Smooth Oscillators Research Articles

KCC theory for a type of nonlinear damped SD oscillators

Smooth and discontinuous (SD) oscillators are the nonlinear models that will exhibit SD dynamics due to continuous variation of the smooth parameter. Kosambi–Cartan–Chern (KCC) theory is a differential geometric theory that describes the deviations from the entire trajectory of the variational equation to the nearby trajectory. This paper presents a completely new dynamical analysis of a type of SD oscillators with nonlinear damping based on the KCC theory. First, the paper gives five KCC geometrical invariants of SD oscillators in the smooth and non-smooth cases, respectively. The results show that the geometric quantities in the non-smooth case cannot be derived directly from the geometric quantities in the smooth case. Second, from these geometric quantities, KCC stability of SD oscillators trajectory at any point except [Formula: see text] is analyzed. Lastly, numerical results show that in some regions that deviate from the equilibrium point of system, the system will exhibit complex and variable dynamical behavior due to small changes in parameters. This paper shows that the KCC theory is also a useful tool in the dynamical analysis of non-smooth systems.

Response solutions for finite smooth harmonic oscillators with quasi-periodic forcing

Response solutions for finite smooth harmonic oscillators with quasi-periodic forcing

On the uniqueness of limit cycles in second-order oscillators

In this paper we shall prove a theorem to guarantee the uniqueness of limit cycles for second-order nonlinear differential equations, and give some sufficient conditions for the existence and nonexistence of the limit cycles, respectively. As applications of these results, we will prove the uniqueness of limit cycles for smooth or nonsmooth van der Pol-Rayleigh oscillators, and give their global dynamics.

Properties of generalized synchronization in smooth and non-smooth identical oscillators

This paper deals with the phenomenon of the GS only in the context of unidirectional connection between identical exciter and receivers. A special attention is focused on the properties of the GS in coupled non-smooth Chua circuits. The robustness of the synchronous state is analyzed in the presence of slight parameter mismatch. The analysis tools are transversal and response Lyapunov exponents and fractal dimension of the attractor. These studies show differences in the stability of synchronous states between smooth (Lorenz system) and non-smooth (Chua circuit) oscillators.

Suppressing homoclinic chaos for a weak periodically excited non-smooth oscillator

In this work, some new effective methods for suppressing homoclinic chaos in a weak periodically excited non-smooth oscillator are studied, and the main idea is to modify slightly the Melnikov function such that the zeros are eliminated. Firstly, a general form of planar piecewise-smooth oscillators is given to approximatively model many nonlinear restoring force of smooth oscillators subjected to all kinds of damping and periodic excitations. In the absence of controls, the Melnikov method for non-smooth homoclinic trajectories within the framework of a piecewise-smooth oscillator is briefly introduced without detailed derivation. This analytical tool is useful to detect the threshold of parameters for the existence of homoclinic chaos in the non-smooth oscillator. After some methods of state feedback control, self-adaptive control and parametric excitations control are, respectively, considered, sufficient criteria for suppressing homoclinic chaos are derived by employing the Melnikov function of non-smooth systems. Finally, the effectiveness of strategies for suppressing homoclinic chaos is analytically and numerically demonstrated through a specific example.

Interplay of the mechanisms of synchronization by common noise and global coupling for a general class of limit-cycle oscillators

We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we consider the case of nonidentical oscillators. For the average oscillator frequency as a function of the natural frequency mismatch, limiting scaling laws are derived; these laws exhibit the nontrivial phenomenon of frequency repulsion accompanying synchronization under negative coupling. The analytical theory is illustrated with examples of Van der Pol and Van der Pol–Duffing oscillators and the neuron-like FitzHugh–Nagumo system; the results are also underpinned by the direct numerical simulation for ensembles of these oscillators.

