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THE SEPARATRIX VALUES OF A PLANAR HOMOCLINIC LOOP

International Journal of Bifurcation and Chaos, 2009
It is well known that the stability of a homoclinic loop for planar vector fields is closely related to the cyclicity of this homoclinic loop. For a planar homoclinic loop consisting of a hyperbolic saddle, the loop values are crucial to the stability. The loop values are divided into two classes: saddle values and separatrix values. The saddle values
Liqin Zhao, Xuexing Wang
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Multidimensional Symplectic Separatrix Maps

Journal of Nonlinear Science, 2002
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Reaction probability for sequential separatrix crossings

Physical Review Letters, 1988
The change of the crossing parameter (essentially the phase) between sequential slow separatrix crossings is calculated for Hamiltonian systems with one degree of freedom. Combined with the previous separatrix crossing analysis, these results reduce the dynamics of adiabatic systems with separatrices to a map.
, Cary, , Skodje
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Application of the separatrix map to study perturbed magnetic field lines near the separatrix

Physics of Plasmas, 1996
In this paper a generalized separatrix map to study perturbed magnetic field lines in a confined magnetic system with the separatrix is discussed. The separatrix map is an iteration transform for the magnetic flux and a toroidal angle at a certain cross section after one period of the unperturbed trajectory.
S. S. Abdullaev, G. M. Zaslavsky
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Quantum dynamics in a chaotic separatrix layer

Physical Review A, 1991
The chaotic dynamics in a separatrix layer are investigated using the ``separatrix map.'' Quantizing this map we show that localization effects inhibit the quantum transport across the separatrix layer. Depending on the size of the nonintegrability parameter, the mechanism of localization is connected with either the remnants of the barely broken ...
, Bubner, , Graham
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Separatrix Chaos

2004
Abstract In low dimensional Hamiltonian systems, the area near a separatrix is the most important domain to study the origin of chaos, since the dynamics near a saddle point is sensitive to small perturbations. The problem of the change of the dynamics near a separatrix under a periodic perturbation can be studied in fairly general way ...
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Stochastic broadening of the separatrix of a tokamak divertor

Physical Review Letters, 1992
The plasma in a modern tokamak is bounded by a separatrix between magnetic field lines that form toroidal magnetic surfaces, on which the plasma is confined, and open field lines that divert the plasma exhaust to so-called divertor plates. This separatrix is sharp in an ideal tokamak, but we show that magnetic perturbations create a stochastic region ...
Punjabi, A., Verma, A., Boozer, A.
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Shadowing in a neighborhood of a separatrix

Differential Equations, 2013
The author analyzes certain type of diffeomorphisms of the plane into itself possessing two hyperbolic fixed points joined by a separatrix. It is proven that, under additional assumptions, there is an \(\varepsilon>0\) such that for every \(\delta>0\), there is a special finite \(\delta\)-pseudo-orbit that is not \(\varepsilon\)-shadowed by any point ...
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Quantum dynamics near a classical separatrix

Physical Review A, 1993
Quantum-mechanical evolution in a slowly varying double-well potential is analyzed in the semiclassical limit to determine the transition probability due to loss of adiabatic invariance of states having energy close to that of a separatrix in the equivalent classical system.
, Cary, , Rusu
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A geometric proof of separatrix crossing results

Nonlinear Analysis: Theory, Methods & Applications, 2004
The authors consider a family of one degree of freedom Hamiltonian systems as follows: \[ q'=H_p(q, p; \lambda),\quad p'=-H_q(q, p; \lambda), \quad \lambda'=\varepsilon, \] where \(\varepsilon>0\) is a small parameter, \(\lambda\in [0, 1]\) is a parameter, \(H(p, q; \lambda)\) is the Hamiltonian function such that for each \(\lambda\) the unperturbed ...
Chow, Shui-Nee, Young, Todd
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