International Journal of Bifurcation and Chaos

This article presents a tribute to L. P. Shilnikov from a fan, a scientist from the next generation who owes much of his career to the results pioneered by this truly great man. The discourse is based on a mixture of personal recollections and conversations with a number of key individuals.

We derive sufficient conditions for the existence of an invariant set in an absorbing region homeomorphic to the product of a multidimensional torus and a ball. This set consists of low dimensional tori labeled by symbolic sequences. It may appear as a result of the breakdown of an attracting multidimensional torus. Trajectories on the set manifest chaotic behavior for some angular coordinates and may behave regularly for others, i.e. the dynamics on the set is of the chimera state type.

Let a system of differential equations possess a saddle periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases of the breakdown of such a homoclinic connection that causes the blue sky catastrophe, as well as the onset of complex dynamics. The birth of an invariant torus and a Klein bottle is also described.

We explore the multifractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities in the parameter space of the Shimizu–Morioka model that exhibits the Lorenz chaotic attractor. We show that complex transformations that underlie the transitions from the Lorenz attractor to wildly chaotic dynamics are intensified by Shilnikov saddle-foci. These transformations are due to the emergence of Shilnikov flames originating from inclination-switch homoclinic bifurcations of codimension-two. We demonstrate how the original computational technique, based on the symbolic description and kneading invariants, can disclose the complexity and universality of parametric structures and their link with nonlocal bifurcations in this representative model.

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincaré maps in models of nonholonomic mechanics.

We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter value, we prove the existence of symmetric homoclinic orbits to saddle invariant 2-tori.

We consider van der Pol–Duffing oscillators coupled via a two-dimensional linear system, representing Huygens coupling. We demonstrate that two identical synchronized oscillators can have chaotic movements and exhibit Shilnikov chaos. We rigorously prove the existence of a homoclinic orbit of a saddle-focus and a symmetrical heteroclinic contour of two saddle-foci, leading to the appearance of chaos via the Shilnikov saddle-focus bifurcation. We also support our analytical results with numerical simulations, revealing the main signature of Shilnikov chaos, the coexistence of chaotic and regular attractors with riddled basins of attraction, including wild attractors.

Meander permutations have been encountered in the context of Gauss words, singularity theory, Sturm global attractors, plane Cartesian billiards, and Temperley–Lieb algebras, among others. In this spirit, we attempt to investigate the difference of orderings of homoclinic orbits on the stable and unstable manifolds of a planar saddle. As an example, we consider reversible linear Anosov maps on the 2-torus, and their relation to continued fraction expansions.

This article contains a review of the brachistochrone problem as initiated by Johann Bernoulli in 1696–1697. As is generally known, the cycloid forms the solutions to this problem. We follow Bernoulli's optical solution based on the Fermat principle of least time and later rephrase this in terms of Hamilton's 1828 paper. Deliberately an anachronistic style is maintained throughout. Hamilton's solution recovers the cycloid in a way that is reminiscent of how Newton's mathematical principles imply Kepler's laws.

The field of dynamical systems had been revolutionized by the seminal work of Leonid Shil'nikov. As a tribute to his genius we analyze in this paper the response of dynamical systems to systematic variations of a control parameter in time, using a normal form approach. Explicit expressions of the normal forms and of their parameter dependences are derived for a class of systems possessing multiple steady-states associated to collective choices between several options in group-living organisms, giving rise to bifurcations of the pitchfork and of the limit point type. Depending on the conditions, delays in the transitions between states, stabilization of metastable states, or on the contrary enhancement of the choice of the most rewarding option induced by the time dependence of the parameter are identified.

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a two-dimensional torus with a fast frequency vector , with ω = (1, Ω) where Ω is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré–Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.

In this work, we classify the bifurcations of chaotic attractors in 1D piecewise smooth maps from the point of view of underlying homoclinic bifurcations of repelling cycles which are located before the bifurcation at the boundary of the immediate basin of the chaotic attractor.

We study the synchronization of bursting cells that are coupled through both excitatory and inhibitory connections. We extend our recent results on networks of Hindmarsh–Rose bursting neurons [Belykh et al., 2014] to coupled Sherman β-cell models and show that the addition of repulsive inhibition to an excitatory network can induce synchronization. We discuss the mechanism of this purely synergenic phenomenon and demonstrate that the inhibition leads to the disappearance of a homoclinic bifurcation that governs the type of synchronized bursting. As a result, the inhibition causes the transition from square-wave to easier-to-synchronize plateau bursting, so that weaker excitation is sufficient to induce bursting synchrony. We dedicate this paper to the memory of Leonid P. Shilnikov, the pioneer of homoclinic bifurcation theory, and emphasize the importance of homoclinic bifurcations for understanding the emergence of synchronized rhythms in bursting networks.

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

In the survey, we consider bifurcations of attracting (or repelling) invariant sets of some classical dynamical systems with a discrete time.

The statistics of Poincaré recurrences is studied numerically in a one-dimensional cubic map in the presence of harmonic and noisy excitations. It is shown that the distribution density of Poincaré recurrences is periodically modulated by the harmonic force. It is established that the relationship between the Afraimovich–Pesin dimension and Lyapunov exponents is violated in the nonautonomous system.

Asymptotic methods investigate the dynamic properties of the logistic equation with delay and delay control. The possibility to effectively control the characteristics of the relaxation cycle has been illustrated. We introduce a new method of studying the dynamics, provided that the ratio of the delayed control is large enough. The original problem of the dynamics equation with delays has been reduced to the nonlocal special dynamics of nonlinear boundary-value problems of a parabolic type.

In the Pliocene (about two–five million years ago) global climate fluctuated with a period corresponding well to the 41-thousand-year cycle of changes in the Earth's axis obliquity. Then, this period disappeared, despite the fact that the obliquity cycle even slightly increased its swing and, therefore, the climatic response to this external climate forcing would have only strengthened. By analyzing paleoclimatic records covering the whole Pliocene and Pleistocene epoch, we show that the climatic response to the obliquity cycle simply became unstable, and therefore unobservable. At the same time, through the period-doubling bifurcation, which is well-known in dynamical system theory, new stable, and so observable, climatic fluctuations have been excited with an approximately doubled period. Further, these fluctuations experienced several secondary bifurcations, and, as a result, their periods increased even more.

We consider singularly perturbed reaction–diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, ε) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order ε. Another peculiarity of our problem is that — in contrast to the case of Dirichlet boundary conditions — it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions.

Stochastically forced Goodwin model of business cycles is considered. In a multistable zone, we study the phenomena of noise-induced transitions and chaotization. To clarify a probabilistic mechanism of these phenomena found numerically, we apply a semi-analytical method based on the stochastic sensitivity function technique and confidence domains. Using elaborated method, we estimate a critical value of the noise intensity corresponding to the onset of chaos.

In this paper, the local dynamics of differential equations is studied with two asymptotically large proportional delays. Depending on the parameters, critical cases in the problem of the stability of the equilibrium state are identified. In all critical cases, special evolutionary equations (quasinormal forms) are built. Their nonlocal dynamics determine the local behavior of solutions of the original equations.