scholarly journals Homoclinic tangency and variation of entropy

2020 ◽  
Vol 77 (3) ◽  
pp. 383-398
Author(s):  
Marcus Bronzi ◽  
Ali Tahzibi
Keyword(s):  
Homoclinic Tangency

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

HOMOCLINIC BIFURCATION OF MULTIHARMONIC PERTURBED SYSTEMS

2001 ◽  
Vol 11 (02) ◽  
pp. 541-550
Author(s):  
KE WANG ◽  
YUN TANG
Keyword(s):  
Normal Forms ◽  
Singularity Theory ◽  
Melnikov Function ◽  
Homoclinic Bifurcation ◽  
Homoclinic Tangency ◽  
Perturbed System ◽  
Perturbed Systems ◽  
Harmonic Components ◽  
Homoclinic Tangencies

The dynamics of homoclinic tangencies for a multiharmonic perturbed system (MHPS) is studied. By applying the singularity theory to Melnikov function the normal forms at the homoclinic tangency points are obtained, and then the details of the homoclinic bifurcation are quite clear. An important result is that the behavior of homoclinic bifurcation for MHPS is strongly related to the number of harmonic components.

Homoclinic Orbits in the Complex Domain

1997 ◽  
Vol 07 (02) ◽  
pp. 253-274 ◽  
Author(s):  
V. F. Lazutkin ◽  
C. Simó
Keyword(s):  
Homoclinic Orbits ◽  
Homoclinic Tangency ◽  
Complex Domain ◽  
Standard Map ◽  
Homoclinic Points ◽  
Promising Tool ◽  
Theoretical Support ◽  
Area Preserving Maps ◽  
Area Preserving

We consider the standard map, as a paradigm of area preserving map, when the variables are taken as complex. We study how to detect the complex homoclinic points, which cannot dissappear under a homoclinic tangency. This seems a promising tool to understand the stochastic zones of area preserving maps. The paper is mainly phenomenological and includes theoretical support to the observed phenomena. Several conjectures are stated.

scholarly journals Homoclinic points and moduli

1989 ◽  
Vol 9 (2) ◽  
pp. 389-398 ◽  
Author(s):  
Rense A. Posthumus
Keyword(s):  
Dynamical Systems ◽  
Homoclinic Tangency ◽  
Two Dimensional ◽  
Homoclinic Points

AbstractIn this paper we study some conjugacy invariants (moduli) for discrete two dimensional dynamical systems, with a homoclinic tangency. We show that the modulus obtained by Palis in the heteroclinic case also turns up in the case considered here. We also present two new conjugacy invariants.

Chaotic Motions of a Dual-Spin Body

1998 ◽  
Vol 65 (1) ◽  
pp. 150-156 ◽  
Author(s):  
A. C. Or
Keyword(s):  
Steady State ◽  
Numerical Integration ◽  
Coulomb Friction ◽  
Equations Of Motion ◽  
Homoclinic Tangency ◽  
Chaotic Attractors ◽  
Chaotic Motions ◽  
Homoclinic Points ◽  
Melnikov’S Method

The dynamics of a dual-spinner subject to the action of an internal oscillatory torque and Coulomb friction between the two linked bodies is investigated. The conditions for existence of transverse homoclinic points and homoclinic tangency of chaotic motions are obtained using Melnikov’s method. Through long-term numerical integration of the equations of motion, steady-state chaotic attractors are also found and studied numerically by varying the forcing amplitude.

scholarly journals On bifurcations of multidimensional diffeomorphisms having a homoclinic tangency to a saddle-node

2014 ◽  
Vol 19 (4) ◽  
pp. 461-473
Author(s):  
Serey V. Gonchenko ◽  
Olga V. Gordeeva ◽  
Valery I. Lukyanov ◽  
Ivan I. Ovsyannikov
Keyword(s):  
Homoclinic Tangency ◽  
Saddle Node

COMPLEXITY OF HOMOCLINIC BIFURCATIONS AND Ω-MODULI

1996 ◽  
Vol 06 (06) ◽  
pp. 969-989 ◽  
Author(s):  
S. V. GONCHENKO ◽  
O. V. STEN’KIN ◽  
D. V. TURAEV
Keyword(s):  
Parameter Analysis ◽  
Homoclinic Tangency ◽  
Topological Conjugacy ◽  
Two Dimensional ◽  
Control Parameters ◽  
Homoclinic Bifurcations ◽  
Complete Study ◽  
Splitting Parameter ◽  
Two Parameter ◽  
Nonwandering Set

Bifurcations of two-dimensional diffeomorphisms with a homoclinic tangency are studied in one-and two-parameter families. Due to the well-known impossibility of a complete study of such bifurcations, the problem is restricted to the study of the bifurcations of the so-called low-round periodic orbits. In this connection, the idea of taking Ω-moduli (continuous invariants of the topological conjugacy on the nonwandering set) as the main control parameters (together with the standard splitting parameter) is proposed. In this way, new bifurcational effects are found which do not occur at a one-parameter analysis. In particular, the density of cusp-bifurcations is revealed.

On dynamic properties of diffeomorphisms with homoclinic tangency

2005 ◽  
Vol 126 (4) ◽  
pp. 1317-1343 ◽  
Author(s):  
S. V. Gonchenko ◽  
D. V. Turaev ◽  
L. P. Shil’nikov
Keyword(s):  
Dynamic Properties ◽  
Homoclinic Tangency
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