Dynamics of a delayed two-coupled oscillator with excitatory-to-excitatory connection

In this paper, we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory-to-excitatory connection. We study how the strength of the connections between the oscillators affects the dynamics of the neural network. We give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system. We also investigate the spatio-temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Moreover, the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory. Some numerical simulations are provided to illustrate the theoretical results.     

Heteroclinic Ratchets in Networks of Coupled Oscillators

We analyze an example system of four coupled phase oscillators and discover a novel phenomenon that we call a “heteroclinic ratchet”; a particular type of robust heteroclinic network on a torus where connections wind in only one direction. The coupling structure has only one symmetry, but there are a number of invariant subspaces and degenerate bifurcations forced by the coupling structure, and we investigate these. We show that the system can have a robust attracting heteroclinic network that responds to a specific detuning Δ between certain pairs of oscillators by a breaking of phase locking for arbitrary Δ>0 but not for Δ≤0. Similarly, arbitrary small noise results in asymmetric desynchronization of certain pairs of oscillators, where particular oscillators have always larger frequency after the loss of synchronization. We call this heteroclinic network a heteroclinic ratchet because of its resemblance to a mechanical ratchet in terms of its dynamical consequences. We show that the existence of heteroclinic ratchets does not depend on symmetry or number of oscillators but depends on the specific connection structure of the coupled system.     

Isotropy of Angular Frequencies and Weak Chimeras with Broken Symmetry

The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here, we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector—for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving a result for the existence of weak chimeras without instantaneous or setwise symmetries for coupled phase oscillators. Moreover, we construct a coupling function that gives rise to chaotic weak chimeras without symmetry in weakly coupled populations of phase oscillators with generalized coupling.     

Winding Numbers and Average Frequencies in Phase Oscillator Networks

We study networks of coupled phase oscillators and show that network architecture can force relations between average frequencies of the oscillators. The main tool of our analysis is the coupled cell theory developed by Stewart, Golubitsky, Pivato, and Torok, which provides precise relations between network architecture and the corresponding class of ODEs in R^M and gives conditions for the flow-invariance of certain polydiagonal subspaces for all coupled systems with a given network architecture. The theory generalizes the notion of fixed-point subspaces for subgroups of network symmetries and directly extends to networks of coupled phase oscillators. For systems of coupled phase oscillators (but not generally for ODEs in R^M, where M ≥ 2), invariant polydiagonal subsets of codimension one arise naturally and strongly restrict the network dynamics. We say that two oscillators i and j coevolve if the polydiagonal θ_i = θ_j is flow-invariant, and show that the average frequencies of these oscillators must be equal. Given a network architecture, it is shown that coupled cell theory provides a direct way of testing how coevolving oscillators form collections with closely related dynamics. We give a generalization of these results to synchronous clusters of phase oscillators using quotient networks, and discuss implications for networks of spiking cells and those connected through buffers that implement coupling dynamics.     

Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators

A system of two non-linear differential equations is considered that simulates the dynamics of two completely identical weakly coupled oscillators both in the case of dissipative and active coupling. The system of normal modes is investigated. All the self-similar periodic solutions, including the asymmetric solutions describing the natural ascillations of oscillators with dissimilar amplitude's, are found analytically. The stability is investigated as well as the local bifurcations of the self-similar cycles when there is a change in stability. In particular, the possibility of the creation of two-dimensional invariant tori is pointed out. In the case of active coupling, it is shown that the basic version of the natural oscillations is a stable antiphase cycle that was observed in Huygens experiments.     

Localization of invariant compacta of autonomous systems

A class of problems that may be characterized as localization problems are becoming increasingly popular in qualitative theory of differential equations [1–15]. The specific formulations differ, but geometrically all search for phase space subsets with desired properties, e.g., contain certain solutions of the system of differential equations. Such problems include construction of positive invariant sets that contain certain separatrices of the Lorenz system [1], analysis of asymptotic behavior of solutions of the Lorenz system and determination of sets that contain the Lorenz attractor [2–5, 14], as well as determination of sets containing all periodic trajectories [6–13], separatrices, and other trajectories [10, 11]. Such sets may be naturally called localizing sets and it is obviously interesting to study methods and results that produce exact or nearly exact localizing sets for each phase space structure. In this article we focus on localization of the invariant compact sets in the phase space of a differential equation system, specifically the problem of finding phase space subsets that contain all the invariant compacta of the system. Invariant compact sets are equilibria, periodic trajectories, separatrices, limit cycles, invariant tori, and other sets and their finite unions. These sets and their properties largely determine the phase space structure and the qualitative behavior of solutions of the differential equation system.     

