Chaotic Repellor Research Articles

Border collision bifurcations in a piecewise linear duopoly model

We consider a duopoly model characterized by a two-dimensional non-invertible continuous map T given by piecewise linear functions, with several partitions, defining a duopoly game. The structure of the game is such that it has separate second iterate so that its dynamics can be studied via a one-dimensional composite function, that is piecewise linear with multiple partitions in which the definition of the map changes. The number of partitions may change from 2 to 5, depending on the parameters. The dynamics are characterized by degenerate bifurcations and border collision bifurcations, which are typical in maps having kink points. Here the peculiarity is the multiplicity of the partitions, which leads to bifurcations different from those occurring in maps with only one kink point. We show several bifurcations, coexistence of cycles, attracting and superstable, as well chaotic attractors and chaotic repellors, related to the outcome of particular border collision bifurcations.

Noise-induced behavioral change driven by transient chaos

We study behavioral change in the context of a stochastic, non-linear consumption model with preference adjusting, interdependent agents. Changes in long-run consumption behavior are modelled as noise induced transitions between coexisting attractors. A particular case of multistability is considered: two fixed points, whose immediate basins have smooth boundaries, coexist with a periodic attractor, with a fractal immediate basin boundary. If a trajectory leaves an immediate basin, it enters a set of complexly intertwined basins for which final state uncertainty prevails. The standard approach to predicting transition events rooted in the stochastic sensitivity function technique due to Mil'shtein and Ryashko (1995) does not apply since the required exponentially stable attractor, for which a confidence region could be constructed, does not exist. To solve the prediction problem we propose a heuristic based on the idea that a vague manifestation of a non-attracting chaotic set (chaotic repellor) - could serve as a surrogate for an attractor. A representation of the surrogate is generated via an algorithm for generating the boundary of an absorbing area due to Mira et al. (1996). Then a confidence domain for the surrogate is generated using the approach due to Bashkirtseva and Ryashko (2019). The intersections between this confidence region and the immediate basins of the coexisting attractors can then be used to make predictions about transition events. Preliminary assessments show that the heuristic indeed explains the transition probabilities observed in numerical experiments.

Constructing chaotic repellors

Abstract The introduction of surfaces of equilibria in a dynamical system is a useful tool for constructing a chaotic repellor. To transform an attractor to a repellor, there are infinitely many available functions for introducing a surface of equilibria. Chaotic repellors can be constructed thereafter from a single chaotic attractor, a symmetric pair of chaotic attractors or even from those systems with attractor doubling and self-reproducing. Offset boosting of a variable driven by an embedded function or extra supplementary functions shows a flexible control on system attractors along with those coexisting repellors, which also rescales the frequency of oscillation even without destroying the amplitude of them.

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: divergence and bounded dynamics

In this work we continue the study of a family of 1D piecewise smooth maps, defined by a linear function and a power function with negative exponent, proposed in engineering studies. The range in which a point on the right side is necessarily mapped to the left side, and chaotic sets can only be unbounded, has been already considered. In this work we are characterizing the remaining ranges, in which more iterations of the right branch are allowed and in which divergent trajectories occur. We prove that in some regions a bounded chaotic repellor always exists, which may be the only non-divergent set, or it may coexist with an attracting cycle. In another range, in which divergence cannot occur, we prove that unbounded chaotic sets always exist. The role of particular codimension-two points is evidenced, associated with fold bifurcations and border collision bifurcations (BCBs), related to cycles having the same symbolic sequences. We prove that they exist related to the border collision of any admissible cycle. We show that each BCB, each fold bifurcation and each homoclinic bifurcation is a limit set of infinite families of other BCBs.

Understanding the sub-critical transition to turbulence in wall flows

In contrast with free shear flows presenting velocity profiles with inflection points which cascade to turbulence in a relatively mild way, wall bounded flows are deprived of (inertial) instability modes at low Reynolds numbers and become turbulent in a much wilder way, most often marked by the coexistence of laminar and turbulent domains at intermediate Reynolds numbers, well below the range where (viscous) instabilities can show up. There can even be no unstable mode at all, as for plane Couette flow (pCf) or for Poiseuille pipe flow (Ppf) that are currently the subject of intense research. Though the mechanisms involved in the transition to turbulence in wall flows are now better understood, statistical properties of the transition itself are yet unsatisfactorily assessed. A widely accepted interpretation rests on non-trivial solutions of the Navier-Stokes equations in the form of unstable travelling waves and on transient chaotic states associated to chaotic repellors. Whether these concepts typical of the theory of temporal chaos are really appropriate is yet unclear owing to the fact that, strictly speaking, they apply when confinement in physical space is effective while the physical systems considered are rather extended in at least one space direction, so that spatiotemporal behaviour cannot be ruled out in the transitional regime. The case of pCf will be examined in this perspective through numerical simulations of a model with reduced cross-stream (y) dependence, focusing on the in-plane (x, z) space dependence of a few velocity amplitudes. In the large aspect-ratio limit, the transition to turbulence takes place via spatiotemporal intermittency and we shall attempt to make a connection with the theory of first-order (thermodynamic) phase transitions, as suggested long ago by Pomeau.

