Singular fractal dimension at periodicity cascades in parameters spaces

Carlos E. P. Abreu; Joelson D. V. Hermes; Diogo Ricardo da Costa; Everton S. Medeiros; Rene O. Medrano-T

Home/Archive/arXiv

arXiv · 2401.07648

Singular fractal dimension at periodicity cascades in parameters spaces

Carlos E. P. Abreu·Joelson D. V. Hermes·Diogo Ricardo da Costa·Everton S. Medeiros·Rene O. Medrano-T

View original source ↗Read source document ↗Export metadata ↓

Abstract

In the parameter spaces of nonlinear dynamical systems, we investigate the boundaries between periodicity and chaos and unveil the existence of fractal sets characterized by a singular fractal dimension. This dimension stands out from the typical fractal dimensions previously considered universal for these parameter boundaries. We show that the singular fractal sets dwell along parameter curves, called extreme curves, that intersect periodicity cascades at their center of stability in all scales of parameters spaces. The results reported here are generally demonstrated for the class of one-dimensional maps with at least two control parameters, generalizations to other classes of systems are possible.

Explore related subjects

nlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Carlos E. P. Abreu, Joelson D. V. Hermes, Diogo Ricardo da Costa, Everton S. Medeiros, Rene O. Medrano-T. 2024-01-15. Singular fractal dimension at periodicity cascades in parameters spaces. https://arxiv.org/abs/2401.07648

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Final state sensitivity and fractal basin boundaries from coupled Chialvo neurons

We investigate and quantify the basin geometry and extreme final state uncertainty of two identical electrically asymmetrically coupled Chialvo neurons. The system's diverse behaviors are presented, along with the mathematical reasoning behind its chaotic and nonchaotic dynamics as determined by the structure of the coupled equations. The system is found to be multistable with two qualitatively different attractors. Although each neuron is individually nonchaotic, the chaotic basin takes up the vast majority of the coupled system's state space, but the nonchaotic basin stretches to infinity due to chance synchronization. The boundary between the basins is found to be fractal, leading to extreme final state sensitivity. This uncertainty and its potential effect on the synchronization of biological neurons may have implications for understanding neuronal biology.

nlin.CD↗

arXiv2026

Jordan-Block Degeneracy and Cubic-Order Bifurcating Periodic Orbits in Minimum-Energy Optimal Control of Hamiltonian Equilibria

Equilibria of the Hamiltonian system associated with Pontryagin's minimum principle exhibit an exact doubling of the natural spectrum and, under a simple pairing condition, a Jordan block at every simple purely imaginary eigenvalue. Consequently, the classical Lyapunov Center Theorem does not apply to the augmented system, and no periodic orbit with nonzero optimal control bifurcates at linear order. We establish this mechanism in general and show that an optimal-control-induced periodic family emerges at cubic order in a Lindstedt--Poincaré expansion. The mechanism is illustrated in closed form for the pendulum and evaluated numerically for the planar $L_2$ equilibrium of Hill's restricted three-body problem, where the third-order approximation is validated against an independently computed family of periodic orbits.

nlin.CD↗

arXiv2026

Hypersensitivity and Turnpikes in Optimal Control of Inverted Pendulum: A Dynamical Systems Perspective

The hypersensitivity and turnpike phenomena in the optimal control of an inverted pendulum are investigated from a dynamical-systems perspective. We show that, for a fixed terminal time and a fixed terminal state optimal control problem, (1) the hypersensitivity originates from the fractal structure of the set of initial adjoint variables in the associated Hamiltonian dynamics, (2) the turnpike arises from slow dynamics in the vicinity of a degenerate center manifold, and (3) the escape channels are formed by normally hyperbolic invariant manifolds (NHIMs). As a consequence, small perturbations in the initial adjoint variables lead to qualitatively distinct extremal trajectories, resulting in severe numerical instability in trajectory optimization. Both the fractal structure and the invariant sets are characterized numerically and analytically.

nlin.CD↗