scholarly journals Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

2018 ◽  
Vol 38 (9) ◽  
pp. 4483-4507 ◽  
Author(s):  
Amadeu Delshams ◽  
◽  
Marina Gonchenko ◽  
Sergey V. Gonchenko ◽  
J. Tomás Lázaro ◽  
...  
Keyword(s):  
Homoclinic Tangencies

CHUA’S CIRCUIT: RIGOROUS RESULTS AND FUTURE PROBLEMS

1994 ◽  
Vol 04 (03) ◽  
pp. 489-519 ◽  
Author(s):  
LEONID P. SHIL’NIKOV
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complex Dynamics ◽  
Multidimensional Systems ◽  
Chua’S Circuit ◽  
Rigorous Results ◽  
Chua's Circuit ◽  
Mathematical Problems ◽  
Homoclinic Tangencies

Mathematical problems arising from the study of complex dynamics in Chua’s circuit are discussed. An explanation of the extreme complexity of the structure of attractors of Chua’s circuit is given. This explanation is based upon recent results on systems with homoclinic tangencies. A number of new dynamical phenomena is predicted for those generalizations of Chua’s circuits which are described by multidimensional systems of ordinary differential equations.

scholarly journals On bifurcations of area-preserving and nonorientable maps with quadratic homoclinic tangencies

2014 ◽  
Vol 19 (6) ◽  
pp. 702-717 ◽  
Author(s):  
Amadeu Delshams ◽  
Marina Gonchenko ◽  
Sergey V. Gonchenko
Keyword(s):  
Homoclinic Tangencies ◽  
Area Preserving

Persistent homoclinic tangencies for conservative maps near the identity

2000 ◽  
Vol 20 (2) ◽  
pp. 393-438 ◽  
Author(s):  
PEDRO DUARTE
Keyword(s):  
Hamiltonian Systems ◽  
Time Variable ◽  
Integrable Hamiltonian Systems ◽  
Homoclinic Tangencies

For families of conservative maps near the identity we prove the existence of open sets of parameters with persistence of homoclinic tangencies between stable and unstable leaves of ‘thick’ horse-shoes. Such families are obtained, for instance, by perturbing integrable Hamiltonian systems in $\mathbb{R}^2$ with a rapidly periodic oscillatory term and then performing a slowing change in the time variable.

Homoclinic tangencies and routes to chaos in a dripping faucet experiment

2001 ◽  
Vol 291 (1-4) ◽  
pp. 244-254 ◽  
Author(s):  
R.D. Pinto ◽  
J.C. Sartorelli ◽  
W.M. Gonçalves
Keyword(s):  
Routes To Chaos ◽  
Homoclinic Tangencies

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.

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