Twentieth European Dynamics Days

University of Surrey

Guildford

GU2 5XH

Sunday 25th June to Thursday 29th June 2000

Invited Speakers: Abstracts

Speaker Title and Abstract

V Babitsky (Loughborough) Nonlinear phenomena and smart dynamical structures.
Discovery of new nonlinear phenomena and development of computation and control facilities permit to bring into engineering the new types of intelligent dynamical structures with adaptive behaviour. The adaptive mechanisms for such structures are embedded with specifics of their nonlinear characteristics. The systems use the natural ability of nonlinear systems to transform of applied energy into new desired forms of response.
The lecture is devoted to the description of nonlinear phenomena, methodology of synthesis of the smart dynamical structures and demonstration of such structures developed for applications in vibration protection and vibration processing.
The effects of strongly nonlinear resonant behaviour are presented with particular interest to stabilisation of operating modes and their practical implementation into engineering structures. The special methods of excitation, filtration and control of nonlinear vibration are advanced with the help of smart nonlinear structures and electronic feedbacks.
The concept of a technological machine as an autoresonant homeostat is developed on these foundations. It is based on keeping up an entire mechatronic system behind the boundary of dynamic stability and active matching between the vibrating system and the technological environment. As a result, the resonant conditions are maintained under deviations of parameters and loads, and only the weak control of the system is required. The functioning of autoresonant machines are demonstrated by videofilm.

P Bressloff (Loughborough) Geometric hallucinations, Euclidean symmetry and pattern formation in primary visual cortex.
We show that the circuitry of the primary visual cortex has approximate Euclidean symmetry with a novel group action. This action gives rise to the spontaneous formation of cortical patterns that generate images in the visual field (via a conformal mapping) consistent with some of the common forms of geometric visual hallucinations. Various techniques from group theory and nonlinear analysis will be applied to the problem of computing the types of patterns and their stability.

T Bridges (Surrey) Stiefel manifolds, exterior algebra, and the computation of Lyapunov exponents and the Evans function.
Two problems that are notoriously difficult for numerical solution are the computation of a few Lyapunov exponents (LEs) of large dimension dynamical systems, and the computation of the Evans function, whose zeros are stability exponents associated with the linearization about pulses or solitary waves (SWs). In this talk we show how differential geometry leads to new algorithms for these problems.
For computing LEs, a differential geometric view of continuous orthogonalization suggests that one restrict the linearized vectorfield to a Stiefel manifold. However, the Stiefel manifold is not in general an attracting submanifold of the ambient Euclidean space: it is a constraint manifold with a weak numerical invariant. New numerical algorithms for this problem are then designed which use the fibre-bundle characterization of Stiefel manifolds, and these algorithms preserve orthonormality to machine accuracy. The new algorithms are straightforward to implement and are natural for large-dimension dynamical systems. An example is presented where a few Lyapunov exponents are computed for an array of coupled oscillators.
In the second part of the talk, a novel algorithm for computing the Evans function is presented based on numerical exterior algebra. It turns out that Hodge duality and the Hodge star operator come into the analysis in an essential way. When combined with a geometric integrator, such as Gauss-Legendre RK method, this algorithm has the advantage that numerical integration can be done stably, and analyticity is preserved. The numerical algorithm will be applied to show that the Hocking-Stewartson pulse solution of the cGL equation is unstable.

J Brindley (Leeds) Fizzle or frizzle: Mathematical challenges posed by ignition.
The phenomenon of ignition, in which a potentially reactive material changes from a state of very slow, essentially negligible, reactivity, to a state of very rapid reactivity characterised by burning or explosion is commonplace. The ignition may or may not be desirable, and may be externally forced or occur spontaneously.
In mathematical terms the phenomenon is usually one of blow-up in a priori unknown finite time and at some unknown location in a reaction diffusion system, and exact prediction of criticality is often not possible, though many qualitative results have been rigorously established.
Two particular examples of unwanted ignition will be discussed, one relating to spontaneous ignition of a reactant liquid in a porus matrix, the other to a hotspot in a reactant solid or powder. Each displays unexpected features, on which some light is thrown by simple modelling of the highly nonlinear dynamics.

