Synchronization of time-delay coupled pulse oscillators

We present a detailed study of the dynamics of pulse oscillators with time-delayed coupling. We get the return maps, obtain strict solutions and analyze their stability. For the case of two oscillators, a periodical structure of synchronization regions is found in parameter space, and the regions corresponding to in-phase and antiphase regimes alternate with growth of time delay. Two types of switching between in-phase and antiphase regimes are studied. We also show that for different parameters coupling delay may have synchronizing or desynchronizing effect. Another novel result is that phase locked regimes exist for arbitrary large values. The specificity of system dynamics with large delay is studied.     

Synchronization of two flow excited pendula

Two aerodynamically excited pendula are considered as a simple example of two linearly coupled, self sustained mechanical oscillators, modelled by two coupled Van-der-Pol equations. The considered mechanical application admits of a systematic survey of synchronized regimes within the framework of standard nonlinear stability analysis. Using normal form theory and the prevailing direct averaging approach the occurring Hopf bifurcation with two distinct pairs of purely imaginary eigenvalues is studied in the non-resonant case and in the 1:1-resonance corresponding, respectively, to strong and weak coupling. In particular, for the resonant case a graphical approach permits a comprehensive interpretation relating the stable stationary solutions of the averaged system with synchronized regimes and allows an analytical computation of the oscillation amplitudes and the synchronous frequency.     

Switching dynamics of multiple linear oscillators

In this paper, the switching dynamics of linear oscillators with arbitrary discontinuous forcing are investigated through the concept of switching systems, and such switching systems consist of countable prescribed linear oscillators with different external excitations. The traditional treatments are to smoothen the discontinuity at switching points of two subsystems in a switching system, which can provide an approximate solution only. Therefore, an alternative method is presented to obtain an exact solution of the resultant switching linear system. Under periodic piecewise forcing and random forcing, the corresponding exact solutions and stochastic responses of switching linear systems are developed. For any periodic forcing, the periodic responses and stability of the resultant system composed of multiple linear oscillators in different time intervals are presented. In addition, the resultant switching system consisting of two oscillators are discussed, and the corresponding stability analysis is carried out.     

Synchronization in nonlinear oscillators with conjugate coupling

In this work, we investigate the synchronization in oscillators with conjugate coupling in which oscillators interact via dissimilar variables. The synchronous dynamics and its stability are investigated theoretically and numerically. We find that the synchronous dynamics and its stability are dependent on both coupling scheme and the coupling constant. We also find that the synchronization may be independent of the number of oscillators. Numerical demonstrations with Lorenz oscillators are provided.     

Various synchronization phenomena in bidirectionally coupled double scroll circuits

In this paper, the various cases of synchronization phenomena investigated in a system of two bidirectionally coupled double scroll circuits, were studied. Complete synchronization, inverse lag synchronization, and inverse π-lag synchronization are the observed synchronization phenomena, as the coupling factor is varied. The inverse lag synchronization phenomenon in mutually coupled identical oscillators is presented for the first time. As the coupling factor is increased, the system undergoes a transition from chaotic desynchronization to chaotic complete synchronization, while inverse lag synchronization and inverse π-lag synchronization are observed for greater values of the coupling factor, depending on the initial conditions of the state variables of the system. Inverse π-lag synchronization in coupled nonlinear oscillators is a special case of lag synchronization, which is also presented for the first time.     

Investigating the Spectrum of Self-Similar Solutions of the Nonlinear Heat Equation

Self-similar solutions of the nonlinear heat equation with a three-dimensional source and density that varies as a power function of the radius are considered in planar, cylindrical, and spherical geometries. The self-similar solutions evolve in a blow-up setting and constitute time-dependent dissipative structures. The eigenfunction spectrum of the self-similar problem is investigated for various values of the model parameters by computational methods involving continuation in a parameter and bifurcation analysis. It is shown that the spectral problem may have a nonunique solution. We establish the number of eigenfunctions and their existence domain in the parameter space. The evolution of the eigenfunctions with changes in the parameter is examined. The stability of the self-similar solutions is shown to depend on the parameter values, the eigenfunction index, and the eigenfunction parity. New structurally stable and metastable self-similar solutions are obtained. The metastable solutions follow the self-similar law almost during the entire blow-up time and preserve their complex structure as the temperature is increased by two orders of magnitude.__________Translated from Prikladnaya Matematika i Informatika, No. 16, pp. 27–65, 2004.     

