scholarly journals Algrithm for detecting homoclinic orbits of time-continuous dynamical system and its application

2013 ◽  
Vol 62 (10) ◽  
pp. 100501
Author(s):  
Yang Fang-Yan ◽  
Hu Ming ◽  
Yao Shang-Ping
Keyword(s):  
Dynamical System ◽  
Homoclinic Orbits ◽  
Continuous Dynamical System

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.

Stationary waves on an inclined sheet of viscous fluid at high Reynolds and moderate Weber numbers

1996 ◽  
Vol 307 ◽  
pp. 191-229 ◽  
Author(s):  
Jeng-Jong Lee ◽  
Chiang C. Mei
Keyword(s):  
Dynamical System ◽  
Phase Speed ◽  
Propagation Speed ◽  
Reynolds Numbers ◽  
Homoclinic Orbits ◽  
Finite Amplitude ◽  
Stationary Waves ◽  
Asymptotic Equations ◽  
Wave Propagation Speed ◽  
High Reynolds

A theory is described for the nonlinear waves on the surface of a thin film flowing down an inclined plane. Attention is focused on stationary waves of finite amplitude and long wavelength at high Reynolds numbers and moderate Weber numbers. Based on asymptotic equations accurate to the second order in the depth-to-wavelength ratio, a third-order dynamical system is obtained after changing to the frame of reference moving at the wave propagation speed. By examining the fixed-point stability of the dynamical system, parametric regimes of heteroclinc orbits and Hopf bifurcations are delineated. Extensive numerical experiments guided by the linear analyses reveal a variety of bifurcation scenarios as the phase speed deviates from the Hopf-bifurcation thresholds. These include homoclinic bifurcations which lead to homoclinic orbits corresponding to well separated solitary waves with one or several humps, some of which occur after passing through chaotic zones generated by period-doublings. There are also cases where chaos is the ultimate state following cascades of period-doublings, as well as cases where only limit cycles prevail. The dependence of bifurcation scenarios on the inclination angle, and Weber and Reynolds numbers is summarized.

scholarly journals Multiple Bifurcations of a Cylindrical Dynamical System

2016 ◽  
Vol 46 (1) ◽  
pp. 33-52
Author(s):  
Ning Han ◽  
Qingjie Cao
Keyword(s):  
Dynamical System ◽  
Limit Cycles ◽  
Homoclinic Orbits ◽  
Complex Dynamic ◽  
Pendulum System ◽  
Sd Oscillator ◽  
Bistable State ◽  
Different Types ◽  
Coupling Dynamics ◽  
Dynamic Bifurcation

Abstract This paper focuses on multiple bifurcations of a cylindrical dynamical system, which is evolved from a rotating pendulum with SD oscillator. The rotating pendulum system exhibits the coupling dynamics property of the bistable state and conventional pendulum with the ho- moclinic orbits of the first and second type. A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected by using analytical analysis and nu- merical calculation. It is found that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles, coexisting with the oscillating limit cycle. Additionally, the results obtained herein, are helpful to explore different types of limit cycles and the complex dynamic bifurcation of cylindrical dynamical system.

HOMOCLINIC BIFURCATIONS IN A PLANAR DYNAMICAL SYSTEM

2001 ◽  
Vol 11 (04) ◽  
pp. 1183-1191 ◽  
Author(s):  
FOTIOS GIANNAKOPOULOS ◽  
TASSILO KÜPPER ◽  
YONGKUI ZOU
Keyword(s):  
Dynamical System ◽  
Numerical Methods ◽  
Bifurcation Diagram ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Intersection Point ◽  
Homoclinic Bifurcation ◽  
Numerical Techniques ◽  
Planar Dynamical System ◽  
Homoclinic Bifurcations

The homoclinic bifurcation properties of a planar dynamical system are analyzed and the corresponding bifurcation diagram is presented. The occurrence of two Bogdanov–Takens bifurcation points provides two local existing curves of homoclinic orbits to a saddle excluding the separatrices not belonging to the homoclinic orbits. Using numerical techniques, these curves are continued in the parameter space. Two further curves of homoclinic orbits to a saddle including the separatrices not belonging to the homoclinic orbits are calculated by numerical methods. All these curves of homoclinic orbits have a unique intersection point, at which there exists a double homoclinic orbit. The local homoclinic bifurcation diagram of both the double homoclinic orbit point and the points of homoclinic orbits to a saddle-node are also gained by numerical computation and simulation.

scholarly journals The influence of autotoxicity on the dynamics of vegetation spots

2020 ◽  
Author(s):  
Annalisa Iuorio ◽  
Frits Veerman
Keyword(s):  
Dynamical System ◽  
Reaction Diffusion ◽  
Homoclinic Orbits ◽  
Spatial Dimension ◽  
Travelling Wave ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Scaling Analysis ◽  
Water Models ◽  
Vegetation Water

AbstractPlant autotoxicity has proved to play an essential role in the behaviour of local vegetation. We analyse a reaction-diffusion-ODE model describing the interactions between vegetation, water, and autotoxicity. The presence of autotoxicity is seen to induce movement and deformation of spot patterns in some parameter regimes, a phenomenon which does not occur in classical biomass-water models. We aim to analytically quantify this novel feature, by studying travelling wave solutions in one spatial dimension. We use geometric singular perturbation theory to prove the existence of symmetric, stationary and non-symmetric, travelling pulse solutions, by constructing appropriate homoclinic orbits in the associated 5-dimensional dynamical system. In the singularly perturbed context, we perform an extensive scaling analysis of the dynamical system, identifying multiple asymptotic scaling regimes where (travelling) pulses may or may not be constructed. We discuss the agreement and discrepancy between the analytical results and numerical simulations. Our findings indicate how the inclusion of an additional ODE may significantly influence the properties of classical biomass-water models of Klausmeier/Gray–Scott type.

scholarly journals Spectrum of continuous two-contours system

2019 ◽  
Vol 24 ◽  
pp. 01014 ◽  
Author(s):  
Alexander G. Tatashev ◽  
Marina V. Yashina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Common Point ◽  
Continuous Dynamical System ◽  
Point Node ◽  
System States

A deterministic continuous dynamical system is considered. This system contains two contours. The length of theith contour equalsci,i= 1, 2. There is a moving segment (cluster) on each contour. The length of the cluster, located on theith contour, equalsli,i= 1, 2. If a cluster moves without delays, then the velocity of the cluster is equal to 1. There is a common point (node) of the contours. Clusters cannot cross the node simultaneously, and therefore delays of clusters occur. A set of repeating system states is called a spectral cycle. Spectral cycles and values of average velocities of clusters have been found. The system belongs to a class of contour systems. This class of dynamical systems has been introduced and studied by A.P. Buslaev.

Generalized synchronization of continuous dynamical system

2007 ◽  
Vol 28 (2) ◽  
pp. 157-162 ◽  
Author(s):  
Gang Zhang ◽  
Zeng-rong Liu ◽  
Zhong-jun Ma
Keyword(s):  
Dynamical System ◽  
Generalized Synchronization ◽  
Continuous Dynamical System
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