A Champneys (Bristol) The Indian Rope Trick for a continuously flexible rod; a nonlinear analysis.
Acheson and Mullin have demonstrated both experimentally and theoretically that it is possible to balance a system of $N$ coupled rigid pendulums upside down by oscillating its support. In the limit $N \to \infty$ the theoretical region of
stability in frequency vs. amplitude disappears. Nevertheless experiments of Mullin on a continuously flexible rod (a piece of curtain wire just long enough to fall over under its own weight) suggest it can be stabilised using the right frequency and amplitude of excitation.
We propose a theory for this. First, a dimensionless PDE is derived from rod theory that is fourth-order in arclength and second-order in time. The stability of the vertical solution subject to parametric excitation is then studied using Floquet theory, which is the infinite-dimensional analogue of the equivalent analysis for the Mathieu equation. The results of double-scale asymptotics and numerics agree and lead to an upside down stability region, albeit one which is punctured by thin Arnold tongues corresponding to higher-order resonances.
The asymptotic analysis is extended to include geometric nonlinearities and viscous damping. The stability of various bifurcating states at the simplest resonances is calculated. Emphasis is placed on the distinction between planar and circular motions which bifurcate into different symmetry subgroups of the problem. A qualitative agreement is found with the experimental data on the observed stability boundaries being caused by an interaction between a fundamental and sub-harmonic instability.

M Dellnitz (Paderborn) Set Oriented Numerical Methods for Dynamical Systems.
Over the past few years so-called set oriented numerical methods have been developed for the numerical analysis of dynamical systems. These methods allow to compute directly chain recurrent sets or invariant manifolds but also statistical quantities such as invariant measures, Lyapunov exponents or almost invariant sets can be approximated.
In this talk an overview about recent accomplishments in this field will be given. In particular, concrete areas of application of the set oriented numerical techniques will be presented: the design of energy efficient spacecraft trajectories, the approximation of molecular dynamics and also problems in global optimization.

B Eckhardt (Marburg) Transition to turbulence in shear flows.
Several shear flows show a transition to turbulence despite linear stability of the laminar flow. This transition is accompagnied by a strong dependence on initial conditions and does not seem to have a sharp transitional Reynolds number. These observations are consistent with the formation of a strange repellor in phase space. Experimental and numerical insights into the structure of the repellor will be discussed for the specific example of plane Couette flow.

M Hasler (Lausanne) Transmission of information over a noisy channel using chaotic signals.
It is explained how chaotic signals can be used to transmit information. The transmitted signal is always perturbed by noise. The problem of reliable extraction of the information from the noise perturbed chaotic signal at the receiver side is discussed in some detail. The most simple channel model, additive white gaussian noise, is used, and the corresponding optimal receiver algorithm is derived. Then suboptimal algorithms such as chaos synchronization and correlation as well as hidden Markov models are discussed.

P Holmes (Princeton) Models for insect locomotion, or why cockroaches get away.
I will discuss joint work with John Schmitt in which nonlinear mechanics meets biology. Motivated by experimental studies of insects, we propose a mechanical model for the dynamics of legged locomotion in the horizontal plane. Our three-degree-of freedom rigidbody model with massless, compliant legs in intermittent contact with the ground allows for passive and prescribed (active muscle) force and torque generation. We focus here on energetically conservative bipedal models, each leg corresponding to the front/rear/opposite-middle tripod used in rapid running by many insect species, and we consider both fixed and moving center of pressure models. We show that the (piecewise holonomic) mechanics due to intermittent foot contacts can confer strong asymptotic stability in heading and body orientation. We discuss the relevance of our idealised models to experiments and simulations on insect running and turning, show that their gait and force characteristics match observations reasonably well, and consider the implications for the neural control of locomotion.

S Luzzatto (Warwick) Recent results on Lorenz attractors.
I will describe recent ideas and results on the application of probabilistic methods to the problem of proving the existence of strange attractors with some (non-uniform) hyperbolic characteristics. Interestingly similar methods also yield powerful results about the dynamical properties of these same attractors.

R Murray (Waikato) Ergodic theory and rigorous computation in dynamical systems
This talk will discuss one of the lingering issues in the modern study of dynamical systems: how can we be confident that numerical calculations involving complicated systems reproduce correct dynamical behaviour, and not numerical artifacts? I will use an example from computational ergodic theory to illustrate that careful attention to the mathematics underlying the objects being computed can yield pleasingly rigorous numerical methods.
Invariant probability measures are the basic objects in the ergodic theory of dynamical systems. Via the famous Birkhoff Ergodic Theorem, invariant measures can be used to compute the time averages of interesting dynamical quantities (time spent in given regions of phase space, largest Lyapunov exponent, etc.). Thus, it is of great interest to be able to find invariant measures for a given dynamical system.
Algorithms developed by Dellnitz and co--workers have recently achieved spectacular success in the computation of invariant measures, invariant manifolds, attractors etc. Many of these algorithms rely on an approximation scheme known as Ulam's method. In this talk, I will use simple examples to show why Ulam's method is extremely reliable, and thus provides a very satisfactory basis for rigorous computational methods in ergodic theory.

