scholarly journals Discretizations of dynamical systems with a saddle-node homoclinic orbit

1996 ◽  
Vol 2 (3) ◽  
pp. 351-365 ◽  
Author(s):  
W.-J. Beyn ◽  
◽  
Y.-K Zou ◽  
Keyword(s):  
Dynamical Systems ◽  
Homoclinic Orbit ◽  
Saddle Node

GEOMETRIC LIFT OF PATHS OF HAMILTONIAN EQUILIBRIA AND HOMOCLINIC BIFURCATION

2012 ◽  
Vol 22 (12) ◽  
pp. 1250304 ◽  
Author(s):  
THOMAS J. BRIDGES
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Homoclinic Orbit ◽  
Nonlinear Coefficient ◽  
Homoclinic Bifurcation ◽  
Intrinsic Curvature ◽  
Planar System ◽  
Classical View ◽  
Four Dimensions ◽  
New Formula

A saddle-center transition of eigenvalues in the linearization about Hamiltonian equilibria, and the attendant planar homoclinic bifurcation, is one of the simplest and most well-known bifurcations in dynamical systems theory. It is therefore surprising that anything new can be said about this bifurcation. In this tutorial, the classical view of this bifurcation is reviewed and the lifting of the planar system to four dimensions gives a new view. The principal practical outcome is a new formula for the nonlinear coefficient in the normal form which generates the homoclinic orbit. The new formula is based on the intrinsic curvature of the lifted path of equilibria.

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

2017 ◽  
Vol 33 (1) ◽  
pp. 113-135 ◽  
Author(s):  
Fabíolo Moraes Amaral ◽  
Luís Fernando C. Alberto ◽  
Josaphat R. R. Gouveia
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Points ◽  
Stability Boundary ◽  
Saddle Node ◽  
The Stability ◽  
Autonomous Dynamical Systems

scholarly journals A non-transverse homoclinic orbit to a saddle-node equilibrium

1996 ◽  
Vol 16 (3) ◽  
pp. 431-450 ◽  
Author(s):  
Alan R. Champneys ◽  
Jörg Härterich ◽  
Björn Sandstede
Keyword(s):  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Bifurcation Curve ◽  
Numerical Computations ◽  
Parameter Region ◽  
Codimension Two ◽  
Saddle Node ◽  
Unstable Manifolds ◽  
The Creation ◽  
Shift Dynamics

AbstractA homoclinic orbit is considered for which the center-stable and center-unstable manifolds of a saddle-node equilibrium have a quadratic tangency. This bifurcation is of codimension two and leads generically to the creation of a bifurcation curve defining two independent transverse homoclinic orbits to a saddle-node. This latter case was shown by Shilnikov to imply shift dynamics. It is proved here that in a large open parameter region of the codimension-two singularity, the dynamics are completely described by a perturbation of the Hénon-map giving strange attractors, Newhouse sinks and the creation of the shift dynamics. In addition, an example system admitting this bifurcation is constructed and numerical computations are performed on it.

A Note on Phase Coherence of Chaotic Sets

2016 ◽  
Vol 26 (14) ◽  
pp. 1650243
Author(s):  
Xiao-Song Yang ◽  
Tiantian Wu
Keyword(s):  
Dynamical Systems ◽  
Mild Condition ◽  
Homoclinic Orbit ◽  
Continuous Time ◽  
Chaotic Motion ◽  
Phase Coherence ◽  
Heteroclinic Cycle ◽  
Geometric Description ◽  
Geometric Setting

The phase coherence phenomenon of chaotic motion is unique to continuous time dynamical systems and of significance in many disciplines such as nonlinear physics and biology. In this paper, we present a geometric description of phase coherence of chaotic motion and show that chaotic sets near a homoclinic orbit or heteroclinic cycle are phase coherent in this geometric setting under a mild condition.

Automatic Detection of Saddle-Node–Transcritical Interactions

2019 ◽  
Vol 29 (08) ◽  
pp. 1950104
Author(s):  
Lennaert van Veen ◽  
Marvin Hoti
Keyword(s):  
Dynamical Systems ◽  
Automatic Detection ◽  
Special Structure ◽  
Test Functions ◽  
Test Function ◽  
Predator Prey ◽  
Codimension Two ◽  
Codimension One ◽  
Saddle Node ◽  
Transcritical Bifurcations

Dynamical systems with special structure can exhibit transcritical bifurcations of codimension one. In such systems, the interactions of transcritical bifurcations of codimension two can act as organizing centers. We consider saddle-node–transcritical interactions with either one or two zero eigenvalues and show that, using default test functions, the widely used continuation packages MatCont and AUTO classify these interactions as cusp and Bogdanov–Takens bifurcations, respectively. We propose a new test function that distinguishes these singularities and demonstrate its use in the analysis of a predator–prey-nutrient model strained by a toxicant. The details of the implementation are provided, along with test codes for MatCont.

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