12,939 research outputs found
Convergence Time Towards Periodic Orbits in Discrete Dynamical Systems
We investigate the convergence towards periodic orbits in discrete dynamical
systems. We examine the probability that a randomly chosen point converges to a
particular neighborhood of a periodic orbit in a fixed number of iterations,
and we use linearized equations to examine the evolution near that
neighborhood. The underlying idea is that points of stable periodic orbit are
associated with intervals. We state and prove a theorem that details what
regions of phase space are mapped into these intervals (once they are known)
and how many iterations are required to get there. We also construct algorithms
that allow our theoretical results to be implemented successfully in practice.Comment: 17 pages; 7 figure
Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
Magnetorotational dynamo action in Keplerian shear flow is a three-dimensional, non-linear magnetohydrodynamic process whose study is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics of transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to turbulent injection of both kinetic and magnetic energy in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to better understand the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows
Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients
The foundations of the chaotic scattering theory for transport and
reaction-rate coefficients for classical many-body systems are considered here
in some detail. The thermodynamic formalism of Sinai, Bowen, and Ruelle is
employed to obtain an expression for the escape-rate for a phase space
trajectory to leave a finite open region of phase space for the first time.
This expression relates the escape rate to the difference between the sum of
the positive Lyapunov exponents and the K-S entropy for the fractal set of
trajectories which are trapped forever in the open region. This result is well
known for systems of a few degrees of freedom and is here extended to systems
of many degrees of freedom. The formalism is applied to smooth hyperbolic
systems, to cellular-automata lattice gases, and to hard sphere sytems. In the
latter case, the goemetric constructions of Sinai {\it et al} for billiard
systems are used to describe the relevant chaotic scattering phenomena. Some
applications of this formalism to non-hyperbolic systems are also discussed.Comment: 35 pages, compressed file, follow directions in header for ps file.
Figures are available on request from [email protected]
Dynamics of Barred Galaxies
Some 30% of disc galaxies have a pronounced central bar feature in the disc
plane and many more have weaker features of a similar kind. Kinematic data
indicate that the bar constitutes a major non-axisymmetric component of the
mass distribution and that the bar pattern tumbles rapidly about the axis
normal to the disc plane. The observed motions are consistent with material
within the bar streaming along highly elongated orbits aligned with the
rotating major axis. A barred galaxy may also contain a spheroidal bulge at its
centre, spirals in the outer disc and, less commonly, other features such as a
ring or lens. Mild asymmetries in both the light and kinematics are quite
common. We review the main problems presented by these complicated dynamical
systems and summarize the effort so far made towards their solution,
emphasizing results which appear secure. (Truncated)Comment: This old review appeared in 1993. Plain tex with macro file. 82 pages
18 figures. A pdf version with figures at full resolution (3.24MB) is
available at http://www.physics.rutgers.edu/~sellwood/bar_review.pd
The Forced van der Pol Equation II: Canards in the reduced system
This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35].
The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35]. We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system
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