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.     

The index of symmetry of compact naturally reductive spaces

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.     

Magnetic catalysis in a P-even, chiral-invariant three-dimensional model with four-fermion interaction

We study how an external magnetic field H affects the phase structure of a P-even, chiral-invariant three-dimensional model of field theory with a four-fermion interaction. An arbitrarily small, but nonzero, magnetic field induces the spontaneous breaking of the initial symmetry (the magnetic catalysis phenomenon). Depending on the values of the coupling constants, the model vacuum is either P-even or chiral invariant for H≠0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 323–338, August, 2000.     

Bifurcation of periodic orbits in a class of planar Filippov systems

In this paper we discuss the perturbations of a general planar Filippov system with exactly one switching line. When the system has a limit cycle, we give a condition for its persistence; when the system has an annulus of periodic orbits, we give a condition under which limit cycles are bifurcated from the annulus. We also further discuss the stability and bifurcations of a nonhyperbolic limit cycle. When the system has an annulus of periodic orbits, we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching.     

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.     

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.     

Specific locking in populations dynamics: Symmetry analysis for coupled heteroclinic cycles

Population dynamics on two sites of ecological fields are studied. Each site shows oscillatory dynamics with a heteroclinic cycle or a limit cycle attractor, and populations migrate between two sites diffusively. In this system, frequency locking states with specific ratios between the oscillations of two sites are observed. The selection of the ratios are explained with the symmetry of the phase space. Other properties of the locking states as behaviors intrinsic to heteroclinic cycles are also discussed.     

A note on: “Relaxation oscillators with exact limit cycles”

In this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle. This family constitutes a corrected and wider version of an example given in the work [M.A. Abdelkader, Relaxation oscillators with exact limit cycles, J. Math. Anal. Appl. 218 (1998) 308-312]. The result given in this note may be used to construct models of Liénard differential equations exhibiting a desired limit cycle.     

Spatio-temporal symmetries and bifurcations via bi-orthogonal decompositions

Summary A tool for analyzing spatio-temporal complex physical phenomena was recently proposed by the authors, Aubry et al. [5]. This tool consists in decomposing a spatially and temporally evolving signal into orthogonal temporal modes (temporal “structures”) and orthogonal spatial modes (spatial “structures”) which are coupled. This allows the introduction of a temporal configuration space and a spatial one which are related to each other by an isomorphism. In this paper, we show how such a tool can be used to analyze space-time bifurcations, that is, qualitative changes in both space and time as a parameter varies. The Hopf bifurcation and various spatio-temporal symmetry related bifurcations, such as bifurcations to traveling waves, are studied in detail. In particular, it is shown that symmetry-breaking bifurcations can be detected by analyzing the temporal and spatial eigenspaces of the decomposition which then lose their degeneracy, namely their invariance under the symmetry. Furthermore, we show how an extension of the theory to “quasi-symmetries” permits the treatment of nondegenerate signals and leads to an exponentially decreasing law of the energy spectrum. Examples extracted from numerically obtained solutions of the Kuramoto-Sivashinsky equation, a coupled map lattice, and fully developed turbulence are given to illustrate the theory.     

Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. We also consider the problem of existence of adiabatic invariants associated with a slowly perturbed Hamiltonian system. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1513–1528, November, 1999.     

Symmetry in de Branges almost Pontryagin spaces

In many examples of de Branges spaces symmetry appears naturally. Presence of symmetry gives rise to a decomposition of the space into two parts, the ‘even’ and the ‘odd’ part, which themselves can be regarded as de Branges spaces. The converse question is to decide whether a given space is the ‘even’ part or the ‘odd’ part of some symmetric space, and, if yes, to describe the totality of all such symmetric spaces. We consider this question in an indefinite (almost Pontryagin space) setting, and give a complete answer. Interestingly, it turns out that the answers for the ‘even’ and ‘odd’ cases read quite differently; the latter is significantly more complex.     

Sub-Riemannian distance on the Lie group SL(2)

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ? SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.