diagram ◽  
Local Theory ◽  
Homoclinic Snaking

scholarly journals BIFURCATIONS AND CHAOS IN A PERIODIC PREDATOR-PREY MODEL

1992 ◽  
Vol 02 (01) ◽  
pp. 117-128 ◽  
Author(s):  
YU.A. KUZNETSOV ◽  
S. MURATORI ◽  
S. RINALDI
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Diagram ◽  
Local Theory ◽  
Predator Prey ◽  
Varying Parameters ◽  
Routes To Chaos ◽  
Predator Prey Model ◽  
Prey Model ◽  
Continuation Technique

The model most often used by ecologists to describe interactions between predator and prey populations is analyzed in this paper with reference to the case of periodically varying parameters. A complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique. The results perfectly agree with the local theory of periodically forced Hopf bifurcation. The two classical routes to chaos, i.e., cascade of period doublings and torus destruction, are numerically detected.

scholarly journals Complexity and Chimera States in a Network of Fractional-Order Laser Systems

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 341
Author(s):  
Shaobo He ◽  
Hayder Natiq ◽  
Santo Banerjee ◽  
Kehui Sun
Keyword(s):  
Numerical Simulation ◽  
Numerical Solution ◽  
Bifurcation Diagram ◽  
Fractional Order ◽  
Dynamical Model ◽  
Chimera States ◽  
Laser Systems ◽  
Complexity Algorithm ◽  
Derivative Order

By applying the Adams-Bashforth-Moulton method (ABM), this paper explores the complexity and synchronization of a fractional-order laser dynamical model. The dynamics under the variance of derivative order q and parameters of the system have examined using the multiscale complexity algorithm and the bifurcation diagram. Numerical simulation outcomes demonstrate that the system generates chaos with the decreasing of q. Moreover, this paper designs the coupled fractional-order network of laser systems and subsequently obtains its numerical solution using ABM. These solutions have demonstrated chimera states of the proposed fractional-order laser network.

scholarly journals Theorems and Application of Local Activity of CNN with Five State Variables and One Port

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Gang Xiong ◽  
Xisong Dong ◽  
Li Xie ◽  
Thomas Yang
Keyword(s):  
Bifurcation Diagram ◽  
Complex Dynamics ◽  
Nonlinear Dynamical Systems ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Local Activity ◽  
Homogeneous Lattice ◽  
Coupled Cells

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.

Local Theory

1972 ◽  
pp. 1-135
Author(s):  
Roger Godement ◽  
Hervé Jacquet
Keyword(s):  
Local Theory

Geometry of the bifurcations of the normalized reduced Henon–Heiles family

1982 ◽  
Vol 382 (1783) ◽  
pp. 361-371 ◽  
Keyword(s):  
Bifurcation Diagram ◽  
Main Resonance ◽  
Fourth Degree ◽  
Global Geometry ◽  
Geometric Explanation ◽  
Hamiltonian Functions

This paper deals with the global geometry of the bifurcations of a family of Hamiltonian functions that arises from normalizing the Henon–Heiles family to fourth-degree terms and then performing a reduction. This gives a geometric explanation of the bifurcation diagram for the main resonance in the model of axisymmetric galaxies of Braun and Verhulst.

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.

scholarly journals Torsional Characteristics of Carbon Nanotubes: Micropolar Elasticity Models and Molecular Dynamics Simulation

Nanomaterials ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 453
Author(s):  
Razie Izadi ◽  
Meral Tuna ◽  
Patrizia Trovalusci ◽  
Esmaeal Ghavanloo
Keyword(s):  
Carbon Nanotubes ◽  
Molecular Dynamics ◽  
Scale Effects ◽  
Couple Stress Theory ◽  
Dynamics Simulation ◽  
Local Theory ◽  
Beam Model ◽  
Specific Class ◽  
Micropolar Theory ◽  
Material Parameters

Efficient application of carbon nanotubes (CNTs) in nano-devices and nano-materials requires comprehensive understanding of their mechanical properties. As observations suggest size dependent behaviour, non-classical theories preserving the memory of body’s internal structure via additional material parameters offer great potential when a continuum modelling is to be preferred. In the present study, micropolar theory of elasticity is adopted due to its peculiar character allowing for incorporation of scale effects through additional kinematic descriptors and work-conjugated stress measures. An optimisation approach is presented to provide unified material parameters for two specific class of single-walled carbon nanotubes (e.g., armchair and zigzag) by minimizing the difference between the apparent shear modulus obtained from molecular dynamics (MD) simulation and micropolar beam model considering both solid and tubular cross-sections. The results clearly reveal that micropolar theory is more suitable compared to internally constraint couple stress theory, due to the essentiality of having skew-symmetric stress and strain measures, as well as to the classical local theory (Cauchy of Grade 1), which cannot accounts for scale effects. To the best of authors’ knowledge, this is the first time that unified material parameters of CNTs are derived through a combined MD-micropolar continuum theory.

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