Bifurcation analysis of two coupled periodically driven Duffing oscillators

Spatiotemporal spread of perturbations in a driven dissipative Duffing chain: An out-of-time-ordered correlator approach

Out-of-time-ordered correlators (OTOCs) have been extensively used as a major tool for exploring quantum chaos, and recently there has been a classical analog. Studies have been limited to closed systems. In this work, we probe an open classical many-body system, more specifically, a spatially extended driven dissipative chain of coupled Duffing oscillators using the classical OTOC to investigate the spread and growth (decay) of an initially localized perturbation in the chain. Correspondingly, we find three distinct types of dynamical behavior: the sustained chaos, transient chaos, and nonchaotic region, as clearly exhibited by different geometrical shapes in the OTOC heat map. To quantify such differences, we look at instantaneous speed (IS), finite-time Lyapunov exponents (FTLEs), and velocity-dependent Lyapunov exponents (VDLEs) extracted from OTOCs. Introduction of these quantities turns out to be instrumental in diagnosing and demarcating different regimes of dynamical behavior. To gain control over open nonlinear systems, it is important to look at the variation of these quantities with respect to parameters. As we tune drive, dissipation, and coupling, FTLEs and IS exhibit transition between sustained chaos and nonchaotic regimes with intermediate transient chaos regimes and highly intermittent sustained chaos points. In the limit of zero nonlinearity, we present exact analytical results for the driven dissipative harmonic system, and we find that our analytical results can very well describe the nonchaotic regime as well as the late-time behavior in the transient regime of the Duffing chain. We believe that this analysis is an important step forward towards understanding nonlinear dynamics, chaos, and spatiotemporal spread of perturbations in many-particle open systems.

Multi-branched resonances, chaos through quasiperiodicity, and asymmetric states in a superconducting dimer

A system of two identical superconducting quantum interference devices (SQUIDs) symmetrically coupled through their mutual inductance and driven by a sinusoidal field is investigated numerically with respect to dynamical properties such as its multibranched resonance curve, its bifurcation structure and transition to chaos as well as its synchronization behavior. The SQUID dimer is found to exhibit a hysteretic resonance curve with a bubble connected to it through Neimark-Sacker (torus) bifurcations, along with coexisting chaotic branches in their vicinity. Interestingly, the transition of the SQUID dimer to chaos occurs through a torus-doubling cascade of a two-dimensional torus (quasiperiodicity-to-chaos transition). Periodic, quasiperiodic, and chaotic states are identified through the calculated Lyapunov spectrum and illustrated using Lyapunov charts on the parameter plane of the coupling strength and the frequency of the driving field. The basins of attraction for chaotic and non-chaotic states are determined. Bifurcation diagrams are constructed on the parameter plane of the coupling strength and the frequency of the driving field, and they are superposed to maps of the three largest Lyapunov exponents on the same plane. Furthermore, the route of the system to chaos through torus-doubling bifurcations and the emergence of Hénon-like chaotic attractors are demonstrated in stroboscopic diagrams obtained with varying driving frequency. Moreover, asymmetric states that resemble localized synchronization have been detected using the correlation function between the fluxes threading the loop of the SQUIDs.

Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique

A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

Dynamic Analysis of Modified Duffing System via Intermittent External Force and Its Application

Over the past century, a tremendous amount of work on the Duffing system has been done with continuous external force, including analytical and numerical solution methods, and the dynamic behavior of physical systems. However, hows does the Duffing oscillator behave if the external force is intermittent? This paper investigates the Duffing oscillator with intermittent external force, and a modified Duffing chaotic system is proposed. Different from the continuous-control method, an intermittent external force of cosine function was designed to control the Duffing oscillator, such that the modified Duffing (MD) system could behave chaotically. The dynamic characteristics of MD system, such as the strange attractors, Lyapunov exponent spectra, and bifurcation diagram spectra were outlined with numerical simulations. Numerical results showed that there existed a positive Lyapunov exponent in some parameter intervals. Furthermore, by combining it with chaos scrambling and chaos XOR encryption, a chaos-based encryption algorithm was designed via the pseudorandom sequence generated from the MD. Finally, feasibility and validity were verified by simulation experiments of image encryption.

Vibrational resonance in an inhomogeneous medium with periodic dissipation

The role of nonlinear dissipation in vibrational resonance (VR) is investigated in an inhomogeneous system characterized by a symmetric and spatially periodic potential and subjected to nonuniform state-dependent damping and a biharmonic driving force. The contributions of the parameters of the high-frequency signal to the system's effective dissipation are examined theoretically in comparison to linearly damped systems, for which the parameter of interest is the effective stiffness in the equation of slow vibration. We show that the VR effect can be enhanced by varying the nonlinear dissipation parameters and that it can be induced by a parameter that is shared by the damping inhomogeneity and the system potential. Furthermore, we have apparently identified the origin of the nonlinear-dissipation-enhanced response: We provide evidence of its connection to a Hopf bifurcation, accompanied by monotonic attractor enlargement in the VR regime.

