Double grazing bifurcation route in a quasiperiodically driven piecewise linear oscillator

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Bifurcation Theory

Background:

• Piecewise linear oscillators exhibit complex dynamics under quasiperiodic excitation.
• Understanding transitions to chaos and strange nonchaotic attractors (SNAs) is crucial in nonlinear systems.

Purpose of the Study:

• To investigate the route from quasiperiodic torus to strange nonchaotic attractors (SNAs) via double grazing bifurcation.
• To analytically determine the maximum displacement for this bifurcation.
• To characterize the properties of the resulting SNAs.

Main Methods:

• Analysis of a piecewise linear oscillator with quasiperiodic excitation.
• Derivation of analytical expressions for maximum displacement during double grazing bifurcation.
• Numerical verification of SNA properties using Lyapunov exponent, phase sensitivity, power spectrum, and fractal structure.

Main Results:

• Identified the double grazing bifurcation as a pathway to SNAs from quasiperiodic torus.
• Demonstrated that SNAs emerge as the quasiperiodic torus loses smoothness and becomes non-differentiable.
• Confirmed the nonchaotic nature of SNAs via Lyapunov exponent and their strange characteristics through spectral and fractal analyses.
• Observed SNAs in various parameter intervals, including a significant range not leading to quasiperiodic or chaotic orbits.

Conclusions:

• The double grazing bifurcation provides a novel route to SNAs in piecewise linear oscillators.
• SNAs can manifest in diverse parameter regimes, offering a broader understanding of their occurrence.
• The study analytically and numerically characterizes the transition and properties of SNAs, contributing to nonlinear dynamics research.