International Journal of Bifurcation and Chaos

Quantum billiards provide an excellent forum for the analysis of quantum chaos. Toward this end, we consider quantum billiards with time-varying surfaces, which provide an important example of quantum chaos that does not require the semiclassical (ℏ → 0) or high quantum-number limits. We analyze vibrating quantum billiards using the framework of Riemannian geometry. First, we derive a theorem detailing necessary conditions for the existence of chaos in vibrating quantum billiards on Riemannian manifolds. Numerical observations suggest that these conditions are also sufficient. We prove the aforementioned theorem in full generality for one degree-of-freedom boundary vibrations and briefly discuss a generalization to billiards with two or more degrees-of-vibrations. The requisite conditions are direct consequences of the separability of the Helmholtz equation in a given orthogonal coordinate frame, and they arise from orthogonality relations satisfied by solutions of the Helmholtz equation. We then state and prove a second theorem that provides a general form for the coupled ordinary differential equations that describe quantum billiards with one degree-of-vibration boundaries. This set of equations may be used to illustrate KAM theory and also provides a simple example of semiquantum chaos. Moreover, vibrating quantum billiards may be used as models for quantum-well nanostructures, so this study has both theoretical and practical applications.

We consider oscillations of the length and width in rectangular quantum billiards, a two "degree-of-vibration" configuration. We consider several superpositon states and discuss the effects of symmetry (in terms of the relative values of the quantum numbers of the superposed states) on the resulting evolution equations and derive necessary conditions for quantum chaos for both separable and inseparable potentials. We extend this analysis to n-dimensional rectangular parallelepipeds with two degrees-of-vibration. We produce several sets of Poincaré maps corresponding to different projections and potentials in the two-dimensional case. Several of these display chaotic behavior. We distinguish between four types of behavior in the present system corresponding to the separability of the potential and the symmetry of the superposition states. In particular, we contrast harmonic and anharmonic potentials. We note that vibrating rectangular quantum billiards may be used as a model for quantum-well nanostructures of the stated geometry, and we observe chaotic behavior without passing to the semiclassical (ℏ → 0) or high quantum-number limits.

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several first-order Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious solutions and transition to chaos are investigated by presenting analytical and numerical results. The overall performance of the methods for different parameters is discussed.

We analyze a mean-field model for a large array of coupled solid-state lasers with randomly distributed natural frequencies. Using techniques developed previously for coupled nonlinear oscillators, we derive exact formulas for the stability boundaries of the phase locked, incoherent, and off states, as functions of the coupling and pump strength and the spread of natural frequencies. For parameters in the intermediate regime between total incoherence and perfect phase locking, numerical simulations reveal a variety of unsteady collective states in which all the lasers' intensities vary periodically, quasiperiodically, or chaotically.

In this paper we consider a class of optimal control problem involving a chaotic system, where all admissible controls are required to satisfy small boundedness constraints. A numerical approach is developed to seek for an optimal feedback control for the optimal control problem. In this approach, the state space is partitioned into subregions, and the controller is approximated by a linear combination of a modified third order B-spline basis functions. The partition points are also taken as decision variables in this formulation. An algorithm based on this approach is proposed. To show the effectiveness of the proposed method, a control problem involving the Lorenz system is solved by the proposed approach. The numerical results demonstrate that the method is efficient in the construction of a robust, near-optimal control.

This paper studies numerical methods for linear stability analysis of periodic solutions in codes for bifurcation analysis of small systems of ordinary differential equations (ODEs). Popular techniques in use today (including the AUTO97 method) produce very inaccurate Floquet multipliers if the system has very large or small multipliers. These codes compute the monodromy matrix explicitly or as a matrix pencil of two matrices. The monodromy matrix arises naturally as a product of many matrices in many numerical methods, but this is not exploited. In this case, all Floquet multipliers can be computed with very high precision by using the periodic Schur decomposition and corresponding algorithm [Bojanczyk et al., 1992]. The time discretisation of the periodic orbit becomes the limiting factor for the accuracy. We present just enough of the numerical methods to show how the Floquet multipliers are currently computed and how the periodic Schur decomposition can be fitted into existing codes but omit all details. However, we show extensive test results for a few artificial matrices and for two four-dimensional systems with some very large and very small Floquet multipliers to illustrate the problems experienced by current techniques and the better results obtained using the periodic Schur decomposition. We use a modified version of AUTO97 [Doedel et al., 1997] in our experiments.

