Research

General Overview of My Research

Physical systems are nonlinear, and unfortunately, it is nearly impossible to find solutions to nonlinear problems, even in approximate forms. Most of the time, we are interested in the stability and bifurcation of either an equilibrium point or a time-varying solution. We can perform such studies by constructing a variational equation that may have constant or time-varying coefficients and describes the local dynamics. In my research, I develop techniques for the analysis and control of such types of equations. Some of the concepts/ theories that I generally employ include

The Theory of Normal Forms
Center Manifold Theorem
Invariant Manifolds
Floquet Theory
Lyapunov-Floquet Theorem

Some of the Important Outcomes of My Past Work

A Simple Technique for the Computation of Lyapunov-Floquet Transformations

Motivation: Lyapunov-Floquet (L-F) transformations reduce time-periodic systems to a system of equations whose linear parts are time-invariant. They can be useful as various existing techniques applicable to autonomous systems can be applied to study time-periodic systems. These transformations were first developed by my advisor, late Professor Subhash C. Sinha, and his students in the early 1990s using Chebyshev polynomials of the first kind. They demonstrated that the analysis via L-F transformation has several advantages over traditional methods such as perturbation and averaging. Despite this, the usage of L-F transformations has remained unpopular among researchers. This motivated me to develop a relatively simpler approach to construct such transformations.

Contribution: My approach differs from the earlier efforts in the construction of state transition matrices (STMs). To compute STMs, I developed an approach along the lines of time-invariant systems. First, an assumed solution is substituted in the dynamical equation to reduce it to a non-standard eigenvalue problem. Then, the eigenvalue problem is solved to construct the general solution, which is then rearranged to obtain the state transition matrix. Finally, the Lyapunov-Floquet theorem is used to compute Lyapunov-Floquet transformations.

Generalized Resonance Conditions for Various Types of Resonances in Parametrically Excited Systems

Motivation: Existing resonance conditions for parametrically excited systems involve natural frequencies, and therefore, they are valid in the shaded region of the parametric space only (see the figure). Despite this small parameter limitation, existing conditions are used by researchers in all scientific communities. These conditions are insufficient to define resonances that occur for relatively larger values of Parameter 2, the amplitude of the parametric term. Thus, there is a need for the construction of generalized resonance conditions that are valid everywhere in the parametric space and yield true resonance types.

Contributions: I developed generalized resonance conditions using ‘true characteristic exponents,’ which are defined by exploiting the non-uniqueness property of characteristic exponents. The novelty of the work lies in the fact that

the forms of new resonance conditions are similar to the forms of existing resonance conditions involving natural frequencies and
the new resonance conditions are valid in the entire parametric space, unlike existing resonance conditions.

An Approximate Technique for the Analysis of Quasi-Periodic Systems

Motivation: Quasi-periodic motion is one of the solutions of nonlinear systems. It is a dynamic response characterized by two or more two incommensurate frequencies. For such a motion, the variational equation is a set of ordinary differential equations with quasi-periodic coefficients (so-called quasi-periodic systems). Unfortunately, there is no complete mathematical theory for quasi-periodic systems, and this motivated me to develop a theoretical framework to understand their dynamics.

Basic Premise of the Approximate Technique

Contribution 1: Stability of Quasi-Periodic Systems

Based on the above premise, I developed a systematic approach to compute the approximate state transition matrices (STMs) of quasi-periodic systems.

First, the proof of concept was established via numerical integration, and then a symbolic technique was used to compute STMs in terms of system parameters. Subsequently, these symbolic STMs were used to construct stability diagrams.

Contribution 2: Control of Chaos to Desired Quasi-Periodic Motions

In this work, I developed a novel method to drive general nonlinear systems to the desired motion: a fixed point, a periodic orbit, a quasi-periodic motion, or any other desired trajectory such as a logarithmic spiral.

The proposed control system consists of a combination of a nonlinear feedforward controller and a linear feedback controller.

Contribution 3: Lyapunov-Perron (L-P) Transformations

L-P transformations reduce quasi-periodic systems to a system of equations whose linear parts are time-invariant (see the figure).

The computation of approximate STMs of quasi-periodic systems paved the way for the construction of L-P transformations.

These transformations are a very powerful tool for studying quasi-periodic systems as many existing techniques for autonomous systems can be used for such problems.

Page updated

Google Sites

Report abuse