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 a ring of three coupled relaxation oscillators

The dynamics of a ring of three identical relaxation oscillators is shown to exhibit a variety of periodic motions, including clockwise and counter-clockwise wave-like modes, and a synchronous mode in which all three oscillators are in phase. The model involves individual oscillators which exhibit sudden jumps, modeling the relaxation oscillations of van der Pol oscillators. Methods include (i) numerical integration, (ii) a semi-analytical method involving solving transcendental equations numerically, and (iii) perturbation methods. A variety of bifurcations of the periodic motions are identified. This work is motivated by application to the design of a decision-making machine which can sort initial conditions according to their steady state.     

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.     

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 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.     

Dynamics of microbubble oscillators with delay coupling

We investigate the stability of the in-phase mode in a system of two delay-coupled bubble oscillators. The bubble oscillator model is based on a 1956 paper by Keller and Kolodner. Delay coupling is due to the time it takes for a signal to travel from one bubble to another through the liquid medium that surrounds them. Using techniques from the theory of differential-delay equations as well as perturbation theory, we show that the equilibrium of the in-phase mode can be made unstable if the delay is long enough and if the coupling strength is large enough, resulting in a Hopf bifurcation. We then employ Lindstedt’s method to compute the amplitude of the limit cycle as a function of the time delay. This work is motivated by medical applications involving noninvasive localized drug delivery via microbubbles.     

Multistable remote synchronization in a star-like network of non-identical oscillators

In the context of networks of coupled oscillators, remote synchronization happens when phase difference between non-adjacent units become constant, even though there is no global phase-locking. We study such regime considering a star-like network of Stuart-Landau oscillators. As previous works, our setup comprises peripheral nodes with different but close natural frequencies and the central node frequency detuned from them. The main contribution here is to numerically report multistability under intermediate coupling values: some initial condition yield remote synchronization, with quasi-periodic motion; while others do not converge to synchronized states. By using a Gaussian distribution to select the initial phases of the oscillators, we found that relatively small value of the standard deviation and absolute value of the mean of this distribution far from a specific range of values seem to favor remote synchronization in the multistability region. This phenomenon is extensively analyzed for both cases, considering a fixed coupling value.     

Zero singularities in a ring network with two delays

In this paper, we consider a neural network model consisting of three neurons with delayed self- and nearest-neighbor connections. We provide multiple bifurcations of the zero solution of the system near zero eigenvalue singularity. Taking the coupling coefficients as the bifurcation parameters, four kinds of zero singularities are demonstrated through center manifold reduction and normal form calculation.     

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.     

Dynamics of a congestion control model in a wireless access network

In this paper, a congestion control algorithm with heterogeneous delays in a wireless access network is considered. We regard the communication time delay as a bifurcating parameter to study the dynamical behaviors, i.e., local asymptotical stability, Hopf bifurcation and resonant codimension-two bifurcation. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the delay passes through a sequence of critical value. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. In the meantime, the resonant codimension-two bifurcation is also found in this model. Some numerical examples are finally performed to verify the theoretical results.     

Broad-scale small-world network topology induces optimal synchronization of flexible oscillators

The discovery of small-world and scale-free properties of many man-made and natural complex networks has attracted increasing attention. Of particular interest is how the structural properties of a network facilitate and constrain its dynamical behavior. In this paper we study the synchronization of weakly coupled limit-cycle oscillators in dependence on the network topology as well as the dynamical features of individual oscillators. We show that flexible oscillators, characterized by near zero values of divergence, express maximal correlation in broad-scale small-world networks, whereas the non-flexible (rigid) oscillators are best correlated in more heterogeneous scale-free networks. We found that the synchronization behavior is governed by the interplay between the networks global efficiency and the mutual frequency adaptation. The latter differs for flexible and rigid oscillators. The results are discussed in terms of evolutionary advantages of broad-scale small-world networks in biological systems.     

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.     

The KAM theorem with a large perturbation and application to the network of Duffing oscillators

We prove that there is an invariant torus with the given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for application, we prove that a finite network of Duffing oscillators with periodic external forces possesses Lagrange stability for almost all initial data.     

Dynamics of a point in the axisymmetric gravitational potential of a massive fixed ring and center

Theoretical and Mathematical Physics - We consider the problem of three-dimensional motion of a passively gravitating point in the potential created by a homogeneous thin fixed ring and a massive...     

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.     

Periodic solutions for neutral coupled oscillators network with feedback and time-varying delay

This paper is considered to be about the existence of periodic solutions for neutral coupled oscillators network with feedback control and time-varying delay (NCONFT). Based on the systematic method which is firstly applied for NCONFT and consisting of coincidence degree theory, graph theory, and Lyapunov method, some sufficient criteria are obtained to verify the existence of periodic solutions for NCONFT. What’s more, how coupling topology, feedback control, and time-varying delay affect the existence of periodic solutions for NCONFT can be shown by these sufficient criteria. Finally, a numerical simulation is offered to illustrate the effectiveness of our results.     

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.