Dynamics of a ring network of phase-only oscillators

We investigate the existence, stability and bifurcation of phase-locked motions in a ring network consisting of phase-only oscillators arranged in multiple simple rings (sub-rings) which are themselves arranged in a single large ring. In the case of networks with three or four sub-rings, we give approximate expressions for critical coupling coefficients which must be exceeded in order for phase-locking to occur.     

Dynamics of a model of two delay-coupled relaxation oscillators

This paper investigates the dynamics of a new model of two coupled relaxation oscillators. The model replaces the usual DDE (differential-delay equation) formulation with a discrete-time approach with jumps. Existence, bifurcation and stability of in-phase periodic motions is studied. Simple periodic motions, which involve exactly two jumps per period, are found to have large plateaus in parameter space. These plateaus are separated by regions of complicated dynamics, reminiscent of the Devil’s Staircase. Stability of motions in the in-phase manifold are contrasted with stability of motions in the full phase space.     

Dynamics of three coupled limit cycle oscillators with application to artificial intelligence

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can make decisions by identifying a given initial condition with its associated steady state.     

Dynamics of four coupled phase-only oscillators

We study the dynamics of a system of four coupled phase-only oscillators. This system is analyzed using phase difference variables in a phase space that has the topology of a three-dimensional torus. The system is shown to exhibit numerous phase-locked motions. The qualitative dynamics are shown to depend upon a parameter representing coupling strength. This work has application to MEMS artificial intelligence decision-making devices.     

Dynamics of three coupled van der Pol oscillators with application to circadian rhythms

In this work we study a system of three van der Pol oscillators. Two of the oscillators are identical, and are not directly coupled to each other, but rather are coupled via the third oscillator. We investigate the existence of the in-phase mode in which the two identical oscillators have the same behavior. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the computer algebra system MACSYMA and the numerical bifurcation software AUTO.Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator. Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator.     

Dynamics of oscillators coupled by a medium with adaptive impact

In this article, we study the dynamics of coupled oscillators. We use mechanical metronomes that are placed over a rigid base. The base moves by a motor in a one‐dimensional direction and the movements of the base follow some functions of the phases of the metronomes (in other words, it is controlled to move according to a provided function). Because of the motor and the feedback, the phases of the metronomes affect the movements of the base, whereas on the other hand, when the base moves, it affects the phases of the metronomes in return. For a simple function for the base movement (such as y = γ_x[rθ₁ + (1 ? r)θ₂] in which y is the velocity of the base, γ_x is a multiplier, r is a proportion, and θ₁ and θ₂ are phases of the metronomes), we show the effects on the dynamics of the oscillators. Then, we study how this function changes in time when its parameters adapt by a feedback. By numerical simulations and experimental tests, we show that the dynamic of the set of oscillators and the base tends to evolve towards a certain region. This region is close to a transition in dynamics of the oscillators, where more frequencies start to appear in the frequency spectra of the phases of the metronomes. We interpret this as an adaptation towards the edge of chaos.     

Synchronized states in a ring of four mutually coupled oscillators and experimental application to secure communications

Data encryption has become increasingly important for many applications including phone, internet and satellite communications. Considering the desirable properties of ergodicity and high sensitivity to initial conditions and control parameters, chaotic signals are suitable for encryption systems. Chaotic encryption systems generally have high speed with low cost, which makes them better candidates than many traditional ciphers for multimedia data encryption. In this paper, analytical and numerical methods as well as experimental implementation are used to prove partial and complete synchronized states in a ring of four autonomous oscillators in their chaotic states. Application to secure communication is discussed.     

Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay

A system of three coupled van der Pol oscillators with delay is considered. Hopf bifurcations at the zero equilibrium as the delay increases are exhibited. The existence and stability of multiple periodic solutions are established using a symmetric Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799-4838).     

Global synchronization of three coupled chaotic systems with ring connection

This paper studies chaos synchronization of three coupled chaos systems with ring connection. New generic criteria of global chaos synchronization are proposed respectively according to the way of coupling (unidirectional or bidirectional). As an example, The criteria are successfully applied to three coupled identical Lorenz systems. Numerical simulation are shown for demonstration.     

Dynamics in coupled oscillators with recurrent/random forcing, a PDE approach

The current paper is devoted to the study of coupled oscillators with recurrent/random forcing. Special attention is given to the solutions having the same recurrence/randomness as that of the forcing (recurrent/random solutions for short). By embedding coupled oscillators into coupled parabolic equations, it establishes a general theorem on the existence of recurrent/random solutions. It also finds conditions under which such solutions are unique. When the recurrent forcing is actually quasi-periodic or almost periodic, recurrent solutions are refereed to as quasi-periodic or almost periodic solutions in a weak sense and they are quasi-periodic or almost periodic in the classical sense under the uniqueness conditions. In addition, applications of the general theory to coupled Duffing type oscillators and Josephson junctions are considered and the results obtained extend several existing ones for quasi-periodic Duffing oscillators.     

Dynamics of periodically kicked oscillators

We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically kicked oscillators. The key geometric ideas are illustrated via a simple linear shear model.     

A digital model of coupled oscillators

A new model of coupled oscillators is proposed and investigated. All phase variables and parameters are integer-valued. The model is shown to exhibit two types of motions, those which involve periodic phase differences, and those which involve drift. Traditional dynamical concepts such as stability, bifurcation and chaos are examined for this class of integer-valued systems. Numerical results are presented for systems of two and three oscillators. This work has application in digital technology.     

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.     

Weakly coupled oscillators and topological degree

By using the topological degree of Brouwer for mappings along with averaging method, we study the existence of forced periodic solutions for certain weakly coupled periodically perturbed ordinary differential equations.     

Geometrical properties of coupled oscillators at synchronization

We study the synchronization of N nearest neighbors coupled oscillators in a ring. We derive an analytic form for the phase difference among neighboring oscillators which shows the dependency on the periodic boundary conditions. At synchronization, we find two distinct quantities which characterize four of the oscillators, two pairs of nearest neighbors, which are at the border of the clusters before total synchronization occurs. These oscillators are responsible for the saddle node bifurcation, of which only two of them have a phase-lock of phase difference equals ± π/2. Using these properties we build a technique based on geometric properties and numerical observations to arrive to an exact analytic expression for the coupling strength at full synchronization and determine the two oscillators that have a phase-lock condition of ± π/2.