THE FEASIBLE DOMAINS AND THEIR BIFURCATIONS IN AN EXTENDED LOGISTIC MODEL WITH AN EXTERNAL INTERFERENCE

This paper is devoted to the study of the properties of basins of attraction and the domains of feasible trajectories (discrete trajectories having an ecological sense) generated by a family of two-dimensional map T related to a discrete model of populations generation. The inverse of T has vanishing denominator giving rise to nonclassical singularities: a nondefinition line, a focal point and a prefocal line. Furthermore, the differences and relations between the feasible set, the feasible domains and the basins of attraction are presented. A phenomena of coexistence of attractors is shown. The structure of a chaotic repellor is interpreted by use of the singularities.

The influence of noise on a classical chaotic scatterer

A classical, chaotic scatterer consisting of three, equal-sized, equidistant hard discs, also known as the Gaspard–Rice system [Gaspard P, Rice SA. Scattering from a classically chaotic repellor. J Chem Phys 1989;90(4):2225–2241] is studied in the presence of white noise. The fractal dimension of the stable manifold is measured using the uncertainty fraction. The volume of the manifold, and thus of the invariant set, when considered in a possibilistic sense, is found to scale with the magnitude of the noise, thus extending the results of Ott et al. [Ott E, York ED, Yorke JA. A scaling law: How an attractors volume depends on noise level. Physica D 1985;16:62–78] from attracting to non-attracting sets.

Lifetimes of noisy repellors.

We study the effects of additive noise on the lifetimes of chaotic repellors. Using first-order perturbation theory, we argue that noise will increase the lifetime if the escape holes lie in regions where the unperturbed density is higher than that in the immediate vicinity and that it decreases if the density is lower. Numerical experiments support the qualitative conclusions also beyond perturbation theory.

Fractal dimension of higher-dimensional chaotic repellors

Using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in “typical” dynamical systems in terms of the Lyapunov exponents and the characteristic escape time from the repellor. Our main example, a three-dimensional chaotic scattering billiard, yields a new structure for its invariant manifolds. This system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae.

Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.

Erratum: Exact quantization of the scattering from a classically chaotic repellor [J. Chem. Phys. 9 0, 2255 (1989)

First Page

Erratum: Semiclassical quantization of the scattering from a classically chaotic repellor [J. Chem. Phys. 9 0, 2242 (1989)

Erratum: Semiclassical quantization of the scattering from a classically chaotic repellor [J. Chem. Phys. <b>9</b> <b>0</b>, 2242 (1989)

Exact quantization of the scattering from a classically chaotic repellor

The S matrix for the scattering of a point particle from three hard discs fixed on a plane is calculated exactly using Green’s theorem. The S matrix is obtained explicitly from S=I−iCM−1D, where the matrix M describes the multiple scattering between the three discs and the matrices C and D describe the free propagation from the edges of the three discs to large distances. The scattering resonances are located in the complex wave number plane as the zeros of the determinant of the matrix M, and their symmetry representations are determined. The results of this calculation are compared with the results obtained from the semiclassical approximation described in the preceding paper.

Semiclassical quantization of the scattering from a classically chaotic repellor

The scattering of a point particle from three hard discs in a plane is studied in the semiclassical approximation, using the Gutzwiller trace formula. Using a previously introduced coding of the classical dynamics, the needed summation over the classical periodic orbits is performed. The trace function is then given in terms of Ruelle zeta functions. A semiclassical limit upper bound is obtained on the lifetimes of the scattering resonances. This bound is larger than the classical lifetime when the classical repellor is chaotic but coincides with it when the repellor is periodic. We conclude that classical chaos dramatically influences the lifetimes of the scattering resonances. Our upper bound for the resonance lifetime is compared with the results of numerical calculation of the full quantum dynamics. The distribution of the imaginary parts of the complex wave numbers of the resonances is also calculated.

Scattering from a classically chaotic repellor

We report a study of the classical scattering of a point particle from three hard circular discs in a plane, which we propose as a model of an idealized unimolecular fragmentation. The system possesses a fractal and chaotic metastable classical state. On the basis of a coding of the system dynamics, we develop a method to construct the invariant probability measure and to calculate the particle escape rate, the Hausdorff dimension, the Kolmogorov–Sinai entropy per unit time and the mean largest Lyapunov exponent of the repellor. The relations between these characteristics of the system dynamics are discussed. In particular, we show that, in general, chaos inhibits escaping from the metastable state. The theory is compared with numerical simulations. We also introduce the classical tools necessary for the semiclassical quantization of the dynamics; the latter is discussed in the following paper.