Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential

Local control for the collective dynamics of self-propelled particles

Utilizing a paradigmatic model for the motion of interacting self-propelled particles, we demonstrate that local accelerations at the level of individual particles can drive transitions between different collective dynamics, leading to a control process. We find that the ability to trigger such transitions is hierarchically distributed among the particles and can form distinctive spatial patterns within the collective. Chaotic dynamics occur during the transitions, which can be attributed to fractal basin boundaries mediating the control process. The particle hierarchies described in this paper offer decentralized capabilities for controlling artificial swarms.

Drive-specific selection in multistable mechanical networks

Systems with many stable configurations abound in nature, both in living and inanimate matter, encoding a rich variety of behaviors. In equilibrium, a multistable system is more likely to be found in configurations with lower energy, but the presence of an external drive can alter the relative stability of different configurations in unexpected ways. Living systems are examples par excellence of metastable nonequilibrium attractors whose structure and stability are highly dependent on the specific form and pattern of the energy flow sustaining them. Taking this distinctively lifelike behavior as inspiration, we sought to investigate the more general physical phenomenon of drive-specific selection in nonequilibrium dynamics. To do so, we numerically studied driven disordered mechanical networks of bistable springs possessing a vast number of stable configurations arising from the two stable rest lengths of each spring, thereby capturing the essential physical properties of a broad class of multistable systems. We found that there exists a range of forcing amplitudes for which the attractor states of driven disordered multistable mechanical networks are fine-tuned with respect to the pattern of external forcing to have low energy absorption from it. Additionally, we found that these drive-specific attractor states are further stabilized by precise matching between the multidimensional shape of their orbit and that of the potential energy well they inhabit. Lastly, we showed evidence of drive-specific selection in an experimental system and proposed a general method to estimate the range of drive amplitudes for drive-specific selection.

Fractal patterns in the parameter space of a bistable Duffing oscillator

We study the dissipative bistable Duffing oscillator with equal energy wells and observe fractal patterns in the parameter space of driving frequency, forcing amplitude, and damping ratio. Our numerical investigation reveals the Hausdorff fractal dimension of the boundaries that separate the oscillator's intrawell and interwell behaviors. Furthermore, we categorize the interwell behaviors as three steady-state types: switching, reverting, and vacillating. While fractal patterns in the phase space are well known and heavily studied, our results point to another research direction about fractal patterns in the parameter space. Another implication of this study is that the vibration of a continuous bistable system modeled using a single-mode approximation also manifests fractal patterns in the parameter space. In addition, our findings can guide the design of next-generation bistable and multistable mechanical metamaterials.

Ratchet universality in the bidirectional escape from a symmetric potential well

The present work discusses symmetry-breaking-induced bidirectional escape from a symmetric metastable potential well by the application of zero-average periodic forces in the presence of dissipation. We characterized the interplay between heteroclinic instabilities leading to chaotic escape and breaking of a generalized parity symmetry leading to directed ratchet escape to an attractor either at ∞ or at -∞. Optimal enhancement of directed ratchet escape is found to occur when the wave form of the zero-average periodic force acting on the damped driven oscillator matches as closely as possible to a universal wave form, as predicted by the theory of ratchet universality. Specifically, the optimal approximation to the universal force triggers the almost complete destruction of the nonescaping basin for driving amplitudes which are systematically lower than those corresponding to a symmetric periodic force having the same period. We expect that this work could be potentially useful in the control of elementary dynamic processes characterized by multidirectional escape from a potential well, such as forced chaotic scattering and laser-induced dissociation of molecular systems, among others.

Generic Rotating-Frame-Based Approach to Chaos Generation in Nonlinear Micro- and Nanoelectromechanical System Resonators

This Letter provides a low-power method for chaos generation that is generally applicable to nonlinear micro- and nanoelectromechanical systems (MNEMS) resonators. The approach taken is independent of the material, scale, design, and actuation of the device in question; it simply assumes a good quality factor and a Duffing type nonlinearity, features that are commonplace to MNEMS resonators. The approach models the rotating-frame dynamics to analytically constrain the parameter space required for chaos generation. By leveraging these common properties of MNEMS devices, a period-doubling route to chaos is generated using smaller forcing than typically reported in the literature.

Signature identification by Minkowski dimension

In this article, we propose and investigate the possibility of signature identification based on its fractal Minkowski dimension. We consider a signature as a trajectory of a pen tip that obeys the Langevin equations, for which we calculate the fractal Minkowski dimension. This parameter is different for original and intentionally falsified signatures, thus allowing one to reliably distinguish between the signatures left by different persons. The proposed approach together with machine learning techniques is a potentially powerful tool for identification and verification of signatures and any other kind of notations.

