Signatures of strange nonchaotic dynamics in a forced quantum system.

Can strange sets arise out of quantum dynamics? We explore this question using the quantum analog of a classical impact oscillator, which consists of a forced spring-mass-damper system, with a wall it may collide against. The classical impact oscillator is known to undergo a sudden transition to chaos when the mass grazes the wall. We numerically compute the evolution of the wave function in the equivalent quantum system and investigate the dynamical signatures. The entropy of the probability density and the L_{1} norm are used to generate real-valued time series, which are then subjected to diagnostic tests. Even though closed quantum systems with unitary evolution are incapable of showing chaotic behavior, we observe the characteristic features of strange nonchaotic dynamics in the forced system.

Local and global bifurcations in 3D piecewise smooth discontinuous maps.

This paper approaches the problem of analyzing the bifurcation phenomena in three-dimensional discontinuous maps, using a piecewise linear approximation in the neighborhood of a border. The existence conditions of periodic orbits are analytically calculated and bifurcations of different periodic orbits are illustrated through numerical simulations. We have illustrated the peculiar features of discontinuous bifurcations involving a stable fixed point, a period-2 cycle, a saddle fixed point, etc. The occurrence of multiple attractor bifurcation and hyperchaos are also demonstrated.

Nordmark map and the problem of large-amplitude chaos in impact oscillators.

Physical experiments have long revealed that impact oscillators commonly exhibit large-amplitude chaos over a narrow band of parameter values close to grazing bifurcations. This phenomenon is not explained by the square-root singularity of the Nordmark map, which captures the local dynamics to leading order, because this map does not exhibit such dynamics. In this paper, we compare a Poincaré map for a prototypical impact oscillator model with the corresponding Nordmark map. Though the maps agree to leading order, the Poincaré map exhibits a large-amplitude chaotic attractor while the Nordmark map does not because part of the attractor resides in a region of phase space where the two maps differ significantly.

Using YY supermales to destabilize invasive fish populations.

A plausible biocontrol strategy for the eradication of invasive species involves augmenting wild populations with genetically modified supermales. Supermales contain double YY chromosomes. When they are augmented into a wild population, destabilization and eventual extinction occurs over time due to a strongly skewed gender ratio towards males. Here, we employ a mathematical model that considers an Allee effect, but we have discovered through simulation that the presence of supermales leads to an increase in the minimal number of females needed for survival at a value higher than the mathematically defined Allee effect. Using this effect, we focus our research on exploring the sensitivity of the optimized supply rate of supermale fish to the initial gender ratio and density of the wild populations. We find that the eradication strategy with optimized supply rate of supermales can be determined with knowledge of reproductive rate and survival fitness of supermale fish.

Robust chaos in 3-D piecewise linear maps.

A chaotic attractor is called robust if there is no periodic window or any coexisting attractor in some open subset of the parameter space. Such a chaotic attractor cannot be destroyed by a small change in parameter values since a small change in the parameter value can only make small changes in the Lyapunov exponents. Earlier investigations have calculated the existence and the stability conditions of robust chaos in 1D and 2D piecewise linear maps. In this work, we demonstrate the occurrence of robust chaos in 3D piecewise linear maps and derive the conditions of its occurrence by analyzing the interplay between the stable and unstable manifolds.

Does shining light on gold colloids influence aggregation?

In this article we revisit the much-studied behavior of self-assembled aggregates of gold colloidal particles. In the literature, the electrostatic interactions, van der Waals interactions, and the change in free energy due to ligand-ligand or ligand-solvent interactions are mainly considered to be the dominating factors in determining the characteristics of the gold aggregates. However, our light scattering and imaging experiments clearly indicate a distinct effect of light in the growth structure of the gold colloidal particles. We attribute this to the effect of a non-uniform distribution of the electric field in aggregated gold colloids under the influence of light.

Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps.

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, there exist a large number of switching dynamical systems that have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. In this paper we present a systematic approach to the problem of analyzing the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We first motivate the analysis by considering the bifurcations occurring in a familiar physical system-the static VAR compensator used in electrical power systems-and then proceed to formulate the theory needed to explain the bifurcation behavior of such systems. We then integrate the observed bifurcation phenomenology of the compensator with the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps.

Invisible grazings and dangerous bifurcations in impacting systems: the problem of narrow-band chaos.

We discovered a narrow band of chaos close to the grazing condition for a simple soft impact oscillator. The phenomenon was observed experimentally for a range of system parameters. Through numerical stability analysis, we argue that this abrupt onset to chaos is caused by a dangerous bifurcation in which two unstable period-3 orbits, created at "invisible" grazings, take part.

Transitions from phase-locked dynamics to chaos in a piecewise-linear map.

Recent work has shown that torus formation in piecewise-smooth maps can take place through a special type of border-collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle. Transitions from an ergodic to a resonant torus take place via border-collision fold bifurcations. We examine the transition to chaos through torus destruction in such maps. Considering a piecewise-linear normal-form map we show that this transition, by virtue of the interplay of border-collision bifurcations with period-doubling and homoclinic bifurcations, can involve mechanisms that differ qualitatively from those described by Afraimovich and Shilnikov.

Experimental study of impact oscillator with one-sided elastic constraint.

In this paper, extensive experimental investigations of an impact oscillator with a one-sided elastic constraint are presented. Different bifurcation scenarios under varying the excitation frequency near grazing are shown for a number of values of the excitation amplitude. The mass acceleration signal is used to effectively detect contacts with the secondary spring. The most typical recorded scenario is when a non-impacting periodic orbit bifurcates into an impacting one via grazing mechanism. The resulting orbit can be stable, but in many cases it loses stability through grazing. Following such an event, the evolution of the attractor is governed by a complex interplay between smooth and non-smooth bifurcations. In some cases, the occurrence of coexisting attractors is manifested through discontinuous transition from one orbit to another through boundary crisis. The stability of non-impacting and impacting period-1 orbits is then studied using a newly proposed experimental procedure. The results are compared with the predictions obtained from standard theoretical stability analysis and a good correspondence between them is shown for different stiffness ratios. A mathematical model of a damped impact oscillator with one-sided elastic constraint is used in the theoretical studies.

Codimension-three bifurcations: explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps.

Many physical and engineering systems exhibit cascades of periodic attractors arranged in period increment and period adding sequences as a parameter is varied. Such systems have been found to yield piecewise smooth maps, and in some cases the obtained map is discontinuous. By investigating the normal form of such maps, we have detected a type of codimension-three bifurcation which serves as the organizing center of periodic and aperiodic dynamics in the parameter space. The results will help in understanding the occurrence and structure of such cascades observed in many nonsmooth systems in science and engineering.

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation.

Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.

Dangerous bifurcation at border collision: when does it occur?

It has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation where a fixed point remains stable at both sides of the bifurcation point, and yet the orbit becomes unbounded at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size. Such bifurcations have been named "dangerous bifurcations." In this paper we provide explanation of this phenomenon, and develop the analytical conditions on the parameters under which such dangerous bifurcations will occur.

Border collision bifurcations at the change of state-space dimension.

We present the theory of border collision bifurcation for the special case where the state space is piecewise smooth, but two-dimensional in one side of the borderline, and one dimensional in the other side. This situation occurs in a class of switching circuits widely used in power electronic industry. We analyze this particular class of bifurcations in terms of the normal form, where the determinant of the Jacobian matrix at one side of the borderline is greater than unity in magnitude, and in the other side it is zero. (c) 2002 American Institute of Physics.