Transition from mode-locked periodic orbit to chaos in a 2D piecewise smooth non-invertible map

Soma De; Soumitro Banerjee; Akhil Ranjan Roy

Home/Archive/arXiv

arXiv · 0812.3778

Transition from mode-locked periodic orbit to chaos in a 2D piecewise smooth non-invertible map

Soma De·Soumitro Banerjee·Akhil Ranjan Roy

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work we report a new route to chaos from a resonance torus in a piecewise smooth non-invertible map of the plane into itself. The closed invariant curve defining the resonance torus is formed by the union of unstable manifolds of saddle cycle and the points of stable cycle and saddle cycle. We have found that a cusp torus cannot develop before the onset of chaos, though the loop torus appears. The destruction of the two-dimensional torus occurs through homoclinic bifurcation in the presence of an infinite number of loops on the invariant curve. We show that owing to the non-invertible nature of the map, the structure of the basin of attraction changes from simply connected to a nonconnected one. We also describe how the mechanism of transition to chaos differs from the scenario of appearance of chaos in invertible maps as well as in smooth non-invertible maps.

Explore related subjects

nlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Soma De, Soumitro Banerjee, Akhil Ranjan Roy. 2008-12-19. Transition from mode-locked periodic orbit to chaos in a 2D piecewise smooth non-invertible map. https://arxiv.org/abs/0812.3778

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

KEEP EXPLORING

Related papers

arXiv2026

Linear Response Predicts Cusp-Pair Births in Networks with a Localized Cubic

Linear response is cheap to measure; the bistability boundaries it organizes are not. For a passive network with one localized cubic, the driving-point receptance $G$ fixes the period-one cusp set at fundamental-harmonic order: cusps lie on a fixed phase contour of $G$, a tangency of that contour under parameter variation creates a pair, and its curvature separates a gap opening from an isolated loop. For a two-mode absorber the linear prediction locates a benchmark birth coupling to $0.3\%$, and to $0.03\%$ once a third-harmonic correction of scale $|G(3\Omega)/G(\Omega)|$ is included.

nlin.CD↗

arXiv2026

Dynamics Creation through Neural Dynamical Transfer Learning

Data-driven machine learning has established a robust foundation for reconstructing nonlinear dynamical systems from observations, primarily for the purposes of forecasting and control. However, most existing efforts focus on recovering specific observed dynamics rather than the generative synthesis of new ones. Inspired by image fusion and style transfer, we introduce a neural network framework termed Neural Dynamical Transfer Learning (NDTL) to create new systems with prescribed dynamics from pairs of parent nonlinear dynamical systems. By computing fundamental dynamical signatures, including the intrinsic dimension, the Kaplan-Yorke dimension, the invariant measure statistics, and the Lyapunov spectrum, we demonstrate that NDTL preserves key features inherited from the parent models while simultaneously generating novel dynamics. Beyond these validation examples, NDTL induces a criterion for dynamics classification, creates stable oscillatory coexistence in the Hastings-Powell food chain model, produces interpretable epidemiological models, and provides a chaotic source for image encryption.

nlin.CD↗

arXiv2026

The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincar\'e Sections

A Poincar\'e section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the R\"ossler system, a 32-mode Kuramoto--Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$--$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$--$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

nlin.CD↗