Identification of response branches for oscillators with curved and straight contours executing VIV

For oscillators with curved contours, variation of cylinder amplitude with Reynolds number or reduced speed is adequate to accurately identify the extent of resonance branches and critical points marking the transition from one branch to another. However, in presence of straight edges in body contour, the response curve at certain locations of transition, exhibits smooth variation. The same is seen with smooth contoured oscillators of low mass ratio, i.e. value of mass ratio being close to unity. The frequency curves, in contrast, reflect all the transitions that are not captured in the response variation. A simple method is proposed to accurately predict the branching and associated transition points with the aid of oscillation and shedding frequency curves. It is noted that the frequency profile undergoes a change of slope at the location of a transition. Using the proposed method, branching is analyzed for elliptic, square and truncated circular cylinders of mass ratio 10, each of them executing undamped vortex-induced vibrations (VIV) at low Reynolds numbers (=60-150) simultaneously along and across the flow. Illustrations are extended for a square cylinder of mass ratio 1, too. For the first time, VIV results are presented for a truncated circular cylinder.

Computing Melnikov Curves for Periodically Perturbed Piecewise Smooth Oscillators

Curves dividing the parameter plane into regions according to the presence or absence of homoclinic or heteroclinic tangle corresponding to the periodically perturbed saddle of the piecewise smooth oscillator are studied using Melnikov analysis. The analysis is not simplified by choosing the discontinuity plane at a convenient location. Separatrix of the unperturbed system is parametrized exactly in a piecewise manner. Switching times, i.e. parameter values at which the separatrix crosses the discontinuity plane, are obtained. Switching times split the Melnikov integral into various subintegrals which are evaluated either exactly using term-wise integration of the infinite series of the integrand or approximately using a finite-term series approximation of the integrand, the latter being computationally an extensive task. Integral evaluations though approximate, are purely analytical expressions in terms of special functions such as digamma and hypergeometric. Melnikov plots show that the boundary between three regions in the parameter plane differ qualitatively in case of parametric and external excitations, however; adding self-excitation to the external one does not much alter the boundary qualitatively and quantitatively.

Diverting homoclinic chaos in a class of piecewise smooth oscillators to stable periodic orbits using small parametrical perturbations

Abstract This paper investigates the mechanisms of small parametrical perturbations in controlling chaos in a class of non-autonomous piecewise smooth oscillators, which describe a large class of nonlinear dynamical systems in the real world. The analytical expressions of two homoclinic orbits of unperturbed piecewise smooth oscillators, which connect the same hyperbolic saddle point are solved analytically. Firstly, when there are no small parametrical perturbations, by using Melnikov׳s approach, it is rigorously proven that the homoclinic chaos in the Smale horseshoes sense exists when the system׳s parameters are selected above the threshold for chaos occurrence. Secondly, under the small parametrical perturbations, by using Melnikov׳s approach, a sufficient criterion is derived, serving as designing the parameters of the control signal, i.e., amplitude and phase position. In the process of computing Melnikov׳s functions, it is found that the expressions of Melnikov׳s functions could not be solved analytically because the homoclinic orbits are highly complicated. To this end, a numerical algorithm is proposed. Numerical simulations are presented to verify the theoretical results. The results of this paper can be used to explore the underlying chaotic behaviors of the inertial neural network model.

Analysis of Biological Smooth Oscillators Inspired by the Relay Control Tuning Method

Biological oscillation is generated by different genetic circuit structures, such as the smooth oscillator and the relaxation oscillator. Studies on the complex interactions among genes and proteins are essential for understanding the generating mechanism underlying the circadian clock or other biological oscillators. In this paper, the biological oscillation is investigated from a new perspective. It is studied here, through principles in the nonlinear system and control theory, the important factors for the existence of oscillation in the smooth oscillation system. Conclusions have been drawn that both the degradation rates of genes and proteins and the switch-like behavior of the translation process are essential for generating oscillation. The prediction in the Goodwin oscillators model, that the degradations of the clock mRNA and clock protein play important roles in regulating the oscillation period, is discussed here through classic control theory, i.e., the relay control tuning method.

Clustering in delay-coupled smooth and relaxational chemical oscillators

We investigate cluster synchronization in networks of nonlinear systems with time-delayed coupling. Using a generic model for a system close to the Hopf bifurcation, we predict the order of appearance of different cluster states and their corresponding common frequencies depending upon coupling delay. We may tune the delay time in order to ensure the existence and stability of a specific cluster state. We qualitatively and quantitatively confirm these results in experiments with chemical oscillators. The experiments also exhibit strongly nonlinear relaxation oscillations as we increase the voltage, i.e., go further away from the Hopf bifurcation. In this regime, we find secondary cluster states with delay-dependent phase lags. These cluster states appear in addition to primary states with delay-independent phase lags observed near the Hopf bifurcation. Extending the theory on Hopf normal-form oscillators, we are able to account for realistic interaction functions, yielding good agreement with experimental findings.

