Bifurcation Plots Research Articles

HYPERCHAOS IN A NEW FAMILY OF SIMPLE CNNS

In this paper we demonstrate hyperchaotic dynamics in a new family of simple Cellular Neural Networks (CNNs) which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots and several important phase portraits are presented as well.

Convergence analysis of first order reliability method using chaos theory

First order reliability method (FORM) for the computation of reliability index has been used widely because of its advantages of the efficiency and effectiveness as well simplicity for many years. There exists the phenomenon of convergent failure in the FORM in calculating the reliability index iteratively for some limit state functions, for which the essential factor is investigated using chaotic dynamics theory in the present paper. The bifurcation plots of reliability index are presented for several typical limit state functions, and the computational results from those mapping functions due to FORM iterations show the complicated dynamics phenomena such as the periodic oscillation, bifurcation and chaos. Moreover, the Lyapunov exponents of non-linear map from FORM are calculated. From the numerical investigation of presented examples, it is concluded that the convergence of FORM does not depend on the curvature of design points of the limit state function, and the quantitative index for identifying the convergence of FORM iterative computation is the Lyapunov exponents of non-linear map corresponding to that limit state function.

Development of the method and analysis for non-linear lateral stability of railway vehicles in a curved track

This article is the authors’ most recent publication on railway vehicle lateral stability in a curved track. Developed procedures constitute the original and mature method of stability analysis, which is presented here. The method is based on the analysis of bifurcation plots built as a result of numerical simulations for a wide range of curve radii and straight track, as well. Despite the development of the method, new areas of its application are presented. These are studies on the influence of various factors on the stability. These factors apply to the variation of vehicle suspension parameters, wheel–rail profiles, and wear of the wheel–rail profiles in terms of both the magnitude and the type. Also, new forms of result presentation have been employed.

Bifurcations and competing coherent structures in the cubic-quintic Ginzburg–Landau equation I: Plane wave (CW) solutions

Singularity Theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg-Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcations curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space.

Dark-lines in bifurcation plots of nonlinear dynamic systems

Based on the regressive character of chaotic motion in nonlinear dynamic systems, a numerical regression algorithm is developed, which can be used to research the dark-lines passing through chaotic regions in bifurcation plots. The dark-lines of the parabolic mapping are obtained by using the numerical regression algorithm and compared with those that are accurately acquired through dark-line equations. Thus the validity of this algorithm is proved. Furthermore, for the Brussel oscillation system and the piecewise linear dynamic system of a gear pair, the dark-lines are researched by using the regression algorithm. By researching the dark-lines in the bifurcation plots of nonlinear dynamic systems, the periodic windows embedded in chaotic regions can be ascertained by tangential points of dark-lines and the turning points of chaotic attractors can be also obtained by intersected points. The results show that this algorithm is helpful to analyse dynamic behaviour of systems and control chaotic motion.

Hopf bifurcation and chaos analysis of Chen’s system with distributed delays

Abstract In this paper, a general model of nonlinear systems with distributed delays is studied. Chen’s system can be derived from this model with the weak kernel. After the local stability is analyzed by using the Routh–Hurwitz criterion, Hopf bifurcation is studied, where the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical analysis are also presented. Chaotic behavior of Chen’s system with the strong kernel is also found through numerical simulation, in which some waveform diagrams, phase portraits, and bifurcation plots are presented and analyzed.

BIFURCATIONS IN NUMERICAL METHODS FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

We are interested in finding approximate solutions to parameter-dependent Volterra integro-differential equations over long time intervals using numerical schemes. This paper concentrates on changes in qualitative behavior (bifurcations) in the solutions and extends the work of Brunner and Lambert and Matthys (who considered only changes in stability behavior) to consider other bifurcations. We begin by considering a one-parameter equation with fading memory separable convolution kernel: we give an analytical discussion of bifurcations in this case and provide details of the behavior of numerical schemes. We extend our analysis to consider an equation with two-parameter fading memory convolution kernel and show the relationship to the classical test equation studied by the earlier authors. We draw attention to the fact that known stability results may not provide a reliable framework for choice of numerical scheme when other changes in qualitative behavior are also of interest. We give bifurcation plots for a variety of methods and show how, for known values of the parameters, stepsizes h>0 may be chosen to preserve the correct qualitative behavior in the numerical solution of the Volterra integro-differential equation.

