Routes to chaos in phase-modulated resonant tunnelling diode optoelectronic oscillators (RTD-OEOs)

Abstract

Abstract This work uncovers how adaptive phase modulation of an externally injected signal fundamentally restructures the nonlinear dynamical regimes of the classical RTD-OEOs. Starting from a Liénard-type voltage-controlled model, a dynamical phase function is introduced into the harmonic driving term, such that the injected signal takes the form where the phase depends linearly on an internal dynamical variable (voltage, current, carrier density or photon density) through the modulation coefficient, p . This formulation establishes an additional coupling between the electrical and optical subsystems. Unless otherwise stated, the delayed optical feedback is disabled ( α = 0) to isolate the effect of adaptive phase modulation and reduce the governing equations to a finite-dimensional system. We analysed the system dynamics in two-parameter spaces spanned by the driving frequency and amplitude, driving frequency and noise strength and the modulation coefficient and amplitude. Methods of charts of Lyapunov exponent, charts of dynamical regime, and Poincaré bifurcation trees are used to characterize the observed behaviour. Our results reveal the coexistence of periodic, quasiperiodic, and chaotic regimes, organized in structured regions of parameter space. Two distinct routes to chaos are identified. In the absence of the modulation term, the system exhibits a quasiperiodic route to chaos typical of externally driven RTD oscillators. When the modulation term is introduced, a period-doubling cascade emerges due to phase-dependence. The proposed phase-modulated formulation transforms the classical quasiperiodic route observed in the unmodulated RTD-OEO into a period-doubling route as the modulation coefficient increases. The observed regimes are corroborated by representative time series, phase portraits, strange attractors, and broadband continues Fourier spectra in both electrical and optical domains. This offers a new perspective on the control of high-frequency RTD-based optoelectronic oscillators. Beyond advancing their theoretical understanding, the results addressed in this work provide practical strategies for engineering controllable chaos and synchronization, with direct implications for microwave-photonic technologies, neuromorphic dynamics.