Synchronization of electrochemical oscillators with differential coupling

Experiments are presented to describe the effect of capacitive coupling of two electrochemical oscillators during Ni dissolution in sulfuric acid solution. Equivalent circuit analysis shows that the coupling between the oscillators occurs through the difference between the differentials of the electrode potentials. The differential nature of the coupling introduces strong negative nonisochronicity (i.e., phase shear, strong dependence of the period on the amplitude) in the coupling mechanism with smooth oscillators (under conditions just above a Hopf bifurcation point). Because of the negative nonisochronicity, asymmetrically coupled oscillators exhibit anomalous phase synchronization in the form of frequency difference enhancement. At strong coupling bistability is observed between in-phase and antiphase synchronized states. With relaxation oscillators, in contrast to the resistive coupling where antiphase synchronization can occur, the typical system response with weak coupling is out-of-phase synchronization. When the capacitance is applied on the individual resistors attached to the electrodes the oscillators exhibit weak positive nonisochronicity; this is in contrast with the strong negative nonisochronicity obtained with cross coupling. The proposed coupling configurations reveal the importance of the nonisochronicity level of oscillations for the experimentally observed synchronization patterns and also provide efficient ways of tuning the nonisochronicity level of the oscillations. This latter feature can be exploited to design synchronization features with a combination of resistive (difference) and capacitive (differential) coupling.

Enhanced entrainability of genetic oscillators by period mismatch.

Biological oscillators coordinate individual cellular components so that they function coherently and collectively. They are typically composed of multiple feedback loops, and period mismatch is unavoidable in biological implementations. We investigated the advantageous effect of this period mismatch in terms of a synchronization response to external stimuli. Specifically, we considered two fundamental models of genetic circuits: smooth and relaxation oscillators. Using phase reduction and Floquet multipliers, we numerically analysed their entrainability under different coupling strengths and period ratios. We found that a period mismatch induces better entrainment in both types of oscillator; the enhancement occurs in the vicinity of the bifurcation on their limit cycles. In the smooth oscillator, the optimal period ratio for the enhancement coincides with the experimentally observed ratio, which suggests biological exploitation of the period mismatch. Although the origin of multiple feedback loops is often explained as a passive mechanism to ensure robustness against perturbation, we study the active benefits of the period mismatch, which include increasing the efficiency of the genetic oscillators. Our findings show a qualitatively different perspective for both the inherent advantages of multiple loops and their essentiality.

Desynchronization and clustering with pulse stimulations of coupled electrochemical relaxation oscillators

Effects of pulse stimulations on the dynamics of relaxation oscillator populations were experimentally studied in a globally coupled electrochemical system. Similar to smooth oscillations, weakly and moderately relaxational oscillations possess a vulnerable phase, ϕ S ; pulses applied at ϕ S resulted in desynchronization followed by a return to the synchronized state. In contrast to smooth oscillators, weakly and moderately relaxational oscillators exhibited transient and itinerant cluster dynamics, respectively. With strongly relaxational oscillators the pulse applied at a vulnerable phase effected transitions to other cluster configurations without effective desynchronization. Repeated pulse administration resulted in a cluster state that is stable against the perturbation; the cluster configuration is specific to the pulse administered at the vulnerable phase. The pulse-induced transient clusters are interpreted with a phase model that includes first and second harmonics in the interaction function and exhibits saddle type cluster states with strongly stable intra-cluster and weakly unstable inter-cluster modes.

