Showing papers on "Infinite-period bifurcation published in 1988"

Iterates of maps with symmetry

Abstract: In this paper the elementary aspects of bifurcation of fixed points, period doubling, and Hopf bifurcation for iterates of equivariant mappings are discussed. The most interesting of these is an algebraic formulation of the hypotheses of Ruelle’s theorem (D. Ruelle [1973], “Bifurcations in the presence of a symmetry group,” Arch. Rational Mech. Anal., 51, pp. 136–152) on Hopf bifurcation in the presence of symmetry.In the last sections this result is used to show that Hopf bifurcation from standing waves in a system of ordinary differential equations with $O(2)$ symmetry can lead directly to motion on an invariant 3-torus; indeed, depending on the exact symmetry of the standing waves, one might expect to see three invariant 3-tori emanating from such a bifurcation. The unexpected third frequency comes from drift along the torus of standing waves whose existence is forced by the $O(2)$ symmetry.

*Forced Oscillations of a Self-Oscillating Bimolecular Surface Reaction Model

Abstract: The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

A global index for bifurcation of fixed points.

Abstract: 3~ is called the set of trivial fixed points. We are interested in connected branches of fixed points of / bifurcating from y. Let ^--={(λ, x) e 0\\y\\f(^ x) = x} be the set of nontrivial fixed points of /. 28 -— F n y is the set of bifurcation points. If J^ is compact we define an index BI(/) which lies in the (k — l)-stem π^_ χ =co fc_1(pt); ω^ the stable homotopy theory (see G. W. Whitehead [25], Chapter 12). For small values of k these groups are well known. For example, o>0(pt)^Z, co1(pt)^Z2, co2(pt)^Z2. If ΒΙ(/)ΦΟ then a connected subset £f of y exists which bifurcates from y and is not contained in any compact subset of &. There is a striking analogy to the fixed point index (or the Brouwer-Leray-Schauder degree), where the fixed points correspond to the bifurcation points. In particular, BI (/) is homotopy invariant and additive.

Power-series expansion of the potential of the Fokker-Planck equation

Abstract: The potential of Fokker-Planck equations lacking detailed balance is calculated by systematically applying a power-series-expansion approach. With respect to the expansion approach, five kinds of points in variable space, i.e., hyperbolic stable and unstable points, saddles, bifurcation singular points, regular points, and singular points of higher orders, are distinguished.

Unusual bifurcation in a constrained random walk system

Abstract: A random walk with fixed arc length and fixed endpoints which has to pass through k fixed points is considered. It is shown analytically that for the case k=1 and equal distance R between the respective endpoints and the intermediate point a bifurcation occurs with respect to the arc length which the walk traces out between its origin and the intermediate point if R falls below a critical value. From numerical results it follows that this bifurcation also occurs in the following cases: when the intermediate point is not equidistantly positioned between the endpoints; when the walk is not confined to a point but to a hoop; and when the walk has to pass through several points (k)1).

Periodic structure beyond a hope bifurcation

Abstract: A Hopf bifurcation where a stable fixed pointer bifurcates into a stable periodic orbit as a parameter passes through a critical value, occurs frequently in nonlinear problems. Here, the Maynard Smith nonlinear map in population dynamics involving a parameter k is analysed with the help of numerical experiments in order to determine the periodic structure of the map beyond the primary Hopf bifurcation of period 6 which occurs at k = 1. The interesting result obtained is that in the parameter range from k = 1 to the value for blow-up of all initial conditions, there are successive windows of periods 7, 8 and 9, the last containing a secondary Hopf bifurcation of period 4.

Analysis of disturbance-generated bifurcation in an adaptive system with e/sub 1/-modification law

Abstract: A modified adaptation law, the so-called e/sub 1/-modification law, has been proposed by K.S. Narendra and A.M. Annaswamy (1987) for robust model-reference adaptive control in the presence of bounded disturbances. Using this law, the authors show analytically that in the case of regulation, a prototype adaptive control system, with a single unknown pole and an unknown constant disturbance at its input, possesses multiple equilibria and undergoes various bifurcations. The analysis begins by transforming the augmented system equations so that there are only two independent parameters. Then explicit expressions for the equilibria and their bifurcations are given, followed by the proof of the existence of the Hopf bifurcation. As the constant disturbance is varied, saddle-node bifurcation, (subcritical) Hopf bifurcation, and saddle-connection bifurcation occur in that order. Consequently, in this case, the e/sub 1/-modification law is qualitatively equivalent to the sigma -modification law. >

A qualitative and quantitative study of steady-state response of towed floating bodies

Abstract: The problem of multiple equilibria in steady towing of a floating body is considered. The two coordinates of the towing point are the main bifurcation parameters. An approach to bifurcation of steady-state equilibria using singularity theory reveals all qualitatively different bifurcation diagrams that occur locally. It is shown that these bifurcation problems may be viewed as paths in the universal unfolding space of the cusp catastrophe. The organizing centre for the towing problem is the pitchfork singularity. Numerical calculations suggest that results obtained by singularity-theory techniques are valid globally.

Neosteady and neoperiodic solutions

Abstract: This paper considers the secondary and cascading bifurcation of two-dimensional steady and period thermal convection states in a rotating box. Previously developed asymptotic and perturbation methods that rely on the coalescence of two, steady convection, primary bifurcation points of the conduction state as the Taylor number approaches a critical value are employed. A multitime analysis is employed to construct asymptotic expansions of the solutions of the initial-boundary value problem for the Boussinesq theory. The small parameter in the expansion is proportional to the deviation of the Taylor number from its critical value. To leading order, the asymptotic expansion of the solution involves the mode amplitudes of the two interacting steady convection states. The asymptotic analysis yields a first-order system of two coupled ordinary differential equations for the slow-time evolution of these amplitudes. We investigate the steady states of these amplitude equations and their linearized stability. These equations suggest that the following sequence of transitions may occur as the Rayleigh number R is increased, for a fixed value of the Taylor number near the critical value: First, the conduction state loses stability at a primary bifurcation point to steady convection states (rolls) characterized by a single wavenumber. Then, these states lose stability at a secondary bifurcation point to other steady convection states characterized by two different wavenumbers. Finally, these states become unstable at a tertiary Hopf bifurcation point, R = RH, to time-periodic convection states, which are also characterized by the same two wavenumbers. At third order, the Hopf bifurcation is degenerate since for R = RH there is a one-parameter family of periodic solutions, which is bounded in the phase plane by a heteroclinic orbit. For R ? RH, but close to RH, the center of the system's phase plane trajectories is transformed into a focus which is unstable (stable) for R > RH ( RH, and thus escape from it. The new results presented here are: (a) an example of a semibounded convection system with a Hopf bifurcation branch generated by the interaction of two steady state branches; and (b) a detailed study of the transient motion near the "vertical" Hopf bifurcation branch. The "bending" of the branch is described in detail elsewhere.