Transverse Homoclinic Connections for Geodesic Flows

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

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BASIN ORGANIZATION PRIOR TO A TANGLED SADDLE-NODE BIFURCATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127491000087 ◽

1991 ◽

Vol 01 (01) ◽

pp. 107-118 ◽

Cited By ~ 25

Author(s):

MOHAMED S. SOLIMAN ◽

J. M. T. THOMPSON

Keyword(s):

Fractal Structure ◽

Saddle Node Bifurcation ◽

Saddle Node ◽

Homoclinic Connections

Heteroclinic and homoclinic connections of saddle cycles play an important role in basin organization. In this study, we outline how these events can lead to an indeterminate jump to resonance from a saddle-node bifurcation. Here, due to the fractal structure of the basins in the vicinity of the saddle-node, we cannot predict to which available attractor the system will jump in the presence of even infinitesimal noise.

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Expansive geodesic flows on surfaces

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007264 ◽

1993 ◽

Vol 13 (1) ◽

pp. 153-165 ◽

Cited By ~ 9

Author(s):

Miguel Paternain

Keyword(s):

Geodesic Flow ◽

Compact Surface ◽

Conjugate Points ◽

Riemannian Surface ◽

Geodesic Flows ◽

Flows On Surfaces

AbstractWe prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

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Canards in piecewise-linear systems: explosions and super-explosions

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2012.0603 ◽

2013 ◽

Vol 469 (2154) ◽

pp. 20120603 ◽

Cited By ~ 20

Author(s):

Mathieu Desroches ◽

Emilio Freire ◽

S. John Hogan ◽

Enrique Ponce ◽

Phanikrishna Thota

Keyword(s):

Phase Space ◽

Linear Systems ◽

Limit Cycles ◽

Piecewise Linear ◽

Single Parameter ◽

Van Der Pol ◽

Piecewise Linear Systems ◽

Homoclinic Connections ◽

Limiting Case

We show that a planar slow–fast piecewise-linear (PWL) system with three zones admits limit cycles that share a lot of similarity with van der Pol canards, in particular an explosive growth. Using phase-space compactification, we show that these quasi-canard cycles are strongly related to a bifurcation at infinity. Furthermore, we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single-parameter value and with amplitude ranging from an order of ε to an order of 1, a phenomenon truly associated with the non-smooth character of this system and which we call super-explosion .

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