D Sauzin (Paris) Multidimensional splitting in near-integrable Hamiltonian systems.
For near-integrable Hamiltonian systems with three or more degrees of freedom, the KAM theorem does not prevent the existence of "Arnold diffusion", i.e. the drift of the action variables of some trajectories. But because of Nekhoroshev theorem diffusion must be exponentially slow if it exists. Arnold's mechanism is based on the splitting of the stable and unstable manifolds associated to 1-hyperbolic invariant tori; we explain the difficulties encountered when trying to measure this exponentially small phenomenon and a method which makes use of the Hamilton-Jacobi equation and allows to reach lower bounds of the splitting in some cases.

A Scheel (Berlin) Stability, instability, and bifurcation of spiral waves.
Spiral waves arise in many chemical and biological systems. They have also been observed in reaction-diffusion equations that were designed to model patterns in excitable or oscillatory media. We present a method which allows a systematic study of stability properties as well as bifurcations of spiral waves. We relate transport phenomena, induced by the direction of rotation, to stability and robustness of spiral waves in bounded domains --- as compared to the pattern in the unbounded plane. We then list various instability mechanisms, leading to two-frequency meandering motion, spiral break-up, and, finally, spiral turbulence. We illustrate that a careful understanding of the linearized equation leads to an explanation of temporal and spatial patterns observed after bifurcation. This is joint work with Bjoern Sandstede.

W Schiehlen (Stuttgart) Control of chaos in mechanical systems.
A one-body pendulum as well as a mathematical pendulum shows periodic motions only. A two-body pendulum features chaotic behavior in the transition from vibrations in the gravity field with low energy to vibrations in the centrifugal field with high energy.
A three-body pendulum with a properly chosen geometric design may perform chaotic motions in the gravity field. In experiments damping and friction of the bearings have to be compensated by a self-excitation control device to maintain a chaotic motion. The control law for the chaos pendulum will be presented resulting in an especially tuned mechatronically self-existed three-body pendulum. The simulations are verified by an experiment.

M Silber (Northwestern) Weakly damped modes and exotic Faraday wave patterns.
In the Faraday wave experiment, surface waves are parametrically excited on the free surface of a fluid layer that is subjected to a periodic vibration in the vertical direction. Recent experiments employing a forcing function with two rationally related frequency components reveal a variety of exotic standing wave patterns, including so-called superlattice patterns. These wave patterns are spatially periodic but have structure on more than one length scale.
We use methods of equivariant bifurcation theory to investigate this pattern formation problem. Our analysis indicates that the presence of weakly damped modes, that are spatio-temporally resonant with the superlattice patterns, play a decisive role in the pattern selection process. I will also present general results on spatial period-multiplying instablities of hexagonal standing wave patterns, making contact with experimental observations of exotic secondary patterns in the Faraday wave system.

L Tuckermann (Orsay) Numerical Bifurcation Analysis.
For systems making a transition from simple (uniform, laminar, steady) to more complex (non-uniform, periodic, quasiperiodic, chaotic, turbulent) behavior, a bifurcation diagram summarizes the information necessary for understanding the system. A complete bifurcation diagram, including unstable states and limit cycles, is inaccessible to experiment, but is, in principle, obtainable numerically from the governing equations. This is rarely done in practice, however, if the equations are two or three dimensional PDEs. The main barrier comes from linear algebra: the matrices arising from discretisation of PDEs are too large to be inverted or diagonalized. In this talk, we show how to circumvent this obstacle by coupling modern numerical linear algebra to bifurcation theory and computational fluid dynamics. We will show how these techniques can be applied to various hydrodynamic pattern-forming systems.
A system of particular current interest is plane Couette flow, which is linearly stable at all Reynolds numbers, but undergoes sudden transition to turbulence both experimentally and numerically. In an effort to understand transition, intermediate states have been stabilized by inserting a wire into the flow. We report on the bifurcation diagram computed numerically for perturbed plane Couette flow.

