Chaotic dynamics and fractal analysis of nonstandard Hamiltonian systems

Complex dynamical systems governed by nonstandard Lagrangians displaying non-natural forms of kinetic energy terms have recently received particular attention due to their relevance in the theory of differential equations and various fields of science and engineering. Such Lagrangians lead to nonstandard Hamiltonians that have relevance in various nonlinear complex dynamical systems governed by autonomous differential equations. Although guessing the forms of nonstandard Lagrangians requires solid mathematical methodologies, new types of these non-natural Lagrangians have been introduced recently in literature. It is also well-known that nonintegrable Hamiltonian systems with two or more degrees of freedom usually involve chaotic dynamics. Besides, stochasticity and resonance arise in 2-dimensional nonlinear Hamiltonian systems, and the chaos in the stochastic layer is generated by the main resonance interaction. The generation of chaotic trajectories in Hamiltonian systems has Poincaré maps obtained through the Poincaré surface-of-section method used to analyze weakly perturbed Hamiltonian systems. In this study, we study Poincaré maps for two different types of nonstandard Hamiltonians generated from nonstandard Lagrangians, and we analyze some of their relevant chaotic properties based on the largest Lyapunov exponents and the bifurcation diagrams. Additionally, we compute the fractal dimension and Hurst exponent of these Poincaré sections, revealing varying degrees of chaos. The results show that fractal dimension values range between 1.76 and 1.90, while Hurst exponent values remain below 0.5, concerning the presence of anti-persistent chaotic behavior. Several emergent features related to chaotic behavior and fractal structures are observed. Our approach provides a new perspective on assessing the robustness of nonlinear dynamical systems governed by nonstandard Hamiltonians.