Population models of genomic imprinting. I. Differential viability in the sexes and the analogy with genetic dominance.

Many single-locus, two-allele selection models of genomic imprinting have been shown to reduce formally to one-locus Mendelian models with a modified parameter for genetic dominance. One exception is the model where selection at the imprinted locus affects the sexes differently. We present two models of maternal inactivation with differential viability in the sexes, one with complete inactivation, and the other with a partial penetrance for inactivation. We show that, provided dominance relations at the imprintable locus are the same in both sexes, a globally stable polymorphism exists for a range of viabilities that is independent of the penetrance of imprinting. The conditions for a polymorphism are the same as in previous models with differential viability in the sexes but without imprinting and in a model of the paternal X-inactivation system in marsupials. The model with incomplete inactivation is used to illustrate the analogy between imprinting and dominance by comparing equilibrium bifurcation plots for fixed values of dominance and penetrance. We also derive a single expression for the dominance parameter that leaves the frequency and stability of equilibria unchanged for all levels of inactivation. Although an imprinting model with sex differences does not formally reduce to a nonimprinting scheme, close theoretical parallels clearly exist.

Poincaré maps of Duffing-type oscillators and their reduction to circle maps: II. Methods and numerical results

Bifurcation diagrams and plots of Lyapunov exponents in the r- plane for Duffing-type oscillators exhibit a regular pattern of repeating self-similar `tongues' with complex internal structure. We demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a Poincare map of the system in action-angle variables in the limit of large driving force or, equivalently, small driving frequency. This map approaches the one-dimensional form as derived in paper I. This second paper describes our approach to calculating the various constants and functions introduced in paper I. It gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of Duffing-type equations. Finally we show how to extend the range in the parameter space where the map is applicable.

Comments on the numerical investigation of Rayleigh and Marangoni convection in a vertical circular cylinder

Convection in a cylindrical container was simulated with a three-dimensional, time-dependent code. For the case of purely Rayleigh convection, a completely rigid cylinder with adiabatic vertical walls and conducting horizontal walls was considered. The calculations showed that the individual velocity components could be in a transient state, while the total heat transfer was steady. This occurred in cases where the maximum azimuthal component of velocity was very small in magnitude, in comparison to the other two components, and this component decreased in time. On the other hand, the total kinetic energy along with the heat transfer reached a steady value. Consequently the present results have been shown to be at variance with the calculations of Neumann [J. Fluid Mech. 214, 559 (1990)]. Marangoni convection was modeled with a free flat surface on the upper side, assuming the superimposed second layer to be passive. The numerically obtained critical Marangoni numbers and flow patterns were compared favorably to earlier results from linearized stability. In addition, flow structural changes for supercritical Marangoni numbers were illustrated. Interestingly, axisymmetric disturbances led to nonsymmetric bifurcation diagrams, but three-dimensional disturbance calculations led to symmetry in the bifurcation plots. Another very interesting result was the observed transition from three-dimensional to two-dimensional patterns as the Marangoni number was increased. Large computational requirements precluded a detailed parametric study. The special case of Prandtl number equal to 6.7 (corresponding to water), and in the case of Marangoni convection, a surface Biot number of unity was assumed.

Poincare maps of Duffing-type oscillators and their reduction to circle maps. I. Analytic results

Bifurcation diagrams and plots of Lyapunov exponents in the r- Omega -plane for Duffing-type oscillators x+2rx+V'(x, Omega t)=0 exhibit a regular pattern of repeating self-similar 'tongues' with complex internal structure. The authors demonstrate here that this behaviour is easily understood qualitatively and quantitatively from the Poincare map of the system in action-angle variables. This map approaches the one dimensional form phi n+1=A+C e-rT cos phi n;T= pi / Omega ; provided e-rT (but not necessarily Ce-rT), r and Omega are small. The authors derive asymptotic (for small r, Omega ) formulae for A and C for a special class of potentials V. They argue that these special cases contain all the information needed to treat the general case of potentials which obey V">or=0 at all times. The essential tools of the derivation are the use of action-angle variables, the adiabatic approximation and the introduction of a non-oscillating reference solution of Duffing's equation, with respect to which the action-angle variables have to be determined. These allow the explicit construction of the Poincare map in powers of e-rT. To first order, the authors obtain the phi -map, which survives asymptotically. To second order they obtain the two-dimensional I- phi -map. In the I direction it contracts by a factor e-rT upon each iteration.