http://kermani.math.fit.edu/wcna2000/program.html

WCNA-2000

Invited Lecture at main session

Ill-Posed Problems, Multifunctions, and Chaotic Dynamical Systems
A. Sengupta
Department of Mechanical Engineering
Indian Institute of Technology Kanpur, Kanpur 208016, INDIA.
E-mail: osegu@iitk.ac.in
 

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This talk will present a new formulation of one-dimensional discrete chaotic systems based on the concepts of an ill-posed problem f(x) = y where x and y are elements of a topological space X, and a multifunctional limit Multi(X) of a sequence of functions (f_i) in Map(X) that will be constructed as the graphical limit of a sequence of functions. This topological extension of Map(X) is the central ingredient of our appraoch and depends on a pointwise bi- convergence of the sequence. The ill-posedness of importance to us arises from the non- injectivity of the sequence of functions, and the multifunctional inverse of f that is obtained from its generalized inverse G is basic to our theory. We show that a notion of maximal ill-posedness of the sequence of iterates of f is equivalent to chaos of the discrete dynamical system. The following illustration is a typical example of our results.
 

Intermittancy between chaos and order in logistic map for l<l^*=3.828426. The plots display graphical convergence of the iterates of f³

The tangent bifurcation at l^* is characterized by the graphical convergence of the iterates of f³ to a middle horizontal portion as shown in the second figure. The complete formal similarity of these figures to the following set for the graphical convergence of the non- chaotic family of iterates of lx²sin(px)  displaying tangent bifurcation is to be noted.
 

Saddle-node bifurcation for the iteates of lx²sin(px).

References. [1]. A. Sengupta, Multifunctions and generalized inverse, Jour. Inverse and ill-Posed Problems, 5, 265-285(1997).

[2]. A. Sengupta and G. G. Ray, A multifunctional extension of function spaces: Chaotic dynamical spaces are maximally ill-posed,Jour. Inverse and ill-Posed Problems, Accepted.

[3]. A. Sengupta, Ill-posed problems, multifunctions and chaotic dynamical systems, Lectures at Workshop on Nonlinear Analysis and Its Applications, Calcutta 1999.

Organized Session: ILL-POSED PROBLEMS AND COMPLEX DYNAMICAL SYSTEMS

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A. Sengupta
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45 minute Invited Lectures