Synchronization of genetic oscillators

Synchronization of genetic or cellular oscillators is a central topic in understanding the rhythmicity of living organisms at both molecular and cellular levels. Here, we show how a collective rhythm across a population of genetic oscillators through synchronization-induced intercellular communication is achieved, and how an ensemble of independent genetic oscillators is synchronized by a common noisy signaling molecule. Our main purpose is to elucidate various synchronization mechanisms from the viewpoint of dynamics, by investigating the effects of various biologically plausible couplings, several kinds of noise, and external stimuli. To have a comprehensive understanding on the synchronization of genetic oscillators, we consider three classes of genetic oscillators: smooth oscillators (exhibiting sine-like oscillations), relaxation oscillators (displaying jump dynamics), and stochastic oscillators (noise-induced oscillation). For every class, we further study two cases: with intercellular communication (including phase-attractive and repulsive coupling) and without communication between cells. We find that an ensemble of smooth oscillators has different synchronization phenomena from those in the case of relaxation oscillators, where noise plays a different but key role in synchronization. To show differences in synchronization between them, we make comparisons in many aspects. We also show that a population of genetic stochastic oscillators have their own synchronization mechanisms. In addition, we present interesting phenomena, e.g., for relaxation-type stochastic oscillators coupled to a quorum-sensing mechanism, different noise intensities can induce different periodic motions (i.e., inhomogeneous limit cycles).

Control of Complex Dynamics with Time-Delayed Feedback in Populations of Chemical Oscillators: Desynchronization and Clustering

Control of collective dynamics with global time-delayed feedback was experimentally studied with (phase) synchronized populations of smooth, relaxational, and chaotic electrochemical oscillators. The experimental results provide laboratory evidence for the findings in the studies of delayed feedback on phase oscillators; multiple, isolated regions of desynchronization in the parameter plane, hysteresis phenomenon around the boundaries of control regions, synchronization of irregular behavior with very strong feedback strength, and a shift of the synchronization transition curves were reproduced in experiments. In addition, it is observed that clustering accompanies the occurrence of desynchronization in delayed feedback of relaxational oscillators. For populations of elements that are close to Hopf bifurcation, desynchronization, amplitude death, and enhanced synchronization occurs with a low, intermediate, and high delay time, respectively.

Low-dimensional maps for piecewise smooth oscillators

Dynamics of the piecewise smooth nonlinear oscillators is considered, for which, general methodology of reducing multidimensional flows to low-dimensional maps is proposed. This includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution. The global map is comprised of local maps, which are constructed in the smooth sub-regions of phase space. To construct this low-dimensional map, it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory. The dimension reduction is directly related to the dimension of the chosen boundary, and the lower its dimension is, the larger dimension reduction can be achieved. Full details are given for a drifting impact oscillator, where the five-dimensional flow is reduced to one-dimensional (1D) approximate analytical map. First an exact two-dimensional map has been formulated and analysed. A further reduction to 1D approximate map is introduced and discussed. Standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos, and co-existence of multiple attractors. Accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters.

Extracting order parameters from global measurements with application to coupled electrochemical oscillators

The calculation of order parameters to characterize the degree of synchronization usually requires the measurement of time series of observable quantities for all individual dynamical elements. In this article, a new method of extracting the Kuramoto order from global measurements for weakly coupled, nearly identical oscillators is established and successfully tested with electrochemical populations of smooth, relaxational and chaotic oscillators. An experimental study on a larger array of coupled smooth oscillators was performed and the results confirmed the theoretical predictions on size effects including enhanced fluctuations around the synchronization transition. The proposed method opens up the possibility for the study of synchronization in large populations of interacting oscillators in many fields including biology where measurements are limited to global signals and few local sites.

Emerging Coherence of Oscillating Chemical Reactions on Arrays: Experiments and Simulations

We report experiments on emerging coherence through weak global coupling of populations of electrochemical oscillators with heterogeneities. The experiments are done with an array of nickel electrodes undergoing electrodissolution in sulfuric acid. The experiments verify theories of Winfree and Kuramoto that predict that global coupling of a set of smooth limit cycle oscillators with different frequencies produces a phase transition in which some of the elements synchronize. Both the critical point and the predicted dependence of order on coupling are seen in the experiments. We extend the studies to relaxation oscillators and characterize the quantitative similarities and differences between the two types of systems. Numerical simulations using a coupled form of a nickel electrodissolution model support the experimental findings.

Emerging coherence in a population of chemical oscillators.

Coherence of interacting oscillating entities has importance in biological, chemical, and physical systems. We report experiments on populations of chemical oscillators and verify a 25-year-old theory of Kuramoto that predicts that global coupling in a set of smooth limit-cycle oscillators with different frequencies produces a phase transition in which some of the elements synchronize. Both the critical point and the predicted dependence of order on coupling are seen in the experiments. We extend the studies both to relaxation and to chaotic oscillators and characterize the quantitative similarities and differences among the types of systems.