Dynamical Chaos in a Dissipative Driven Quantum Soft Impact Oscillator

TL;DR

This paper investigates chaos in a periodically driven, dissipative soft impact oscillator within a quantum framework using the c-number quantum Langevin equation to access averaged operator dynamics. It develops a semiclassical approach by mapping a classical piecewise-smooth impact model onto an open quantum system with memory and colored noise, enabling dynamical-system diagnostics on the quantum-averaged variables. The main finding is the persistence of grazing-induced chaos under quantum dissipation, with chaos windows identified via bifurcation diagrams, Lyapunov exponents, FFT spectra, and the 0-1 test. The work provides a tractable framework for probing quantum signatures of non-smooth chaos and highlights experimental platforms such as AFM cantilevers and cavity optomechanics for validation.

Abstract

Dynamical chaos in a periodically driven, dissipative soft impact oscillator is investigated in the quantum regime using the complex-number quantum Langevin equation (c-number QLE). The averaged system dynamics are analyzed through a comprehensive suite of time-series diagnostics, including bifurcation diagrams, Lyapunov exponents, Fourier spectra, and the 0-1 test. Systematic variation of the wall position reveals a rich sequence of dynamical transitions and grazing bifurcations, progressing from periodic to multiperiodic motion and culminating in chaotic behavior. These results demonstrate the persistence of impact-induced chaos under quantum dissipation and elucidate how environmental fluctuations influence non-linear dynamics in open quantum systems.

Figures (7)

• Figure 1: Schematic diagram of a soft impact oscillator.
• Figure 2: Graphical representation of the soft impact potential
• Figure 3: (a) Potential, (b) first-order derivative of potential, i.e., force on the particle, and (c) second-order derivative of potential, solid lines represent the original form, and the dots represent the form derived from sigmoid approximation
• Figure 4: Different order contributions to the quantum correction term $Q(t)$ for (a) $\hbar = 0.01$, (b) $\hbar = 0.1$, and (c) $\hbar = 1.0$, evaluated at $x_{\text{wall}} = 0.5$ and $\Omega = 0.5$. In each column, the top, middle, and bottom panels correspond to the second, third, and fourth-order correction terms, respectively.
• Figure 5: (a) Bifurcation diagram obtained by varying the wall position $x_{\text{wall}}$ for $\Omega = 0.5$, using the Poincaré section at $P = 0$. The color indicates the Lyapunov exponent, where green tones correspond to $\lambda \le 0$ (regular motion) and orange tones to $\lambda > 0$ (chaotic motion). Panels (b)–(g) show representative trajectories at selected wall positions, marked in (a) by vertical lines with circular markers: (b) chaotic trajectory at $x_{\text{wall}} = 0.40$ (purple), (c) period-three motion at $x_{\text{wall}} = 0.50$ (red), (d) chaotic trajectory at $x_{\text{wall}} = 1.14$ (blue), (e) period-three motion at $x_{\text{wall}} = 1.18$ (yellow), (f) period-two motion at $x_{\text{wall}} = 1.75$ (gray), and (g) periodic trajectory at $x_{\text{wall}} = 1.90$ (black).