Computational evaluation of invariant sensitivity in nonlinear dynamical
systems using the Feynman path integral formalism
R.C. Venkatesan
Distributed Algorithms Corporation
Pune, India
The preservation of approximate invariance under a group of continuous transformations by numerical algorithms for ODE's and PDE's containing a perturbation parameter 'epsilon' are investigated. A stability of the approximate invariances and cases where the perturbation parameter assumes the role of an independent variable are studied.
A generalized methodology for the transferring invariance properties of given ODE's and PDE's to their grid point values is presented. This is achieved by obtaining the discrete form of the group of continuous transformations corresponding to the original differential equations.
Use of the evolutionary vector field to achieve compression of the computational net space is highlighted. An error control mechanism for deviations from invariance within a prescribed tolerence is formulated. It is shown that the concept of invariant discretization is valid for both fixed as well as adaptive mesh simulations.
In order to investigate into the qualitatively distinct nature of the difference algorithms, salient examples from nonlinear dynamic control are studied. A distributed parameter/path control system for chaotic optical communication based upon the perturbed Duffing-Van der Pol model is examined.
The control paradigm makes use of Feynman path integrals to derive propagators for chaotic communication systems. In this case, the appropriate variational for the system, which is dissipative, is first formulated by means of nonconservative dynamics.
Most typically, such so-called nonconservative systems are described by dissipative and constrained systems, which do not lend themselves to a straightforward variational representation. Nonconservative variational forms are commonly characterized by taking the variational of the derivative NOT TO BE EQUAL TO the derivative of the variational. The path integral is numerically evaluated and the weights assigned to individual paths that would lead to 'useful' regions of communication are obtained.
Use of invariance properties of the difference schemes leads to the identification of the contribution of paths to the Feynman representation of a dynamical system. Finally, a sensitivity study of the deviation tolorance of the paths is performed, and sensitivity invariance criteria are established. Numerical examples for representative cases are provided for PRACTICALLY IMPLEMENTABLE SYSTEMS.