Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems

Laminar chaos in systems with quasiperiodic delay

A type of chaos called laminar chaos was found in singularly perturbed dynamical systems with periodic time-varying delay [Phys. Rev. Lett. 120, 084102 (2018)]0031-900710.1103/PhysRevLett.120.084102. It is characterized by nearly constant laminar phases, which are periodically interrupted by irregular bursts, where the intensity levels of the laminar phases vary chaotically from phase to phase. In this paper, we demonstrate that laminar chaos can also be observed in systems with quasiperiodic delay, where we generalize the concept of conservative and dissipative delays to such systems. It turns out that the durations of the laminar phases vary quasiperiodically and follow the dynamics of a torus map in contrast to the periodic variation observed for periodic delay. Theoretical and numerical results indicate that introducing a quasiperiodic delay modulation into a time-delay system can lead to a giant reduction of the dimension of the chaotic attractors. By varying the mean delay and keeping other parameters fixed, we found that the Kaplan-Yorke dimension is modulated quasiperiodically over several orders of magnitudes, where the dynamics switches quasiperiodically between different types of high- and low-dimensional types of chaos.

Grip force anticipation of nonlinear, underactuated load force

Feedforward internal model-based control enabled by efference copies of motor commands is the prevailing theoretical account of motor anticipation. Grip force control during object manipulation-a paradigmatic example of motor anticipation-is a key line of evidence for that account. However, the internal model approach has not addressed the computational challenges faced by the act of manipulating mechanically complex objects with nonlinear, underactuated degrees of freedom. These objects exhibit complex and unpredictable load force dynamics which cannot be encoded by efference copies of underlying motor commands, leading to the prediction from the perspective of an efference copy-enabled feedforward control scheme that grip force should either lag or fail to coordinate with changes in load force. In contrast to that prediction, we found evidence for strong, precise, anticipatory grip force control during manipulations of a complex object. The results are therefore inconsistent with the internal forward model approach and suggest that efference copies of motor commands are not necessary to enable anticipatory control during active object manipulation.NEW & NOTEWORTHY From the perspective of feedforward internal model-based control, precise, anticipatory grip force (GF) control when manipulating a complex object should not be possible as the object's changing load forces (LFs) cannot be encoded by efference copies of the underlying movements. However, we observed that GF exhibited strong, precise, anticipatory coupling with LF during extended manipulations of a complex object. These findings suggest that an alternative theoretical framework is needed to account for anticipatory GF control.

Oscillating synchronization in delayed oscillators with time-varying time delay coupling: Experimental observation

The time-varying time-delayed (TVTD) systems attract the attention of research communities due to their rich complex dynamics and wide application potentiality. Particularly, coupled TVTD systems show several intriguing behaviors that cannot be observed in systems with a constant delay or no delay. In this context, a new synchronization scenario, namely, oscillating synchronization, was reported by Senthilkumar and Lakshmanan [Chaos 17, 013112 (2007)], which is exclusive to the time-varying time delay systems only. However, like most of the dynamical behavior of TVTD systems, its existence has not been established in an experiment. In this paper, we report the first experimental observation of oscillating synchronization in coupled nonlinear time-delayed oscillators induced by a time-varying time delay in the coupling path. We implement a simple yet effective electronic circuit to realize the time-varying time delay in an experiment. We show that depending upon the instantaneous variation of the time delay, the system shows a synchronization scenario oscillating among lag, complete, and anticipatory synchronization. This study may open up the feasibility of applying oscillating synchronization in engineering systems.

Laminar Chaos in Experiments: Nonlinear Systems with Time-Varying Delays and Noise

A new type of dynamics called laminar chaos was recently discovered through a theoretical analysis of a scalar delay differential equation with time-varying delay. Laminar chaos is a low-dimensional dynamics characterized by laminar phases of nearly constant intensity with periodic durations and a chaotic variation of the intensity from one laminar phase to the next laminar phase. This is in stark contrast to the typically observed higher-dimensional turbulent chaos, which is characterized by strong fluctuations. In this Letter we provide the first experimental observation of laminar chaos by studying an optoelectronic feedback loop with time-varying delay. The noise inherent in the experiment requires the development of a nonlinear Langevin equation with variable delay. The results show that laminar chaos can be observed in higher-order systems, and that the phenomenon is robust to noise and a digital implementation of the variable time delay.

Resonant Doppler effect in systems with variable delay

We demonstrate that a time-varying delay in nonlinear systems leads to a rich variety of dynamical behaviour, which cannot be observed in systems with constant delay. We show that the effect of the delay variation is similar to the Doppler effect with self-feedback. We distinguish between the non-resonant and the resonant Doppler effect corresponding to the dichotomy between conservative delays and dissipative delays. The non-resonant Doppler effect leads to a quasi-periodic frequency modulation of the signal, but the qualitative properties of the solution are the same as for constant delays. By contrast, the resonant Doppler effect leads to fundamentally different solutions characterized by low- and high-frequency phases with a clear separation between them. This is equivalent to time-multiplexed dynamics and can be used to design systems with well-defined multistable solutions or temporal switching between different chaotic and periodic dynamics. We systematically study chaotic dynamics in systems with large dissipative delay, which we call generalized laminar chaos. We derive a criterion for the occurrence of different orders of generalized laminar chaos, where the order is related to the dimension of the chaotic attractor. The recently found laminar chaos with constant plateaus in the low-frequency phases is the zeroth-order case with a very low dimension compared to the known high dimension of turbulent chaos in systems with conservative delay. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Generalized projective synchronization in time-delayed systems: nonlinear observer approach

In this paper, we consider the projective-anticipating, projective, and projective-lag synchronization in a unified coupled time-delay system via nonlinear observer design. A new sufficient condition for generalized projective synchronization is derived analytically with the help of Krasovskii-Lyapunov theory for constant and variable time-delay systems. The analytical treatment can give stable synchronization (anticipatory and lag) for a large class of time-delayed systems in which the response system's trajectory is forced to have an amplitude proportional to the drive system. The constant of proportionality is determined by the control law, not by the initial conditions. The proposed technique has been applied to synchronize Ikeda and prototype models by numerical simulation.

Phase effects on synchronization by dynamical relaying in delay-coupled systems

Synchronization in an array of mutually coupled systems with a finite time delay in coupling is studied using the Josephson junction as a model system. The sum of the transverse Lyapunov exponents is evaluated as a function of the parameters by linearizing the equation about the synchronization manifold. The dependence of synchronization on damping parameter, coupling constant,and time delay is studied numerically. The change in the dynamics of the system due to time delay and phase difference between the applied fields is studied. The case where a small frequency detuning between the applied fields is also discussed.