International Journal of Bifurcation and Chaos

The existence of stable spiral wave solutions in a spatially discrete λ-ω system is proven. The waves are shown to exist only if the coupling parameter d is sufficiently small. There is a critical value d_c such that if d>d_c, the spiral wave solution ceases to exists. This is in agreement with the behavior of such waves in spatially continuous λ-ω systems on finite domains.

The dynamics of a closed thermosyphon are considered. Using an explicit construction, obtained through an inertial manifold, exact low-dimensional models are derived. The behavior of solutions is analyzed for different ranges of the relevant parameters, and the Lorenz model is obtained for a range of parameter values. Numerical experiments are performed for three- and five-mode models.

We study the kinematics of mixing and transport of "passive" particles in two-dimensional thermal convection using a low-order spectral model proposed by Howard and Krishnamurti [1986]. This model allows for a large-scale flow spanning the layer width. Mixing by chaos in the oscillatory flow is demonstrated with the help of numerical Poincaré maps. The chaos induced transport process is characterized from a relation of the form ΔX²(t)~ t^m, for large t, where ΔX²(t) is the mean square distance traveled by a cloud of particles. It is shown that the transport process can be either shear-flow dominated (m=2) or Brownian type (m=1) or an intermediate type characterized by fractional exponents (1<m<2), depending on the external parameters in the problem. The intermediate process has been found to be intermittent in nature. The present results are compared with earlier studies of chaotic advection and also some experimental observations. Directions for future work are also pointed out.

This study provides some connections between bifurcations of one-complex-parameter complex analytic families of maps of the complex plane C and bifurcations of more general two-real-parameter families of real analytic (or C^k or C^∞) maps of the real plane ℛ². We perform a numerical study of local bifurcations in the families of maps of the plane given by

The most striking phenomenon that appears as α is varied is that bifurcation points in the C plane for the quadratic family (α=0) evolve into fascinating bifurcation regions in the C plane for nonzero α. Such points are the cusp of the main cardioid of the Mandelbrot set and contact points between "bulbs" of the Mandelbrot set. Arnold resonance tongues are part of the evolved scenario.

We also provide sufficient conditions for more general perturbations of complex analytic maps of the plane of the form:

If the critical set of a planar quadratic map is bounded and nonempty, then the range is doubly covered except the image of the critical set and the region bounded by it (Fig. 1). For a map of this type, we showed that infinity is a sink and the set of bounded orbits is nonempty hence its boundary may be regarded as a generalization of Julia sets of complex quadratic maps. In this paper, we also provided evidences to conjecture that maps of this type have infinitely many periodic points.

An algorithm to detect homoclinic twist bifurcation points in Z₂-symmetric autonomous systems of ordinary differential equations in ℝ⁴ along curves of symmetric homoclinic orbits to hyperbolic equilibria has been developed. We show convergence of numerical approximations to homoclinic twist bifurcation points in such systems. A test function is defined on the homoclinic solutions, which has a regular zero at the codimension-two bifurcation points. This codimension-two singularity can be continued appending the test function to a three-parameter vector field. We demonstrate the use of the test function on an example of two-coupled Josephson junctions.

We study double bifurcations of a reaction–diffusion problem, and numerical methods for the continuation of bifurcating solution branches. To ensure a correct reflection of the bifurcation scenario in discretizations and to reduce imperfection of bifurcations, we consider a preservation of multiplicities of the bifurcation points in the discrete problems. A continuation-Arnoldi algorithm is exploited to trace the solution branches, and to detect secondary bifurcations. Numerical results on the Brusselator equations confirm our analysis.

We consider a model proposed by us earlier for describing a form of plastic instability found in creep experiments. The model consists of three types of dislocations and some transformations between them. The model is known to reproduce a number of experimentally observed features. The mechanism for the phenomenon has been shown to be Hopf bifurcation with respect to physically relevant drive parameters. Here, we present a mathematical analysis of adiabatically eliminating the fast mode and obtaining a Ginzburg-Landau equation for the slow modes associated with the steps on creep curve. The transition to the instability region is found to be one of subcritical bifurcation over a major part of the interval of one of the parameters while supercritical bifurcation is found in a narrow mid-range of the parameter. This result is consistent with experiments. The dependence of the amplitude and the period of strain jumps on stress and temperature derived from the Ginzburg-Landau equation are also consistent with experiments. On the basis of detailed numerical solution via power series expansion, we show that high order nonlinearities control a large portion of the subcritical domain.

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded single-machine-infinite-busbar power system modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System. The system has five mechanical and two electrical modes. The results show that, as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability in a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. The results show that adding damper windings may induce subsynchronous resonance.

It was shown by Jackson and Grosu [1995a] that entrainment control of a complex dynamic system could be achieved by using an open-plus-closed-loop (OPCL) control action. Jackson [1995] showed that, for multiple-attractor systems, it is also possible to transfer the dynamics of a system between arbitrary attractors, using this control only for a limited period of time ("migration control"). In the case of the Chua system [1986], Jackson and Grosu [1995b] showed that this migration action can be achieved using only a limited pre-recorded signal of one variable of the "target" attractor. Here we study the experimental implementation of this migration action on a Chua circuit with five attractors (fixed points, a limit cycle, and chaotic attractors) [Wu & Pivka, 1994]. It is shown that reliable migrations are possible with a pre-recorded signal of one target variable, provided that the signal is longer than a minimal time, dependent on an adjustable control strength. The minimal time for successful migration is studied for all distinct migration processes. It is also shown that this action is robust to the limited accuracy of the recorded data.

In this paper, determinism in human posture control data is investigated using the approach of nonlinear prediction. We first comment that one should be cautious of using some statistical methods to analyze nonstationary time series. Then we test the predictability of the human posture control data with different prediction techniques, and investigate how nonstationarity and noise affect the prediction results. Different time series are tested, including data from healthy and ill persons, and different predictabilities are found in different time series.