WCNA- 2000

Session: ILL-POSED PROBLEMS AND COMPLEX DYNAMICAL SYSTEMS
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of
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45-minute Invited Lectures organized by
.
A. Sengupta
Department of Mechanical Engineering
IIT Kanpur, Kanpur 208016.
E-mail: osegu@iitk.ac.in

Abstracts of papers invited to this Session are given below

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Paul A. Glendinning
Department of Mathematics
UMIST
P. O. Box. 88
Manchester M60 1QD, U.K.
E-mail: P.A.Glendinning@qmw.ac.uk

Blowout bifurcations occur when a chaotic attractor in an invariant manifold loses transverse stability. In a simple example we show that immediately before blowout there is a simple Milnor (measure- theoretic) attractor which is contained in a larger invariant set containing a dense orbit and on which periodic points are dense (so in some sense this larger attractor is the topological attractor). The implications for the dynamics after the blowout bifurcation are also discussed.

Approximate Parameter Identification for the Wave Equation by Optimal Control Techniques

Vladimir Protopopescu
Center for Engineering Science Advanced Research
Computer Science and Mathematics Division
Oak Ridge National Laboratory
Oak Ridge, TN 37831-6355 ,U.S.A.
E-mail: vvp@ornl.gov

We apply optimal control techniques to find approximate solutions to several inverse problems involving the acoustic wave equation. The generic inverse problem is to determine an unknown parameter in the wave equation from partial and/or noisy measurements of the acoustic signal. In general, the sought parameters may be: the shape and reflection coefficient of the boundary, the dispersive coefficient, the density of the medium, the speed of the signal in the medium, the initial conditions, or any combination thereof. In our approach, these unknown functions are treated as controls and the goal - quantified by a given objective functional - is to drive the model solution close to the experimental observations by adjusting these functions. We prove that, for any "cost of the control", the inverse problem is approximately solved by finding the optimal control pair which minimizes the objective functional. Moreover, by driving the "cost of the control" to zero, we prove that the sequence of (approximate) optimal control pairs converges to a bona fide solution of the inverse problem.

Attractivity and Shadowing Properties in the Chafee- Infante Problem

Sergei Y Pilyugin
Faculty of Mathematics and Mechanics
St Petersburg State University
198904 St Petersburg, RUSSIA.
E-Mail: sp@spil.usr.pu.ru

Consider the semigroup generated by a semilinear parabolic equation with the Dirichlet boundary conditions, where the nonlinearity belongs to the class of Chafee-Infante nonlinearities. Let the equation include a scalar parameter. The following recent results will be discussed.

(1). If the value of parameter is noncritical (i.e., if all fixed points of the semigroup are hyperbolic), then

(2). If the value of parameter is critical, then the semigroup has the usual shadowing property and also the weighted limit shadowing property in a neighborhood of the global attractor. These results are applied to various discretizations of the problem.

A local inverse for nonlinear mappings

Adi Ben-Israel
RUTCOR
Rutgers University
Piscataway, NJ 08854, U.S.A.
E-mail: bisrael@rutcor.rutgers.edu

A local inverse f^-1 is constructed for mappings f: Rⁿ--->R^m, n<=m, with Jacobian of full column-rank. This inverse is analogous to the Moore-Penrose inverse, and reduces to it if the mapping f^-1 is linear. The inverse has volume preserving properties, and its projection trajectories are orthogonal to the graph of f, giving local approximate solutions for f(x)=y, if inconsistent.

Linearized Stability, Bifurcation, and Generalized Inverse Mapping Theorems

Zuhair Nashed
Department of Mathematical Sciences
University of Delaware
Newark, DE 19716, U.S.A.
E-mail: nashed@math.udel.edu

Quantum chaos and quantum transport

Katsuhiro Nakamura
Department of Applied Physics,
Osaka City University, Osaka 558-8585, JAPAN.
E-mail: nakamura@a- phys.eng.osaka- cu.ac.jp

Quantum chaos, i.e., quantum-mechanical signatures of chaos, can be widely observed in mesoscopic "ballistic billiards" where a particle (electron) shows a ballistic motion between successive bouncings at the billiard boundary [1]. While the subject covers both open [2] and closed [3] billiards, the present talk will be limited to quantum transport in open billliards. We shall first talk about the quantum transport in the presence of a weak magnetic field. Recently an important role of quantum chaos is addressed to the quantum interference in classically-chaotic single billiards with a pair of small windows. The quantum correction to the reflection probability depends on statistical features of billiards and thereby on the integrability and nonintegrability of the underlying nonlinear dynamics. We explain the semiclassical theory of the ballistic Aharonov- Bohm-like effect and the ballistic conductance fluctuations in a single Sinai billiard. The remaining half of my talk is concerned with antidot superlattices that simulates the Lorentz gas (ordinary Sinai billiards). We apply the semiclassical theory of Kubo Formula, finding the role of orbit bifurcations in the strong magnetic field region where radius of cyclotron orbit is comparable to lattice periodicity. In conclusion, we emphasize the role of chaos and nonlinear dynamics in mesoscopic quantum transport.

References:

Non-linearity of Thermodynamical Recoverable Processes

Christopher G. Jesudason
M-003 Chemistry Department,
University of Malaya,
50603 Kuala Lumpur, MALAYSIA.
E-mail: jesu@kimia.um.edu.my

Recoverable processes have been described by pure Euler-Lagrange variations [Asian J. Physics, 9(1) 1, (2000)] involving the entropy over a prescribed thermodynamical pathway and constitutes a generalization of the Gibbs' criterion for equilibrium states. These same processes may be described by Linear Irreversible Thermodynamics (LIT) which in turn has been considered to be an approximation of the time reversible Boltzmann equation method. The standard thermocouple is discussed from the LIT and recoverable theory viewpoints, where marked discrepancies appear which cannot be contained by LIT. The Boltzmann method in its original form is therefore analyzed and found to possess difficulties in the light of more recent developments. It is shown that Fourier heat conduction, which has been described by the Boltzmann method, is amenable to analysis using recoverable theory, and since most steady state processes where a temperature gradient exists may be reduced to a generalized form of heat conduction [Apeiron, 6(3-4) 172] which is mechanically irreversible, it follows that recoverable theory can provide an alternate description free from the inherent difficulties of the Boltzmann approach, at least for systems characterized by temperature gradients.

Chaotic bursts and fractals in the dynamics of families of analytic functions

G. P. Kapoor
Department of Mathematics
Indian Institute of Technology Kanpur
Kanpur 208016, INDIA.
E-mail:: gp@iitk.ac.in

ABSTRACT