Integrals and chaos in generalized Hénon-Heiles Hamiltonians

• DOI: 10.1088/1402-4896/adbb2a
• Publication Date: 2025-03-13T16:22:30Z
• Authors: G Contopoulos; A C Tzemos; F Zanias

Abstract We study the approximate (formal) integrals of motion in Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mfenced close=")" open="("> <mml:msup> <mml:mover accent="true"> <mml:mi>x</mml:mi> <mml:mo>̇</mml:mo> </mml:mover> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> </mml:mfenced> <mml:mi>ϵ</mml:mi> <mml:mspace width="0.25em"/> <mml:mi>α</mml:mi> <mml:mn>3</mml:mn> </mml:math> which is an extension usual Hénon-Heiles that has α = − 1/3. compare theoretical surfaces section (at y 0) with exact calculated by integrating numerically many orbits. For small ϵ , invariant curves and are close to each other, but for large there differences. The most important appearance chaos case, becomes dominant as approaches escape perturbation &lt; 0. particular cases 1/3, represents integrable system, Finally, we examine generation through resonance overlap mechanism case 1/3 (the original system) showing both homoclinic heteroclinic intersection asymptotic unstable periodic