[Paper Review] Chaotic Dynamics of a Nonlinear Ring Cavity Driven by an External Multi-frequency Signal

A. A. Balyakin, Nikita M. Ryskin|ArXiv.org|Mar 8, 2005

Chaos control and synchronization17 references3 citations

TL;DR

This paper extends the classic Ikeda map to model a nonlinear ring cavity driven by a multi-frequency external signal, deriving coupled Ikeda maps for double-frequency driving. It demonstrates that a weak secondary signal enables effective control over chaotic dynamics, revealing diverse bifurcation routes to chaos—including Feigenbaum period doubling, Ruelle–Takens quasi-periodicity, and crisis-induced transitions—highlighting potential for optical computing applications via dynamic state manipulation.

ABSTRACT

Complex dynamics of a ring cavity filled by the medium with cubic phase non-linearity driven by an external multi-frequency signal is studied. To describe the dynamics of envelope amplitudes the system of coupled Ikeda maps was derived. The results of numerical simulations in the case of two-frequency external signal are presented. Complex dynamics observed in these numerical simulations is expected to be peculiar for other physical systems that can be treated as nonlinear resonators driven by an external force.

Motivation & Objective

• To investigate chaotic dynamics in a nonlinear ring cavity under multi-frequency external excitation, particularly focusing on double-frequency driving.
• To explore how a weak secondary signal can control the primary system's dynamics, including suppression or induction of chaos.
• To analyze bifurcation sequences leading to chaos and characterize the resulting attractor structures in the coupled system.
• To extend the applicability of the Ikeda map framework beyond single-frequency driving to richer, more complex nonlinear behaviors.
• To assess the feasibility of using such systems as optical logic elements by analyzing controllable transitions between periodic, quasi-periodic, and chaotic states.

Proposed method

• Derives coupled Ikeda maps using Hamiltonian formalism for multi-frequency wave packets in a nonlinear medium with cubic phase nonlinearity.
• Applies plane wave and slow envelope approximations to reduce the system to a set of coupled nonlinear difference equations for complex amplitudes.
• Uses the four-wave interaction model with non-resonant frequencies to ensure no three-wave resonant interactions, enabling derivation of coupled non-linear Schrödinger equations.
• Transforms the PDE system into a discrete map framework by assuming periodic recirculation and time-delayed feedback, leading to the coupled Ikeda map equations.
• Performs stability analysis of fixed points and periodic orbits in the parameter space of input amplitudes and relative phases.
• Employs numerical simulations with bifurcation diagrams (Re(A₁ⁿ) vs. A₀₁) to explore transitions across periodic, quasi-periodic, and chaotic regimes.

Experimental results

Research questions

• RQ1 How does the introduction of a second, weak frequency component alter the chaotic dynamics of a nonlinear ring cavity driven by a primary signal?
• RQ2 What types of bifurcation sequences lead to chaos in a double-frequency driven nonlinear ring cavity, and how do they differ from the single-frequency Ikeda case?
• RQ3 Can the secondary signal be used to control the onset and nature of chaos, including suppression or induction of periodic windows?
• RQ4 What role does the relative phase between the two input frequencies play in shaping the attractor structure and transition pathways?
• RQ5 How do crisis events and hard transitions between chaotic attractors manifest in the multi-frequency driven system?

Key findings

• The system exhibits a rich variety of bifurcation sequences to chaos, including Feigenbaum period doubling, Ruelle–Takens quasi-periodic route, and hard transitions via crisis events.
• At A₀₁ ≈ 1.2, a period-doubling cascade leads to chaos via the Feigenbaum scenario, with subsequent crisis-induced transitions observed at A₀₁ ≈ 1.63.
• Quasi-periodic motion appears softly via Hopf bifurcations at A₀₁ ≈ 0.8–0.9 and disappears at A₀₁ ≈ 1.1–1.2, depending on the relative phase φ₁.
• A hard transition to the Ikeda attractor occurs at A₀₁ ≈ 2.25 (φ₁ = π/6), A₀₁ ≈ 2.3 (φ₁ = 3π/2), and A₀₁ ≈ 2.3 (φ₁ = π), indicating robustness of the final chaotic state.
• Synchronization windows with various periods are observed within quasi-periodic regions, and reverse period-doubling sequences occur during the return to periodic motion.
• The secondary signal acts as an effective control parameter: varying its amplitude and phase allows suppression of chaos or induction of instability, enabling dynamic state switching.