Metamorphoses of basin boundaries in nonlinear dynamical systems

Metamorphoses and explosively remote synchronization in dynamical networks

We uncover a phenomenon in coupled nonlinear networks with a symmetry: as a bifurcation parameter changes through a critical value, synchronization among a subset of nodes can deteriorate abruptly, and, simultaneously, perfect synchronization emerges suddenly among a different subset of nodes that are not directly connected. This is a synchronization metamorphosis leading to an explosive transition to remote synchronization. The finding demonstrates that an explosive onset of synchrony and remote synchronization, two phenomena that have been studied separately, can arise in the same system due to symmetry, providing another proof that the interplay between nonlinear dynamics and symmetry can lead to a surprising phenomenon in physical systems.

Some elements for a history of the dynamical systems theory

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.

Controllable switching between stable modes in a small network of pulse-coupled chemical oscillators

Switching between stable oscillatory modes in a network of four Belousov–Zhabotinsky oscillators unidirectionally coupled in a ring analysed computationally and experimentally.

Interrupted coarsening in the zero-temperature kinetic Ising chain driven by a periodic external field

If quenched to zero temperature, the one-dimensional Ising spin chain undergoes coarsening, whereby the density of domain walls decays algebraically in time. We show that this coarsening process can be interrupted by exerting a rapidly oscillating periodic field with enough strength to compete with the spin-spin interaction. By analyzing correlation functions and the distribution of domain lengths both analytically and numerically, we observe nontrivial correlation with more than one length scale at the threshold field strength.

Edge of chaos and genesis of turbulence

The edge of chaos is analyzed in a spatially extended system, modeled by the regularized long-wave equation, prior to the transition to permanent spatiotemporal chaos. In the presence of coexisting attractors, a chaotic saddle is born at the basin boundary due to a smooth-fractal metamorphosis. As a control parameter is varied, the chaotic transient evolves to well-developed transient turbulence via a cascade of fractal-fractal metamorphoses. The edge state responsible for the edge of chaos and the genesis of turbulence is an unstable traveling wave in the laboratory frame, corresponding to a saddle point lying at the basin boundary in the Fourier space.

Combined effects of LTP/LTD and synaptic scaling in formation of discrete and line attractors with persistent activity from non-trivial baseline

In this article, we analyze combined effects of LTP/LTD and synaptic scaling and study the creation of persistent activity from a periodic or chaotic baseline attractor. The bifurcations leading to the creation of new attractors have been detailed; this was achieved using a mean field approximation. Attractors encoding persistent activity can notably appear via generalized period-doubling bifurcations, tangent bifurcations of the second iterates or boundary crises, after which the basins of attraction become irregular. Synaptic scaling is shown to maintain the coexistence of a state of persistent activity and the baseline. According to the rate of change of the external inputs, different types of attractors can be formed: line attractors for rapidly changing external inputs and discrete attractors for constant external inputs.

Interactions between global and grazing bifurcations in an impacting system

It is well known that the locus of boundary crises in smooth systems contains gaps that give rise to periodic windows. We show that this phenomenon can also be observed in an impacting system, and that the mechanism by which these gaps are created is different. Namely, here gaps are created and disappear at points along the branches of boundary crises where they are intersected by branches of grazing bifurcations. We locate a novel type of double-crisis vertex which we call a grazing-crisis vertex. Additionally, we illustrate several types of basin-boundary metamorphosis that are intricately related with grazing bifurcations.

Modeling the basin of attraction as a two-dimensional manifold from experimental data: applications to balance in humans

We present a method of modeling the basin of attraction as a three-dimensional function describing a two-dimensional manifold on which the dynamics of the system evolves from experimental time series data. Our method is based on the density of the data set and uses numerical optimization and data modeling tools. We also show how to obtain analytic curves that describe both the contours and the boundary of the basin. Our method is applied to the problem of regaining balance after perturbation from quiet vertical stance using data of an elite athlete. Our method goes beyond the statistical description of the experimental data, providing a function that describes the shape of the basin of attraction. To test its robustness, our method has also been applied to two different data sets of a second subject and no significant differences were found between the contours of the calculated basin of attraction for the different data sets. The proposed method has many uses in a wide variety of areas, not just human balance for which there are many applications in medicine, rehabilitation, and sport.

