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Dec 1994

Volume 4, Issue 4, pp. 595-705


Separatrices splitting for Birkhoff’s billiard in symmetric convex domain, closed to an ellipse

M. B. Tabanov

Chaos 4, 595 (1994); http://dx.doi.org/10.1063/1.166037 (12 pages) | Cited 7 times

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An example of a convex domain on the plane with the phenomenon of the transversal intersection of separatrices of the corresponding billiard mapping is presented. This example is constructed as an analytic global symmetric perturbation of an ellipse and we investigate the global symmetric analytic perturbation of the integrable billiard mapping in the ellipse. We establish a theorem on the separatrices splitting of the perturbed billiard mapping and derive the asymptotic formulas for a homoclinic invariant as well as for a ‘‘principal’’ splitting angle of separatrices, arising from the hyperbolic fixed point of the mapping. © 1994 American Institute of Physics.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos

Chaotic domains: A numerical investigation

M. C. Cross, D. Meiron, and Yuhai Tu

Chaos 4, 607 (1994); http://dx.doi.org/10.1063/1.166038 (13 pages) | Cited 35 times

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We study the chaotic domain state in rotating convection using a model equation that allows for a continuous range of roll orientations as in the experimental system. Methods are developed for extracting the domain configuration from the resulting patterns that should be applicable to a wide range of domain states. Comparison with the truncated three mode amplitude equation description is made. © 1994 American Institute of Physics.
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47.27.-i Turbulent flows
05.45.-a Nonlinear dynamics and chaos
47.52.+j Chaos in fluid dynamics

Stability and bifurcations of a stationary state for an impact oscillator

Jan‐Olov Aidanpää, Hayley H. Shen, and Ram B. Gupta

Chaos 4, 621 (1994); http://dx.doi.org/10.1063/1.166039 (10 pages) | Cited 2 times

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The motion of a vibroimpacting one‐degree‐of‐freedom model is analyzed. This model is motivated by the behavior of a shearing granular material, in which a transitional phenomenon is observed as the concentration of the grains decreases. This transition changes the motion of a granular assembly from an orderly shearing between two blocks sandwiching a single layer of grains to a chaotic shear flow of the whole granular mass. The model consists of a mass‐spring‐dashpot assembly that bounces between two rigid walls. The walls are prescribed to move harmonically in opposite phases. For low wall frequencies or small amplitudes, the motion of the mass is damped out, and it approaches a stationary state with zero velocity and displacement. In this paper, the stability of such a state and the transition into chaos are analyzed. It is shown that the state is always changed into a saddle point after a bifurcation. For some parameter combinations, horseshoe‐like structures can be observed in the Poincaré sections. Analyzing the stable and unstable manifolds of the saddle point, transversal homoclinic points are found to exist for some of these parameter combinations. © 1994 American Institute of Physics.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos

Complexity of precipitation patterns: Comparison of simulation with experiment

A. A. Polezhaev and S. C. Müller

Chaos 4, 631 (1994); http://dx.doi.org/10.1063/1.166040 (6 pages) | Cited 16 times

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Numerical simulations show that a simple model for the formation of Liesegang precipitation patterns, which takes into account the dependence of nucleation and particle growth kinetics on supersaturation, can explain not only simple patterns like parallel bands in a test tube or concentric rings in a petri dish, but also more complex structural features, such as dislocations, helices, ‘‘Saturn rings,’’ or patterns formed in the case of equal initial concentrations of the source substances. The limits of application of the model are discussed. © 1994 American Institute of Physics.
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64.75.-g Phase equilibria
64.60.Q- Nucleation
82.40.Ck Pattern formation in reactions with diffusion, flow and heat transfer

Four modes competition and chaos in a shell

Evgeny S. Dekhtyaryuk, Tatyana G. Zakharchenko, Yuri S. Petryna, and Tatyana S. Krasnopolskaya

Chaos 4, 637 (1994); http://dx.doi.org/10.1063/1.166041 (14 pages)

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Steady‐state chaotic vibrations of a shallow shell as a system with a nonsymmetrical restoring force and one equilibrium state are considered. Mode interaction and its effect on a chaotic behavior of the shell is studied. The terms ‘‘natural’’ and ‘‘imposed’’ chaos are introduced for the response of resonant and nonresonant modes. It is shown that such a qualitative difference is important for better understanding of chaos in systems with distributed parameters, and may be very useful for numerical investigations. Some qualitative comparisons with previous papers on chaos in distributed mechanical systems are also made. © 1994 American Institute of Physics.  
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62.30.+d Mechanical and elastic waves; vibrations
05.45.-a Nonlinear dynamics and chaos

Normally attracting manifolds and periodic behavior in one‐dimensional and two‐dimensional coupled map lattices

Claudio Giberti and Cecilia Vernia

Chaos 4, 651 (1994); http://dx.doi.org/10.1063/1.166042 (13 pages) | Cited 3 times