J Yorke (Maryland) Transient Chaos
Transient Chaos occurs in cases where there are horseshoes and chaos in some region but where the chaos does not lie within an attractor. Instead there is a chaotic saddle or chaotic repeller. Trajectories near such sets behave chaotically for awhile and then leave the region. Our group over the years has developed a number of techniques for detecting and locating the chaotic saddles, but they have worked well only in simple cases. Now we have discovered a simple robust method for computing and visualizing these sets. This is joint work with Dr. Helena E. Nusse and David Sweet.

Back to the announcement .

dd2000@surrey.ac.uk .

Last altered: 8th June 2000

Speaker	Title and Abstract
V Babitsky (Loughborough)	Nonlinear phenomena and smart dynamical structures. Discovery of new nonlinear phenomena and development of computation and control facilities permit to bring into engineering the new types of intelligent dynamical structures with adaptive behaviour. The adaptive mechanisms for such structures are embedded with specifics of their nonlinear characteristics. The systems use the natural ability of nonlinear systems to transform of applied energy into new desired forms of response. The lecture is devoted to the description of nonlinear phenomena, methodology of synthesis of the smart dynamical structures and demonstration of such structures developed for applications in vibration protection and vibration processing. The effects of strongly nonlinear resonant behaviour are presented with particular interest to stabilisation of operating modes and their practical implementation into engineering structures. The special methods of excitation, filtration and control of nonlinear vibration are advanced with the help of smart nonlinear structures and electronic feedbacks. The concept of a technological machine as an autoresonant homeostat is developed on these foundations. It is based on keeping up an entire mechatronic system behind the boundary of dynamic stability and active matching between the vibrating system and the technological environment. As a result, the resonant conditions are maintained under deviations of parameters and loads, and only the weak control of the system is required. The functioning of autoresonant machines are demonstrated by videofilm.
P Bressloff (Loughborough)	Geometric hallucinations, Euclidean symmetry and pattern formation in primary visual cortex. We show that the circuitry of the primary visual cortex has approximate Euclidean symmetry with a novel group action. This action gives rise to the spontaneous formation of cortical patterns that generate images in the visual field (via a conformal mapping) consistent with some of the common forms of geometric visual hallucinations. Various techniques from group theory and nonlinear analysis will be applied to the problem of computing the types of patterns and their stability.
T Bridges (Surrey)	Stiefel manifolds, exterior algebra, and the computation of Lyapunov exponents and the Evans function. Two problems that are notoriously difficult for numerical solution are the computation of a few Lyapunov exponents (LEs) of large dimension dynamical systems, and the computation of the Evans function, whose zeros are stability exponents associated with the linearization about pulses or solitary waves (SWs). In this talk we show how differential geometry leads to new algorithms for these problems. For computing LEs, a differential geometric view of continuous orthogonalization suggests that one restrict the linearized vectorfield to a Stiefel manifold. However, the Stiefel manifold is not in general an attracting submanifold of the ambient Euclidean space: it is a constraint manifold with a weak numerical invariant. New numerical algorithms for this problem are then designed which use the fibre-bundle characterization of Stiefel manifolds, and these algorithms preserve orthonormality to machine accuracy. The new algorithms are straightforward to implement and are natural for large-dimension dynamical systems. An example is presented where a few Lyapunov exponents are computed for an array of coupled oscillators. In the second part of the talk, a novel algorithm for computing the Evans function is presented based on numerical exterior algebra. It turns out that Hodge duality and the Hodge star operator come into the analysis in an essential way. When combined with a geometric integrator, such as Gauss-Legendre RK method, this algorithm has the advantage that numerical integration can be done stably, and analyticity is preserved. The numerical algorithm will be applied to show that the Hocking-Stewartson pulse solution of the cGL equation is unstable.
J Brindley (Leeds)	Fizzle or frizzle: Mathematical challenges posed by ignition. The phenomenon of ignition, in which a potentially reactive material changes from a state of very slow, essentially negligible, reactivity, to a state of very rapid reactivity characterised by burning or explosion is commonplace. The ignition may or may not be desirable, and may be externally forced or occur spontaneously. In mathematical terms the phenomenon is usually one of blow-up in a priori unknown finite time and at some unknown location in a reaction diffusion system, and exact prediction of criticality is often not possible, though many qualitative results have been rigorously established. Two particular examples of unwanted ignition will be discussed, one relating to spontaneous ignition of a reactant liquid in a porus matrix, the other to a hotspot in a reactant solid or powder. Each displays unexpected features, on which some light is thrown by simple modelling of the highly nonlinear dynamics.