This paper implements a center-manifold technique to arrive at a normal-form for the natural dynamics of a passive, bipedal rigid-body mechanism in the vicinity of infinite foot width and near-symmetric body geometry. In particular, numerical schemes are developed for finding approximate forms of the relevant invariant manifolds and the near-singular dynamics on these manifolds. The normal-form approximations are found to be highly accurate for relatively large foot widths with a range of validity extending to widths on the order of the mechanisms' height.

In this paper, we investigate the possibility of using interval arithmetic for rigorous investigations of periodic orbits in discrete-time dynamical systems with special emphasis on chaotic systems. We show that methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n. We compare several interval methods for finding periodic orbits. We consider the interval Newton method and methods based on the Krawczyk operator and the Hansen–Sengupta operator. We also test the global versions of these three methods. We propose algorithms for computation of the invariant part and nonwandering part of a given set and for computation of the basin of attraction of stable periodic orbits, which allow reducing greatly the search space for periodic orbits.

As examples we consider two-dimensional chaotic discrete-time dynamical systems, defined by the Hénon map and the Ikeda map, with the "standard" parameter values for which the chaotic behavior is observed. For both maps using the algorithms presented in this paper, we find very good approximation of the invariant part and the nonwandering part of the region enclosing the chaotic attractor observed numerically. For the Hénon map we find all cycles with period n ≤ 30 belonging to the trapping region. For the Ikeda map we find the basin of attraction of the stable fixed point and all periodic orbits with period n ≤ 15. For both systems using the number of short cycles, we estimate its topological entropy.

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

We present one-parameter families of rational functions of arbitrary degree d which are globally generalized polynomial-like of degree d and roughly speaking locally quadratic-like everywhere, where the parameter appears not only as a purely multiplicative factor but also in a more complicated nonlinear way. The connectedness locus of these families contains homeomorphic copies of the Mandelbrot set. Main emphasis is put on the explicit construction (and not as usual on the existence only) of the sets on which generalized polynomial-likeness and quadratic-likeness are given as well as on the explicit description of the regions where the homeomorphic copies of the Mandelbrot set are located.

The singularity theorem is used to derive the deformations of the static bifurcation diagram of the Gray–Scott model.

In this Letter, we study the bifurcation of the Kuramoto–Sivashinsky (K–S) equation in one-spatial dimension with three kinds of boundary value conditions. Using the Liapunov–Schmidt reduction technique, the original equation is first reduced to one or two bifurcation equations, so that bifurcation analysis of the original equation can be transformed to that of the reduced-order systems, and can therefore be carried out in detail.

Kink solutions in coupled sine circle map lattices demonstrate interesting bifurcation behavior. These are illustrated by the study of spatial period two kink solutions for this system. Different types of spatiotemporal solutions such as temporally frozen kinks, spatiotemporally synchronized solutions and kink induced temporally intermittent solutions appear in different regions of parameter space for this system and bifurcations are seen from one type of solution to another. The upper boundaries of the regions where the kinks are stable can be picked up by linear stability analysis. However, the eigenvalues of the stability matrix do not cross the unit circle along the lower stability boundaries, although the nature of the solution changes. Thus linear stability analysis is not sufficient to identify these lower boundaries. Hence we have proposed new characterisers which are capable of identifying such boundaries. Our identifiers successfully pick up the lower boundaries missed by linear stability analysis as well as the upper boundaries. Our characterisers could be of utility in other situations as well.

This paper studies bifurcations in a three node power system when excitation limits are considered. This is done by approximating the limiter by a smooth function to facilitate bifurcation analysis. Spectacular qualitative changes in the system behavior induced by the limiter are illustrated by two case studies. Period doubling bifurcations and multiple attractors are shown to result due to the limiter. Detailed numerical simulations are presented to verify the results and illustrate the nature of the attractors and solutions involved.

In this letter it is proven that any piecewise-linear (PWL) function can be realized as the sum of a definite number of basic PWL functions, like that implemented in the classical Cellular Neural Network (CNN) structure by L. Chua. This allows us to consider the basic Chua's PWL function as a fundamental unit to build up a generic PWL function and, thus, to extend the class of nonlinear dynamics that can be realized by CNNs to a more wide class including PWL approximation of any nonlinearity. Moreover, an example of the cell realization of such a CNN is reported.