Resonant vibrations in harmonically excited weakly coupled mechanical systems with cyclic symmetry

The resonant vibrations in weakly coupled nonlinear cyclic symmetric structures are studied. These structures consist of weakly coupled identical nonlinear oscillators. A careful bifurcation analysis of the amplitude equations is performed in the fundamental resonance case for an illustrative example consisting of a three particle system. In case of a uniformly distributed excitation, a localized response is identified in which one of the particles exhibits large amplitude motions compared to those of the other particles. In case of single-particle excitation, it is found that for very small coupling strength and large external mistuning, a large stable localized periodic response coexists with an extended small response. With an increase in the coupling strength, multiple extended solutions arise near the exact external resonance via saddle-node bifurcations. Further increase in coupling strength and a decrease in damping results in isolated asymmetric solution branches, which bifurcate from the symmetric solutions via symmetry-breaking bifurcations. The role of coupling strength in creating/destroying localized solutions is discussed.     

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.     

The dynamics ofn weakly coupled identical oscillators

Summary We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.     

Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation

Solutions of problems for the system of equations describing weakly nonlinear quasi-transverse waves in an elastic weakly anisotropic medium are studied analytically and numerically. It is assumed that dissipation and dispersion are important for small-scale processes. Dispersion is taken into account by terms involving the third derivatives of the shear strains with respect to the coordinate, in contrast to the previously considered case when dispersion was determined by terms with second derivatives. In large-scale processes, dispersion and dissipation can be neglected and the system of equations is hyperbolic. The indicated small-scale processes determine the structure of discontinuities and a set of admissible discontinuities (with a steady-state structure). This set is such that the solution of a self-similar Riemann problem constructed using solutions of hyperbolic equations and admissible discontinuities is not unique. Asymptotics of non-self-similar problems for equations with dissipation and dispersion were numerically found, and it appeared that they correspond to self-similar solutions of the Riemann problem. In the case of nonunique self-similar solutions, it is shown that the initial conditions specified as a smoothed step lead to a certain self-similar solution implemented as the asymptotics of the unsteady problem depending on the smoothing method.     

Nonlinear oscillations in a MEMS energy scavenger

The dynamics of an electret-based, capacitive, vibration-to-electric micro-converter (energy scavenger) is described by a set of ODEs where a second-order equation is coupled to two first-order equations through strongly-nonlinear terms. The nonlinear regimes of forced oscillations are analyzed with a semi-analytical approach, finding that the system exhibits features typical of Duffing-like nonlinear oscillators, such as jumps and multivalued frequency-response curves, with both stable and unstable periodic solutions. It is also proved that, for appropriate combinations of parameters, the system acts as a linear, damped oscillator, independently of the oscillation amplitude: in this case, the nonlinear coupling term reduces to a viscous-like term, physically interpretable as electromechanical damping.     

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.     

Limit Cycle-Strange Attractor Competition

To understand the competition between what are known as limit cycle and strange attractor dynamics, the classical oscillators that display such features were coupled and studied with and without external forcing. Numerical simulations show that, when the Duffing equation (the strange attractor prototype) forces the van der Pol oscillator (the limit cycle prototype), the limit cycle is destroyed. However, when the van der Pol oscillator is coupled to the Duffing equation as linear forcing, the two traditionally stable steady states are destabilized and a quasi-periodic orbit is born. In turn, this limit cycle is eventually destroyed because the coupling strength is increased and eventually gives way to strange attractor or chaotic dynamics. When two van der Pol oscillators are coupled in the absence of external periodic forcing, the system approaches a stable, nonzero steady state when the coupling strengths are both unity; trajectories approach a limit cycle if coupling strengths are equal and less than 1. Solutions grow unbounded if the coupling strengths are equal and greater than 1. Quasi-periodic solutions give way to chaos as the coupling strength increases and one oscillator is strongly coupled to the other. Finally, increasing the nonlinearity in both the oscillators is stabilizing whereas increasing the nonlinearity in a single oscillator results in subcritical instability.     

