非线性接触哈密顿系统中几何诱导的自适应耗散：理论及在杜芬振子中的应用
Geometry Induced Adaptive Dissipation in Non Linear Contact Hamiltonian Systems Theory and Application to Duffing Oscillator
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中文总结 AI 辅助
研究通过扩展正则接触哈密顿力学开发广义接触哈密顿框架，以杜芬振子为应用，表明非线性接触几何控制有效耗散，为构建和分析非线性耗散哈密顿系统提供几何框架。
中文摘要 AI 辅助
我们通过用光滑非线性接触势取代经典线性接触势，将正则接触哈密顿力学扩展到非线性耗散系统，从而开发了一个广义接触哈密顿框架。所提出的公式建立了广义能量耗散定律以及可允许接触诱导阻尼的结构表征，引入有效接触耗散作为自适应耗散的内在几何度量。作为应用，推导了广义接触杜芬振子，其中耗散从接触几何中固有产生而非现象学引入。对广义接触杜芬振子的数值研究表明，非线性接触几何控制有效耗散并在相空间结构、能量衰减、动力学稳定性和长期非线性行为中产生显著变化。因此，所提出的理论为在接触哈密顿力学中构建和分析非线性耗散哈密顿系统提供了一个系统的几何框架。
英文摘要
We develop a generalized contact Hamiltonian framework by extending canonical contact Hamiltonian mechanics to nonlinear dissipative systems through the replacement of the classical linear contact potential with a smooth nonlinear contact potential. The proposed formulation establishes a generalized energy dissipation law together with a structural characterization of admissible contact-induced damping, introducing effective contact dissipation as an intrinsic geometric measure of adaptive dissipation. As an application, a generalized contact Duffing oscillator is derived, in which dissipation emerges intrinsically from the contact geometry rather than being introduced phenomenologically. Numerical investigations of the generalized contact Duffing oscillator demonstrate that nonlinear contact geometry governs the effective dissipation and produces significant changes in the phase-space structure, energy decay, dynamical stability, and long-term nonlinear behavior. The proposed theory therefore provides a systematic geometric framework for constructing and analyzing nonlinear dissipative Hamiltonian systems within contact Hamiltonian mechanics.
