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 A NUMERICAL TOOLBOX FOR HOMOCLINIC BIFURCATION ANALYSIS

1996 ◽  
Vol 06 (05) ◽  
pp. 867-887 ◽  
Author(s):  
A.R. CHAMPNEYS ◽  
YU. A. KUZNETSOV ◽  
B. SANDSTEDE
Keyword(s):  
Numerical Analysis ◽  
Chemical Kinetics ◽  
Bifurcation Analysis ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Homoclinic Bifurcation ◽  
Codimension Two ◽  
Codimension One ◽  
Homoclinic Bifurcations ◽  
Theoretical Results

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

Two Simplest Quadratic Chaotic Maps Without Equilibrium

2018 ◽  
Vol 28 (12) ◽  
pp. 1850144 ◽  
Author(s):  
Shirin Panahi ◽  
Julien C. Sprott ◽  
Sajad Jafari
Keyword(s):  
Dynamical System ◽  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic Maps ◽  
Basin Of Attraction ◽  
Hidden Attractor ◽  
Return Map ◽  
Right Hand ◽  
Dynamical Properties ◽  
The Right

Two simple chaotic maps without equilibria are proposed in this paper. All nonlinearities are quadratic and the functions of the right-hand side of the equations are continuous. The procedure of their design is explained and their dynamical properties such as return map, bifurcation diagram, Lyapunov exponents, and basin of attraction are investigated. These maps belong to the hidden attractor category which is a newly introduced category of dynamical system.

scholarly journals Numerical daemons of hydrological models are summoned by extreme precipitation

2021 ◽  
Author(s):  
Peter T. La Follette ◽  
Adriaan J. Teuling ◽  
Nans Addor ◽  
Martyn Clark ◽  
Koen Jansen ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Extreme Precipitation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Computational Cost ◽  
Numerical Error ◽  
Hydrological Models ◽  
Multiple Model ◽  
Numerical Techniques ◽  
Modeling Framework

Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on approximate numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation like that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation events. In this experiment, a large number of hydrographs is generated with the modular modeling framework FUSE, using eight numerical techniques across a variety of forcing datasets. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational expense and numerical error associated with each hydrograph were recorded. It was found that numerical error (root mean square error) usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both low numerical error and low computational cost. A basic literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be sub-optimal. We conclude that relatively large numerical errors might be common in current models, and because these will likely become larger as the climate changes, we advocate for the use of low cost, low error numerical methods.

scholarly journals Numerical methods for kinetic equations

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 369-520 ◽  
Author(s):  
G. Dimarco ◽  
L. Pareschi
Keyword(s):  
Computational Complexity ◽  
Numerical Methods ◽  
State Of The Art ◽  
Kinetic Equations ◽  
Numerical Techniques ◽  
Asymptotic Preserving ◽  
Hybrid Schemes ◽  
Velocity Models ◽  
Current State ◽  
Number Of Particles

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

scholarly journals Note on the Sensitivity of Simulated Solutions of the Nonlinear Singularly Perturbed Dynamical System on the Used Different Rewriting into a System of First Order Differential Equations

2016 ◽  
Vol 24 (39) ◽  
pp. 51-55
Author(s):  
Vladimír Liška ◽  
Zuzana Šútova ◽  
Dušan Pavliak
Keyword(s):  
Dynamical System ◽  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Numerical Scheme ◽  
Singularly Perturbed ◽  
First Order ◽  
First Order Differential Equations

Abstract In this paper we analyze the sensitivity of solutions to a nonlinear singularly perturbed dynamical system based on different rewriting into a System of the First Order Differential Equations to a numerical scheme. Numerical simulations of the solutions use numerical methods implemented in MATLAB.

scholarly journals Interaction of homoclinic solutions and Hopf points in functional differential equations of mixed type

2009 ◽  
Vol 139 (1) ◽  
pp. 123-155 ◽  
Author(s):  
Marc Georgi
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Homoclinic Orbit ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
Homoclinic Orbits ◽  
Homoclinic Solution ◽  
Equations Of Mixed Type ◽  
Functional Differential ◽  
Equation Of Mixed Type

We study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are diffcult to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov–Schmidt reduction is not possible.As one of the main results we prove the existence of centre-stable and centre-unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the case for ordinary differential equations and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions that exist on account of the Hopf bifurcation, for exactly one asymptotic direction t→8 or t→−∞.

BIFURCATIONS OF HOMOCLINIC ORBIT CONNECTING TWO NONLEADING EIGENDIRECTIONS

2007 ◽  
Vol 17 (03) ◽  
pp. 823-836 ◽  
Author(s):  
TIANSI ZHANG ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Period Doubling ◽  
Homoclinic Bifurcation ◽  
Period Doubling Bifurcation ◽  
Dimensional System ◽  
Bifurcation Surface

Bifurcations of homoclinic orbit connecting the strong stable and strong unstable directions are investigated for four-dimensional system. The existence, numbers, co-existence and incoexistence of 1-homoclinic orbit, 2n-homoclinic orbit, 1-periodic orbit and 2n-periodic orbit are obtained, and the bifurcation surfaces (including codimension-1 homoclinic bifurcation surfaces, double periodic orbit bifurcation surfaces, homoclinic-doubling bifurcation surfaces, period-doubling bifurcation surfaces and codimension-2 triple periodic orbit bifurcation surface, and homoclinic and double periodic orbit bifurcation surface) and the existence regions are also located.

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.

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