Interaction of four rarefaction waves in the bi-symmetric class of the pressure-gradient system

The global existence and structure of solutions to multi-dimensional pressure-gradient system has some open problems. In this paper, we construct global classical solutions to the interaction of four planar rarefaction waves with two axes of symmetry for the pressure-gradient system in two space dimensions. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar.     

Lie symmetry analysis of electron–electromagnetic wave interaction under condition of the anomalous Doppler effect

Lie symmetry analysis is applied for a problem of interaction of electron cyclotron oscillators with a slow electromagnetic wave under condition of the anomalous Doppler effect. This analysis reveals scaling invariance of the system and existence of self-similar solutions which describe amplification of a short electromagnetic pulse with its subsequent compression. The results of theoretical analysis are confirmed by numerical simulations.     

带梯度吸收项的快扩散方程的自相似奇性解

研究一类带有非线性梯度吸收项的快速扩散方程的自相似奇性解．通过自相似变换,该自相似奇性解满足一个非线性常微分方程的边值问题,再利用打靶法技巧研究该常微分方程初值问题解的存在唯一性并根据初值的取值范围对其解进行了分类．通过对这些解类的性质的分析研究,得出了自相似强奇性解存在唯一性的充分必要条件,此时自相似奇性解就是强奇性解．     

Synchronization analysis through coupling mechanism in realistic neural models

Synchronization which relates to the system’s stability is important to many engineering and neural applications. In this paper, an attempt has been made to implement response synchronization using coupling mechanism for a class of nonlinear neural systems. We propose an OPCL (open-plus-closed-loop) coupling method to investigate the synchronization state of driver-response neural systems, and to understand how the behavior of these coupled systems depend on their inner dynamics. We have investigated a general method of coupling for generalized synchronization (GS) in 3D modified spiking and bursting Morris–Lecar (M-L) neural models. We have also presented the synchronized behavior of a network of four bursting Hindmarsh–Rose (H-R) neural oscillators using a bidirectional coupling mechanism. We can extend the coupling scheme to a network of N neural oscillators to reach the desired synchronous state. To make the investigations more promising, we consider another coupling method to a network of H-R oscillators using bidirectional ring type connections and present the effectiveness of the coupling scheme.     

Symmetry and phaselocking in chains of weakly coupled oscillators

Weakly coupled chains of oscillators with nearest-neighbor interactions are analyzed for phaselocked solutions. It is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound. It is also shown that qualitative behavior of these solutions depends on whether the coupling is “diffusive” or “I synaptic”. terms defined in the paper. The methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two-point (continuum) boundary value problem that is easier to analyze; the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law. An application to the neurophysiology of motor behavior is discussed briefly.     

Stability and bifurcation analysis of a buckled beam via active control

In this paper, we focus on applying active control to nonlinear dynamical beam system to eliminate its vibration. We analyzed stability using frequency-response equations and bifurcation. The analytical solution of the nonlinear differential equations describing the above system is investigated using multiple time scale method (MTSM). All resonance cases were extracted from second order approximations. Numerical solutions of the system are included. The effects of most system parameters were investigated. The results demonstrated that proposed controller is efficient to suppress the vibrations. Increasing the quadratic stiffness coefficient term vanished the multi-valued solution. Bifurcation diagrams refiled the effects of various system parameters on its stability showing different bifurcation cases. Finally, we conclude that for low values of natural frequencies dynamical system, the controller is more effective. The results show that the analytical solutions of the system are in good agreement with the numerical solutions.     

Further Remarks on the Luo–Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

It is shown that the self-similar ansatz proposed by T. Hou and G. Luo to describe a singular solution of the 3D axisymmetric Euler equations leads, without assuming any asymptotic condition on the self-similar profiles, to an overdetermined system of partial differential equations that produces two families of solutions: a class of trivial solutions in which the vorticity field is identically zero, and a family of solutions that blow-up immediately, where the vorticity field is governed by a stationary regime. In any case, the analytical properties of these solutions are not consistent with the numerical observations reported by T. Hou and G. Luo. Therefore, this result is a refinement of the previous work published by D. Chae and T.-P. Tsai on this matter, where the authors found the trivial class of solutions under a certain decay condition of the blow-up profiles.