[Henon Map] Research in Progress

Global Behavior of Nonlinear Systems - abstract.
Nonlinear Aeroelasticity and Control - abstract..
Controlling Chaos - abstract..
Alternative Experimental Techniques and New Phenomena in an Impacting System - abstract.
Global Behavior of Nonlinear Systems
This research considers some global behavior aspects of nonlinear systems, and in particular, the dynamics associated with an indeterminate saddle-node bifurcation. Certain nonlinear oscillators display the well-known jump to resonance under various excitation parameter combinations; such jumps lead to a region of hysteresis in which stable, competing attractors exist. At certain levels of excitation, the catchment regions of these attractors are non-fractal, but as parameters are varied, it has been shown that completely determinate basins give way to fractal basin boundaries and ultimately an erosion of the entire basin such that jumps to resonance become indeterminate, since initial conditions can never be known accurately enough to overcome the fractal nature of the basin.

Experimentally, this study is considering two different nonlinear systems, one governed by a quadratic nonlinearity (the "escape" equation) and another governed by a cubic nonlinearity (Duffing's equation). Taking advantage of the "ball-rolling-on-a-hill" concept, tracks in the shape of potential energy surfaces have been constructed upon which a cart can roll, and the entire system can be excited over a range of forcing frequencies and amplitudes. Initial condition maps of the cart are generated in attempts to experimentally demonstrate the indeterminate bifurcation; these maps are constructed through slow parameter sweeps and through the relatively new method of stochastic interrogation, which allows an entire basin of attraction to be visited from one time series that can be perturbed.

Nonlinear Aeroelasticity and Control
The nonlinear behavior and flutter characteristics of an aeroelastic system with control surface freeplay are being studied. A low-order, state-space numerical model of the aeroelastic typical section has been created which allows for the inclusion of structural freeplay and preload in the control surface degree of freedom. A physical model of the system has also been constructed for use in the Duke University low-speed wind tunnel.

Previous work included the design of a linear control law to significantly raise the flutter speed of the linear system. Current work focuses on the nonlinear behavior induced by the freeplay in the control surface. A variable structure (sliding mode) control is also being developed.

Controlling Chaos
A signature of chaos that can be found in a nonlinear system is the system's sensitive dependence on its initial conditions. These systems can also display a sensitive dependence on small parameter changes. Sensitive dependence, among other things, makes the analysis of nonlinear systems exceedingly complicated and has convinced many to restrict their study to linear systems. Recently, however, researchers have capitalized on this intriguing feature of a nonlinear system's dynamics in order to stabilize chaotic behavior.

At Duke we are applying a controlling scheme originally created by Ott, Grebogi and Yorke to a model of the forced Duffing Oscillator. The goal is to stabilize a selected unstable chaotic response using only very small parameter perturbations. Part of the attraction to this method of control is that there are theoretically an infinite amount of chaotic orbits embedded within the systems global response. This suggests that there is some flexibility in choosing the desired behavior of the system. The validity of the theoretical concepts as well as their practical limitations will be determined when the control scheme is applied to our experimental apparatus.

Alternative Experimental Techniques and New Phenomena in an Impacting System
This project encompasses two main foci: new non-invasive data acquisition techniques and new dynamic phenomena, namely grazing bifurcations. The system under study is an impacting pendulum under sinusoidal base excitation. Both numerical and experimental approaches are used to study the system.

In the past, a potentiometer or similar device has been used experimentally to provide continuous monitoring of the displacement of the impacting pendulum. This techniques has the disadvantage that it significantly affects the damping (and thereby the dynamic behavior) of the system under study. We investigate two non-invasive methods of data acquisition. The first involves monitoring the occurrence of impact with a piezoceramic material, and the second involves recording the sound of the impacts. A nearly one-dimensional map can be constructed from the inter-spike intervals, and information about the behavior of the system can be extracted.

The second part of this project investigates the case when the impact does not occur at the equilibrium point of the system. For various forcing parameters, this means that the pendulum may never impact, may always impact, or may undergo grazing or low-velocity impact. This third case produces a new class of bifurcations which occur at the transition between the 'never-impacting' range and the 'always-impacting' range of parameters.