Avoiding escapes in open dynamical systems using phase control

In this paper we study how to avoid escapes in open dynamical systems in the presence of dissipation and forcing, as it occurs in realistic physical situations. We use as a prototype model the Helmholtz oscillator, which is the simplest nonlinear oscillator with escapes. For some parameter values, this oscillator presents a critical value of the forcing for which all particles escape from its single well. By using the phase control technique, weakly changing the shape of the potential via a periodic perturbation of suitable phase varphi , we avoid the escapes in different regions of the phase space. We provide numerical evidence, heuristic arguments, and an experimental implementation in an electronic circuit of this phenomenon. Finally, we expect that this method might be useful for avoiding escapes in more complicated physical situations.

Dynamics of scattering on a classical two-dimensional artificial atom

A classical two-dimensional (2D) model for an artificial atom is used to make a numerical "exact" study of elastic and nonelastic scattering. Interesting differences in the scattering angle distribution between this model and the well-known Rutherford scattering are found in the small energy and/or small impact parameter scattering regime. For scattering off a classical 2D hydrogen atom different phenomena such as ionization, exchange of particles, and inelastic scattering can occur. A scattering regime diagram is constructed as function of the impact parameter (b) and the initial velocity (v) of the incoming particle. In a small regime of the (b,v) space the system exhibits chaos, which is studied in more detail. Analytic expressions for the scattering angle are given in the high impact parameter asymptotic limit.

Locus of boundary crisis: expect infinitely many gaps

Boundary crisis is a mechanism for destroying a chaotic attractor when one parameter is varied. In a two-parameter setting the locus of the boundary crisis is associated with curves of homoclinic or heteroclinic bifurcations of periodic saddle points. It is known that this locus has nondifferentiable points. We show here that the locus of boundary crisis is far more complicated than previously reported. It actually contains infinitely many gaps, corresponding to regions (of positive measure) where attractors exist.

Edge of chaos in a parallel shear flow

We study the transition between laminar and turbulent states in a Galerkin representation of a parallel shear flow, where a stable laminar flow and a transient turbulent flow state coexist. The regions of initial conditions where the lifetimes show strong fluctuations and a sensitive dependence on initial conditions are separated from the ones with a smooth variation of lifetimes by an object in phase space which we call the "edge of chaos." We describe techniques to identify and follow the edge, and our results indicate that the edge is a surface. For low Reynolds numbers we find that the surface coincides with the stable manifold of a periodic orbit, whereas at higher Reynolds numbers it is the stable set of a higher-dimensional chaotic object.

Persistence of chaos in a time-delayed-feedback controlled Duffing system

This paper concerns global phase structures of a time-delayed-feedback controlled two-well Duffing system. The remains of a global stretch and fold structure along an unstable manifold, which develops from an unstable fixed point in function space, reveals that the global chaotic dynamics is inherited from the original system by the controlled system. The remains of the original chaotic dynamics causes a highly complicated domain of attraction for target orbits and a long chaotic transient before convergence.

Bistable chaos without symmetry in generalized synchronization

Frequently, multistable chaos is found in dynamical systems with symmetry. We demonstrate a rare example of bistable chaos in generalized synchronization (GS) in coupled chaotic systems without symmetry. Bistable chaos in GS refers to two chaotic attractors in the response system which both synchronize with the driving dynamics in the sense of GS. By choosing appropriate coupling, the coupled system could be symmetric or asymmetric. Interestingly, it is found that the response system exhibits bistability in both cases. Three different types of bistable chaos have been identified. The crisis bifurcations which lead to the bistability are explored, and the relation between the bistable attractors is analyzed. The basin of attraction of the bistable attractors is extensively studied in both parameter space and initial condition space. The fractal basin boundary and the riddled basin are observed and they are characterized in terms of the uncertainty exponent.