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We consider diffusively coupled logistic maps in one‐ and two‐dimensional lattices. We investigate periodic behaviors as the coupling parameter varies, i.e., existence and bifurcations of some periodic orbits with the largest domain of attraction. Similarity and differences between the two lattices are shown. For small coupling the periodic behavior appears to be characterized by a number of periodic orbits structured in such a way to give rise to distinct, reverse period‐doubling sequences. For intermediate values of the coupling a prominent role in the dynamics is played by the presence of normally attracting manifolds that contain periodic orbits. The dynamics on these manifolds is very weakly hyperbolic, which implies long transients. A detailed investigation allows the understanding of the mechanism of their formation. A complex bifurcation is found which causes an attracting manifold to become unstable. © 1994 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

The kinetics of liquid–gas phase transitions of a Van der Waals substance with fluctuations taken into account

A. N. Malakhov and N. V. Agudov

Chaos 4, 665 (1994); http://dx.doi.org/10.1063/1.166043 (7 pages) | Cited 1 time

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The isotherm of a Van der Waals substance, containing only stable points, is obtained on the basis of using the thermodynamic potential for nonequilibrium states and taking fluctuations into account. It is shown that in the vicinity of two‐phase states this isotherm is close to the horizontal phase equilibrium line, defined by Maxwell’s rule. The lifetimes of the metastable states of the Van der Waals substance, which depend on the intensity of the external fluctuations and the number of particles in the system, are estimated. © 1994 American Institute of Physics.
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64.70.F- Liquid-vapor transitions
64.60.My Metastable phases
05.70.Jk Critical point phenomena
05.70.Ln Nonequilibrium and irreversible thermodynamics

Regular and chaotic transport of impurities in steady flows

A. A. Vasiliev and A. I. Neishtadt

Chaos 4, 673 (1994); http://dx.doi.org/10.1063/1.166044 (8 pages) | Cited 3 times

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This paper considers the properties of the transport of impurity particles in steady fluid flows and describes the principal modes of particle motion. An impurity consisting of particles with a lower density than that of the medium is localized at stationary points of the flow, whereas a heavy impurity can perform a spatially unbounded motion. The conditions for the transition from the bounded motion of a heavy impurity to the long‐range transport mode, which occurs as a result of a loss of the stability of the heteroclinic trajectory, are obtained for a model two‐dimensional flow having an eddy‐cell structure. A mode is found in which a particle, after being transported over a long distance, is trapped forever within the confines of one cell. The transition from regular to chaotic particle transport is analyzed. The question of the effect of a small noise (for example, molecular diffusion) on the character of the motion of a heavy impurity is investigated. It is shown that this effect is important at high viscosity and leads to a transition from bounded motion of the impurity particle to diffusion‐type chaotic motion. © 1994 American Institute of Physics.
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47.52.+j Chaos in fluid dynamics
47.55.Kf Particle-laden flows

Chaotic capture of vortices by a moving body. II. Bound pair model

Harry H. Luithardt, James B. Kadtke, and Gianni Pedrizzetti

Chaos 4, 681 (1994); http://dx.doi.org/10.1063/1.166045 (11 pages) | Cited 3 times

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Previously, we have presented a simple model for the interaction of a fluid vortex structure with a moving bluff body, and demonstrated the existence of a trapping mechanism related to chaotic scattering. This single point vortex model required explicit perturbation to generate chaos and the subsequent complex dynamics. Here, we present a model which attempts to introduce internal degrees‐of‐freedom in the vortex structure in the simplest manner, by replacing the single vortex with a like‐signed pair. We show that this model exhibits chaotic trapping without the need of explicit perturbation, however, the region of parameter space for which trapping occurs is exceedingly small due to the spatially dependent form of the perturbation. We claim that this result explains some the behavior observed in Navier–Stokes simulations of the same vortex–body system, where we find close correspondence between the dynamics of an extended vorticity distribution and the single vortex model. Finally, we generalize the model to unequal strength vortex pairs, and find more complex behavior which includes ‘‘partial’’ capture of the weaker vortex by the body. © 1994 American Institute of Physics.
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47.32.C- Vortex dynamics
47.52.+j Chaos in fluid dynamics
47.15.ki Inviscid flows with vorticity

Unsteady processes in machines

Friedrich Pfeiffer

Chaos 4, 693 (1994); http://dx.doi.org/10.1063/1.166048 (13 pages) | Cited 2 times

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Couplings in machines and mechanisms always have play and friction. While under loading, stick–slip phenomena and impact events can take place. Such processes are modeled as multibody systems whose structure is time variant or unsteady. The time‐variant number of degrees of freedom is due to stick–slip contacts. The coupling characteristics become unsteady, for instance there exist jumps in the loads, if impacts occur. For establishing a uniform theory for such phenomena we use a Lagrangian approach connecting the additional constraint equations and the equations of motion by Lagrange multipliers, which are proportional to the constraint forces. Stick–slip and impact events are evaluated by indicator functions leading to special numerical algorithms for the search of switching points. Contact problems are formulated as a complementarity problem which can be solved by efficient algorithms. The theory is applied to rattling in gears, impact drilling machines, turbine blade dampers, and a woodpecker toy. In some of these applications, chaos as a result of bifurcations is possible, which results from variations in the parameters. © 1994 American Institute of Physics.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos
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