A Champneys (Bristol)	The Indian Rope Trick for a continuously flexible rod; a nonlinear analysis. Acheson and Mullin have demonstrated both experimentally and theoretically that it is possible to balance a system of $N$ coupled rigid pendulums upside down by oscillating its support. In the limit $N \to \infty$ the theoretical region of stability in frequency vs. amplitude disappears. Nevertheless experiments of Mullin on a continuously flexible rod (a piece of curtain wire just long enough to fall over under its own weight) suggest it can be stabilised using the right frequency and amplitude of excitation. We propose a theory for this. First, a dimensionless PDE is derived from rod theory that is fourth-order in arclength and second-order in time. The stability of the vertical solution subject to parametric excitation is then studied using Floquet theory, which is the infinite-dimensional analogue of the equivalent analysis for the Mathieu equation. The results of double-scale asymptotics and numerics agree and lead to an upside down stability region, albeit one which is punctured by thin Arnold tongues corresponding to higher-order resonances. The asymptotic analysis is extended to include geometric nonlinearities and viscous damping. The stability of various bifurcating states at the simplest resonances is calculated. Emphasis is placed on the distinction between planar and circular motions which bifurcate into different symmetry subgroups of the problem. A qualitative agreement is found with the experimental data on the observed stability boundaries being caused by an interaction between a fundamental and sub-harmonic instability.
M Dellnitz (Paderborn)	Set Oriented Numerical Methods for Dynamical Systems. Over the past few years so-called set oriented numerical methods have been developed for the numerical analysis of dynamical systems. These methods allow to compute directly chain recurrent sets or invariant manifolds but also statistical quantities such as invariant measures, Lyapunov exponents or almost invariant sets can be approximated. In this talk an overview about recent accomplishments in this field will be given. In particular, concrete areas of application of the set oriented numerical techniques will be presented: the design of energy efficient spacecraft trajectories, the approximation of molecular dynamics and also problems in global optimization.
B Eckhardt (Marburg)	Transition to turbulence in shear flows. Several shear flows show a transition to turbulence despite linear stability of the laminar flow. This transition is accompagnied by a strong dependence on initial conditions and does not seem to have a sharp transitional Reynolds number. These observations are consistent with the formation of a strange repellor in phase space. Experimental and numerical insights into the structure of the repellor will be discussed for the specific example of plane Couette flow.
M Hasler (Lausanne)	Transmission of information over a noisy channel using chaotic signals. It is explained how chaotic signals can be used to transmit information. The transmitted signal is always perturbed by noise. The problem of reliable extraction of the information from the noise perturbed chaotic signal at the receiver side is discussed in some detail. The most simple channel model, additive white gaussian noise, is used, and the corresponding optimal receiver algorithm is derived. Then suboptimal algorithms such as chaos synchronization and correlation as well as hidden Markov models are discussed.
P Holmes (Princeton)	Models for insect locomotion, or why cockroaches get away. I will discuss joint work with John Schmitt in which nonlinear mechanics meets biology. Motivated by experimental studies of insects, we propose a mechanical model for the dynamics of legged locomotion in the horizontal plane. Our three-degree-of freedom rigidbody model with massless, compliant legs in intermittent contact with the ground allows for passive and prescribed (active muscle) force and torque generation. We focus here on energetically conservative bipedal models, each leg corresponding to the front/rear/opposite-middle tripod used in rapid running by many insect species, and we consider both fixed and moving center of pressure models. We show that the (piecewise holonomic) mechanics due to intermittent foot contacts can confer strong asymptotic stability in heading and body orientation. We discuss the relevance of our idealised models to experiments and simulations on insect running and turning, show that their gait and force characteristics match observations reasonably well, and consider the implications for the neural control of locomotion.
S Luzzatto (Warwick)	Recent results on Lorenz attractors. I will describe recent ideas and results on the application of probabilistic methods to the problem of proving the existence of strange attractors with some (non-uniform) hyperbolic characteristics. Interestingly similar methods also yield powerful results about the dynamical properties of these same attractors.
