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Classical chaos

Spatiotemporal chaos in terms of unstable recurrent patterns

Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

P. Cvitanovic, V Putkaradze (and F. Christiansen)

Trace Formula for Dynamic Systems in the Presence of Noise

Evolution operator methods for evaluation of averages in chaotical dynamical systems require arbitrarily fine symbolic dynamics partition of the phase space; at the same time, the attainable accuracy in physical dynamical systems is always limited by external noise. We use external noise as a parameter which defines the smallest phase-space partitions that need to be taken into account in a given computation, and generate finite Markov diagram approximation to the dynamics.

P. Cvitanovic, (R. Mainieri and G. Vattay)

Symbolic Dynamics and Markov Partitions for the Stadium Billiard

We investigate the Bunimovich stadium dynamics and show that in the limit of infinitely long stadium the symbolic dynamics is a subshift of finite type. For a stadium of finite length the Markov partitions are infinite, but the inadmissible symbol sequences can be determined exactly by means of the appropriate pruning front. We outline a construction of a sequence of finite Markov graph approximations by means of approximate pruning fronts with finite numbers of steps.

P. Cvitanovic and K.T. Hansen

Topology and Bifurcation Structure of Maps of Hénon type

We construct a series of n-modal approximations to maps of the Hénon type and apply the corresponding symbolic dynamics to description of the admissible orbits and possible bifurcation structures for such maps.

P. Cvitanovic and K.T. Hansen

Shock Lines and Bound States in the Complex Ginzburg-Landau Equation

Cellular patterns of shock lines appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field. In the case of the complex Ginzburg-Landau equation, disordered cells of effectively frozen spirals appear, separated by thin walls (shocks), on a scale much larger than the basic wavelength of the spirals. We show that these structures can be understood in very simple terms. In particular, we show that the walls are, to a good approximation, segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex. The fact that the phase is only defined up to an integer multiple of tex2html_wrap_inline1004 introduces a quantization condition on the sizes of the smallest spiral domains. The transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations. The structure depends on the angle the wall makes with the local phase contours, and the behavior can be either monotonic or oscillatory depending on the parameters.

T. Bohr, (G. Huber, E. Ott and M. Bazhenov)

Classical Dynamics of Billiards with Step

We study the classical dynamics of a billiard system with two different potentials in the interiour. The step changes the direction of the classical path or give a reflection. The classical dynamics is partly chaotic and partly regular and we describe the structure of the system.

K.T. Hansen and (A. Köhler)

Classical Description of the Limacon Billiards

We describe the dynamics of the limacon billiards using symbolic dynamics.

K.T. Hansen and (A. Bäcker and H. Dullin)

Periodic Fluctuations in Northern Microtines

The size of nonlinear terms and the dependence on the geographical gradient in the description of the population of northern microtines, is investigated using reconstruction of circle maps.

K.T. Hansen and (W. Falck and N.C. Stenseth)

Nonlinear Dynamics of a Vectored Thrust Aircraft

Thrust vectoring allows an aircraft to venture into regions of operation that cannot be reached with conventional controls. Using relations for the aerodynamic coefficients obtained by NASA in wind tunnel experiments with the F/A-18, we have performed a detailed simulation study of the longitudinal dynamics in the post-stall regime. Under variation of the the thrust magnitude and the thrust vectoring angle, the equilibrium state exhibits three Hopf and two saddle-node bifurcations. If, in an attempt to stabilize the oscillatory behavior, the thrust deflection is periodically adjusted, a complicated structure of overlapping saddle-node, period-doubling and torus bifurcations arise. Responding to a request from SAAB staff members, we have investigated some of the main features of this structure by means of one- and two-dimensional continuation techniques.

(P. Gránásy), E. Mosekilde, and C.B. Soerensen

Bifurcation Structure of an Optical Ring Cavity

One- and two-dimensional continuation techniques have been applied to determine the main bifurcation structure of an optical ring cavity with an absorbing element. Variation of the speed of light in this element with the light intensity causes the system to exhibit a series of resonances, leading to multistability and to families of similar solutions in parameter space. Within the individual family, the organization of solutions displays an infinite number of regulatory arranged domains of overlapping period-doubling and saddle-node bifurcations.

C. Kubstrup and E. Mosekilde

Bifurcations in Two Coupled Roessler Systems

We have performed a bifurcation analysis of two symmetrically coupled Roessler systems. The imposed symmetry does not allow any one direction to become preferred, and the coupled systems is therefore an example of a higher order system that cannot effectively be reduced to a one-dimensional system. A result hereof is the exchange of some of the period-doubling bifurcations of the individual system with Hopf bifurcations in the coupled system. This again leads to a replacement of the Feigenbaum transition to chaos by a quasiperiodic transition. Moreover, the system exhibits hyperchaos in a significant part of parameter space. A more detailed theoretical analysis of this problem was given in a paper by Reick and Mosekilde last year. Here, it was also proved that the replacement of some of the period-doubling bifurcations by Hopf bifurcations is robust to a certain degree of asymmetry between the two subsystems.

E. Mosekilde, J. Rasmussen and (C. Reick)

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Klaus Lindemann
Fri Feb 21 15:17:28 MET 1997