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.

Impulse-induced optimum control of escape from a metastable state by periodic secondary excitations

We characterize the role of the impulse transmitted (time integral over a half-period) by resonant secondary excitations at controlling (suppressing and enhancing) escape from a potential well, which is induced by periodic primary excitations. By using the universal model of a dissipative Helmholtz oscillator, we demonstrate numerically that optimum control of escape occurs when the impulse transmitted by the chaos-controlling excitations is maximum while keeping their amplitude and period fixed. These findings are in complete agreement with analytical predictions from two independent methods: Melnikov analysis and energy-based analysis. Additional numerical results corresponding to other alternative escape-controlling excitations demonstrate the generality of the essential role of the excitation's impulse.

Controlling escape from a potential well by reshaping periodic secondary excitations

The role of the wave form of periodic secondary excitations at controlling (suppressing and enhancing) escape from a potential well is investigated. We demonstrate analytically (by Melnikov analysis) and numerically that a judicious choice of the excitation's wave form greatly improves the effectiveness of the escape-controlling excitations while keeping their amplitude and period fixed. These predictions are confirmed by an energy-based analysis that provides the same optimal values of the escape-controlling parameters. The example of a dissipative Helmholtz oscillator is used to illustrate the accuracy of these results.

Exploring partial control of chaotic systems

In this paper we make a thorough exploration of the technique of partial control of chaotic systems. This control technique allows one to keep the trajectories of a dynamical system close to a chaotic saddle even if the control applied is smaller than the effects of environmental noise in the system, provided that the chaotic saddle is due to the existence of a horseshoelike mapping in phase space. We state this here in a mathematically precise way using the Conley-Moser conditions, and we prove that they imply that our partial control strategy can be applied. We also give an upper bound of the control-noise ratio needed to achieve this goal, and we describe how this technique can be applied for large noise values. Finally, we study in detail the effect of imperfect targeting in our control technique. All these results are illustrated numerically with the paradigmatic Hénon map.

Partial control of chaotic systems

In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exist what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise. We call this type of control partial control of chaos.

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.

Basins of attraction in piecewise smooth Hamiltonian systems

Piecewise smooth Hamiltonian systems arise in physical and engineering applications. For such a system typically an infinite number of quasi-periodic "attractors" coexist. (Here we use the term "attractors" to indicate invariant sets to which typically initial conditions approach, as a result of the piecewise smoothness of the underlying system. These "attractors" are therefore characteristically different from the attractors in dissipative dynamical systems.) We find that the basins of attraction of different "attractors" exhibit a riddled-like feature in that they mix with each other on arbitrarily fine scales. This practically prevents prediction of "attractors" from specific initial conditions and parameters. The mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems. We demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation.

Fluctuational transitions across different kinds of fractal basin boundaries

We study fluctuational transitions in discrete and continuous dynamical systems that have two coexisting attractors in phase space, separated by a fractal basin boundary which may be either locally disconnected or locally connected. Theoretical and numerical evidence is given to show that, in each case, the transition occurs via a unique accessible point on the boundary, both in discrete systems and in flows. The complicated structure of the escape paths inside the locally disconnected fractal basin boundary is determined by a hierarchy of homoclinic points. The interrelation between the mechanism of transitions and the hierarchy is illustrated by consideration of fluctuational transitions in dynamical systems demonstrating "fractal-fractal" basin boundary metamorphosis at some value of a control parameter. The most probable escape path from an attractor, which can be either regular or chaotic, is found for each type of boundary using both statistical analysis of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Fluctuational transitions through a fractal basin boundary

Fluctuational transitions between two coexisting chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the transition mechanism is determined by a hierarchy of homoclinic points. The most probable escape path from a chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Inhibition of chaotic escape from a potential well by incommensurate escape-suppressing excitations

Theoretical results are presented concerning the reduction of chaotic escape from a potential well by means of a harmonic parametric excitation that satisfies an ultrasubharmonic resonance condition with the escape-inducing excitation. The possibility of incommensurate escape-suppressing excitations is demonstrated by studying rational approximations to the irrational escape-suppressing frequency. The analytical predictions for the suitable amplitudes and initial phases of the escape-suppressing excitation are tested against numerical simulations based on a high-resolution grid of initial conditions. These numerical results indicate that the reduction of escape is reliably achieved for small amplitudes and at, and only at, the predicted initial phases. For the case of irrational escape-suppressing frequencies, the effective escape-reducing initial phases are found to lie close to the accumulation points of the set of suitable initial phases that are associated with the complete series of convergents up to the convergent giving the chosen rational approximation.

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.