Why are chaotic attractors rare in multistable systems?

We show that chaotic attractors are rarely found in multistable dissipative systems close to the conservative limit. As we approach this limit, the parameter intervals for the existence of chaotic attractors as well as the volume of their basins of attraction in a bounded region of the state space shrink very rapidly. An important role in the disappearance of these attractors is played by particular points in parameter space, namely, the double crises accompanied by a basin boundary metamorphosis. Scaling relations between successive double crises are presented. Furthermore, along this path of double crises, we obtain scaling laws for the disappearance of chaotic attractors and their basins of attraction.

Entropy and bifurcations in a chaotic laser

We compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection. We consider the Poincaré return map to a fixed plane, and are able to compute the stable and unstable manifolds of periodic points globally, even though it is impossible to find a plane on which the Poincaré map is globally smoothly defined. In this way, we obtain the information that forms the input of the entropy calculations, and characterize the boundary crisis in which the chaotic attractor is destroyed. This boundary crisis involves a periodic point with negative eigenvalues, and the entropy associated with the chaotic attractor persists in a chaotic saddle after the bifurcation.

Countable and uncountable boundaries in chaotic scattering

We study the topological structure of basin boundaries of open chaotic Hamiltonian systems in general. We show that basin boundaries can be classified as either type I or type II, according to their topology. Let B be the intersection of the boundary with a one-dimensional curve. In type I boundaries, B is a Cantor set, whereas in type II boundaries B is a Cantor set plus a countably infinite set of isolated points. We show that the occurrence of one or the other type of boundary is determined by the topology of the accessible configuration space, and also by the chosen definition of escapes. We show that the basin boundary may undergo a transition from type I to type II, as the system's energy crosses a critical value. We illustrate our results with a two-dimensional scattering system.

Catastrophic bifurcation from riddled to fractal basins

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

Output functions and fractal dimensions in dynamical systems

We present a novel method for the calculation of the fractal dimension of boundaries in dynamical systems, which is in many cases many orders of magnitude more efficient than the uncertainty method. We call it the output function evaluation (OFE) method. We show analytically that the OFE method is much more efficient than the uncertainty method for boundaries with D<0.5, where D is the dimension of the intersection of the boundary with a one-dimensional manifold. We apply the OFE method to a scattering system, and compare it to the uncertainty method. We use the OFE method to study the behavior of the fractal dimension as the system's dynamics undergoes a topological transition.

Theory of oscillations in average crisis-induced transient lifetimes

Analytical and numerical study of the roughly periodic oscillations emerging on the background of the well-known power law governing the scaling of the average lifetimes of crisis induced chaotic transients is presented. The explicit formula giving the amplitude of "normal" oscillations in terms of the eigenvalues of unstable orbits involved in the crisis is obtained using a simple geometrical model. We also discuss the commonly encountered situation when normal oscillations appear together with "anomalous" ones caused by the fractal structure of basins of attraction.

Chaotic phenomena triggering the escape from a potential well

This paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold. The aim is to show how the well-known atoms of dissipative dynamics (saddle-node folds, period-doubling flips, cascades to chaos, boundary crises, etc.) assemble to form molecules of overall response (hierarchies of cusps, incomplete Feigenbaum trees, etc.). Particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle, the latter being accurately predicted by a Melnikov analysis. After escape, chaotic transients are shown to conform to recent scaling laws. Analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible.

Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics

Recently research has shown that many simple nonlinear deterministic systems can behave in an apparently unpredictable and chaotic manner. This realization has broad implications for many fields of science. Basic developments in the field of chaotic dynamics of dissipative systems are reviewed in this article. Topics covered include strange attractors, how chaos comes about with variation of a system parameter, universality, fractal basin boundaries and their effect on predictability, and applications to physical systems.