Higher-order Melnikov theory for adiabatic systems

Journal Article · Sun Dec 01 04:00:00 EST 1996 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.531751· OSTI ID:397462

Soto-Trevino, C; Kaper, T J ^[1]

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1. Department of Mathematics, Boston University, Boston, Massachusetts 02215 (United States)

In this paper, we study adiabatic Hamiltonian systems including those subject to small-amplitude forcing and damping. It is known that simple zeroes of the adiabatic Poincare{endash}Arnold{endash}Melnikov function imply the existence of primary intersection points of the stable and unstable manifolds of hyperbolic orbits. Here, we present an {ital N}th-order Melnikov function whose simple zeroes correspond to {ital N}th-order transverse intersection points and hence to {ital N}-pulse homoclinic orbits. Using this function, it can be shown that {ital N}-pulse homoclinic orbits arise in a plethora of adiabatic models, including systems with slowly varying potentials. The theory is illustrated on a damped Hamiltonian system with a slowly varying cubic potential. In addition, the {ital N}th-order adiabatic Melnikov function is useful for showing the existence of multi-pulse resonant periodic orbits in the special class of slow, time-periodic systems. {copyright} {ital 1996 American Institute of Physics.}

• OSTI ID: 397462
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 37; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
ADIABATIC INVARIANCE
BIFURCATION
CLASSICAL MECHANICS
MATHEMATICAL MANIFOLDS
PERTURBATION THEORY
PHASE SPACE