Global dynamics and stochastic resonance of the forced FitzHugh-Nagumo neuron model

We have analyzed the responses of an excitable FitzHugh-Nagumo neuron model to a weak periodic signal with and without noise. In contrast to previous studies which have dealt with stochastic resonance in the excitable model when the model with periodic input has only one stable attractor, we have focused our attention on the relationship between the global dynamics of the forced excitable neuron model and stochastic resonance. Our results show that for some parameters the forced FitzHugh-Nagumo neuron model has two attractors: the small-amplitude subthreshold periodic oscillation and the large-amplitude suprathreshold periodic oscillation. Random transitions between these two periodic oscillations are the essential mechanism underlying stochastic resonance in this model. Differences of such stochastic resonance to that in a classical bistable system and the excitable system are discussed. We also report that the state of the basin of attraction has a significant effect on the stability of neuronal firings, in the sense that the fractal basin boundary of the system enhances the noise-induced transitions.

Simple dynamical system with discrete bound states

We study numerically the dynamical system of a two-electron atom with the Darwin interaction as a model to investigate scale-dependent effects of the relativistic action-at-a-distance electrodynamics. This dynamical system consists of a small perturbation of the Coulomb dynamics for energies in the atomic range. The key properties of the Coulomb dynamics are (i) a peculiar mixed-type phase space with sparse families of stable nonionizing orbits and (ii) scale-invariance symmetry, with all orbits defined by an arbitrary scale parameter. The combination of this peculiar chaotic dynamics [(i) and (ii)], with the scale-dependent relativistic corrections (Darwin interaction), generates the phenomenon of scale-dependent stability: We find numerical evidence that stable nonionizing orbits can exist only for a discrete set of resonant energies. The Fourier transform of these nonionizing orbits is a set of sharp frequencies. The energies and sharp frequencies of the nonionizing orbits we study are in the quantum atomic range.

Dynamic stabilization in the double-well duffing oscillator

Bifurcations associated with stability of the saddle fixed point of the Poincare map, arising from the unstable equilibrium point of the potential, are investigated in a forced Duffing oscillator with a double-well potential. One interesting behavior is the dynamic stabilization of the saddle fixed point. When the driving amplitude is increased through a threshold value, the saddle fixed point becomes stabilized via a pitchfork bifurcation. We note that this dynamic stabilization is similar to that of the inverted pendulum with a vertically oscillating suspension point. After the dynamic stabilization, the double-well Duffing oscillator behaves as the single-well Duffing oscillator, because the effect of the central potential barrier on the dynamics of the system becomes negligible.

Nonlinearly saturated dynamical state of a three-wave mode-coupled dissipative system with linear instability

Linearly unstable dissipative systems with quadratic nonlinearity occurring in plasma physics, optics, fluid mechanics, etc. are often modeled by a general set of three-wave mode-coupled ordinary differential equations for complex variables. Bounded attractors of the set approximate nonlinearly saturated turbulent states of real physical systems. Exact criteria for boundedness of the attractors are found. Fundamentally different kinds of asymptotic behavior of the wave triad are classified in the parameter space and quantitatively assessed.

Fractal basin boundaries generated by basin cells and the geometry of mixing chaotic flows

Experiments and computations indicate that mixing in chaotic flows generates certain coherent spatial structures. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifold of some periodic orbit) then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We demonstrate an amazing property for certain global structures: A basin has a basin cell if and only if every diverging curve comes close to every basin boundary point of that basin.

Basin explosions and escape phenomena in the twin-well Duffing oscillator: compound global bifurcations organizing behaviour

The sinusoidally drive, twin-well Duffing oscillator has become a central archetypal model for studies of chaos and fractal basin boundaries in the nonlinear dynamics of dissipative ordinary differential equations. It can also be used to illustrate and elucidate universal features of the escape from a potential well, the jumps from one-well to cross-well motions displaying similar characteristics to those recently charted for the cubic one-well potential. We identify here some new codimension-two global bifurcations which serve to organize the bifurcation set and structure the related basin explosions and escape phenomena.

Steady states in the twin-well potential oscillator: Computer simulations and approximate analytical studies

The paper is focused on the phenomena of various steady-state oscillations exhibited by the twin-well potential system. Regions of existence of different attractors in the system parameter domain are examined and a picture book of different steady states for fixed damping and forcing is presented: 20 different combinations of single or coexisting, small orbit or large orbit, periodic and chaotic attractors are displayed. Computer simulations are followed by an approximate analytical analysis: A study of various forms of instability of periodic solutions gives close form approximate criteria for occurrence of T-periodic small orbit and large orbit oscillations, and for cross-well chaos.

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.