Nonlinear normal modes and localization in two bubble oscillators

We investigated a bifurcation structure of coupled nonlinear oscillation of two spherical gas bubbles subject to a stationary sound field by means of nonlinear modal analysis. The goal of this paper is to describe an energy localization phenomenon of coupled two-bubble oscillators, resulting from symmetry-breaking bifurcation of the steady-state oscillation. Approximate asymptotic solutions of nonlinear normal modes (NNMs) and steady state oscillation are obtained based on the method of multiple scales. It is found that localized oscillation arises in a neighborhood of the localized normal modes. The analytical solutions of the amplitude and the phase shift of the steady-state oscillation are compared to numerical results and found to be in good agreement within the limit of small-amplitude oscillation. For larger amplitude oscillation, a bifurcation diagram of the localized solution as a function of the driving frequency and the separation distance between the bubbles is provided in the presence of the thermal damping. The numerical results show that the localized oscillation can occur for a fairly typical parameter range used in practical experiments and simulations in the early literatures.

Analysis of vibrational resonance in bi-harmonically driven plasma

The phenomenon of vibrational resonance (VR) is examined and analyzed in a bi-harmonically driven two-fluid plasma model with nonlinear dissipation. An equation for the slow oscillations of the system is analytically derived in terms of the parameters of the fast signal using the method of direct separation of motion. The presence of a high frequency externally applied electric field is found to significantly modify the system's dynamics, and consequently, induce VR. The origin of the VR in the plasma model has been identified, not only from the effective plasma potential but also from the contributions of the effective nonlinear dissipation. Beside several dynamical changes, including multiple symmetry-breaking bifurcations, attractor escapes, and reversed period-doubling bifurcations, numerical simulations also revealed the occurrence of single and double resonances induced by symmetry breaking bifurcations.

Spectral element method and the delayed feedback control of chaos

A spectral element approach is introduced to determine the Floquet exponents (FEs) of unstable periodic orbits (UPOs) stabilized by extended delayed feedback control (EDFC). The spectral approach does not require solving time-dependent eigenproblems that existing methods require. Instead, the spectral approach determines the stability of the delay differential equations of the system by numerical approximation. The method is capable of analyzing systems whose UPOs arise from bifurcations other than period-doubling. Results are presented for stabilizing UPOs in Duffing systems. The FEs calculated by the spectral approach are compared to published results for two examples. In both cases, the spectral method results agree well with those determined by previous methods. In addition, the spectral method was used to analyze a high-dimensional, asymmetrical system with a UPO in chaos arising from tori doubling following a Hopf bifurcation.

Nonadiabatic dynamics of two strongly coupled nanomechanical resonator modes

The Landau-Zener transition is a fundamental concept for dynamical quantum systems and has been studied in numerous fields of physics. Here, we present a classical mechanical model system exhibiting analogous behavior using two inversely tunable, strongly coupled modes of the same nanomechanical beam resonator. In the adiabatic limit, the anticrossing between the two modes is observed and the coupling strength extracted. Sweeping an initialized mode across the coupling region allows mapping of the progression from diabatic to adiabatic transitions as a function of the sweep rate.

Stochastic resonance in coupled underdamped bistable systems

We study onset and control of stochastic resonance (SR) phenomenon in two driven bistable systems, mutually coupled and subjected to independent noises, taking into account the influence of both the inertia and the coupling. In the absence of coupling, we found two critical damping parameters: one for the onset of SR and another for which SR is optimum. We then show that in weakly coupled systems, emergence of SR is governed by chaos. A strong coupling between the two oscillators induces coherence in the system; however, the systems do not synchronize no matter what the coupling is. Moreover, a specific coupling parameter is found for which the SR of each subsystem is optimum. Finally, a scheme for controlling SR in such coupled systems is proposed by introducing a phase difference between the two coherent driving forces.

Period doubling and spatiotemporal chaos in periodically forced CO oxidation on Pt(110)

Periodic forcing of chemical turbulence in the catalytic CO oxidation on Pt(110) can induce a period doubling cascade to chaos. Using a forcing frequency near the second harmonic of the system's natural frequency, and carefully increasing the forcing amplitude, the system successively exhibits spiral wave turbulence, resonant pattern formation, and chaotic oscillations. In the latter case, global coupling induces strong spatial correlation. Experimental results are presented as well as numerical simulations using a realistic model. Good agreement is found between experiment and theory. The results give further insight into the complex nature of reaction-diffusion systems and are of high importance regarding control strategies on such systems. The presented setup enhances the range of achievable dynamical states and allows for new experimental investigations on the dynamics of extended oscillatory systems.

Bifurcation and chaos in coupled ratchets exhibiting synchronized dynamics

The bifurcation and chaotic behavior of unidirectionally coupled deterministic ratchets is studied as a function of the driving force amplitude and frequency . A classification of the various types of bifurcations likely to be encountered in this system was done by examining the stability of the steady state in linear response as well as constructing a two-parameter phase diagram in the plane. Numerical explorations revealed varieties of bifurcation sequences including quasiperiodic route to chaos. Besides, the familiar period-doubling and crises route to chaos exhibited by the one-dimensional ratchet were also found. In addition, the coupled ratchets display symmetry-breaking, saddle-nodes and bubbles of bifurcations. Chaotic behavior is characterized by using the Lyapunov exponent spectrum; while a perusal of the phase space projected in the Poincaré cross section confirms some of the striking features.