R Murray (Waikato)	Ergodic theory and rigorous computation in dynamical systems This talk will discuss one of the lingering issues in the modern study of dynamical systems: how can we be confident that numerical calculations involving complicated systems reproduce correct dynamical behaviour, and not numerical artifacts? I will use an example from computational ergodic theory to illustrate that careful attention to the mathematics underlying the objects being computed can yield pleasingly rigorous numerical methods. Invariant probability measures are the basic objects in the ergodic theory of dynamical systems. Via the famous Birkhoff Ergodic Theorem, invariant measures can be used to compute the time averages of interesting dynamical quantities (time spent in given regions of phase space, largest Lyapunov exponent, etc.). Thus, it is of great interest to be able to find invariant measures for a given dynamical system. Algorithms developed by Dellnitz and co--workers have recently achieved spectacular success in the computation of invariant measures, invariant manifolds, attractors etc. Many of these algorithms rely on an approximation scheme known as Ulam's method. In this talk, I will use simple examples to show why Ulam's method is extremely reliable, and thus provides a very satisfactory basis for rigorous computational methods in ergodic theory.
D Sauzin (Paris)	Multidimensional splitting in near-integrable Hamiltonian systems. For near-integrable Hamiltonian systems with three or more degrees of freedom, the KAM theorem does not prevent the existence of "Arnold diffusion", i.e. the drift of the action variables of some trajectories. But because of Nekhoroshev theorem diffusion must be exponentially slow if it exists. Arnold's mechanism is based on the splitting of the stable and unstable manifolds associated to 1-hyperbolic invariant tori; we explain the difficulties encountered when trying to measure this exponentially small phenomenon and a method which makes use of the Hamilton-Jacobi equation and allows to reach lower bounds of the splitting in some cases.
A Scheel (Berlin)	Stability, instability, and bifurcation of spiral waves. Spiral waves arise in many chemical and biological systems. They have also been observed in reaction-diffusion equations that were designed to model patterns in excitable or oscillatory media. We present a method which allows a systematic study of stability properties as well as bifurcations of spiral waves. We relate transport phenomena, induced by the direction of rotation, to stability and robustness of spiral waves in bounded domains --- as compared to the pattern in the unbounded plane. We then list various instability mechanisms, leading to two-frequency meandering motion, spiral break-up, and, finally, spiral turbulence. We illustrate that a careful understanding of the linearized equation leads to an explanation of temporal and spatial patterns observed after bifurcation. This is joint work with Bjoern Sandstede.
W Schiehlen (Stuttgart)	Control of chaos in mechanical systems. A one-body pendulum as well as a mathematical pendulum shows periodic motions only. A two-body pendulum features chaotic behavior in the transition from vibrations in the gravity field with low energy to vibrations in the centrifugal field with high energy. A three-body pendulum with a properly chosen geometric design may perform chaotic motions in the gravity field. In experiments damping and friction of the bearings have to be compensated by a self-excitation control device to maintain a chaotic motion. The control law for the chaos pendulum will be presented resulting in an especially tuned mechatronically self-existed three-body pendulum. The simulations are verified by an experiment.
M Silber (Northwestern)	Weakly damped modes and exotic Faraday wave patterns. In the Faraday wave experiment, surface waves are parametrically excited on the free surface of a fluid layer that is subjected to a periodic vibration in the vertical direction. Recent experiments employing a forcing function with two rationally related frequency components reveal a variety of exotic standing wave patterns, including so-called superlattice patterns. These wave patterns are spatially periodic but have structure on more than one length scale. We use methods of equivariant bifurcation theory to investigate this pattern formation problem. Our analysis indicates that the presence of weakly damped modes, that are spatio-temporally resonant with the superlattice patterns, play a decisive role in the pattern selection process. I will also present general results on spatial period-multiplying instablities of hexagonal standing wave patterns, making contact with experimental observations of exotic secondary patterns in the Faraday wave system.
L Tuckermann (Orsay)	Numerical Bifurcation Analysis. For systems making a transition from simple (uniform, laminar, steady) to more complex (non-uniform, periodic, quasiperiodic, chaotic, turbulent) behavior, a bifurcation diagram summarizes the information necessary for understanding the system. A complete bifurcation diagram, including unstable states and limit cycles, is inaccessible to experiment, but is, in principle, obtainable numerically from the governing equations. This is rarely done in practice, however, if the equations are two or three dimensional PDEs. The main barrier comes from linear algebra: the matrices arising from discretisation of PDEs are too large to be inverted or diagonalized. In this talk, we show how to circumvent this obstacle by coupling modern numerical linear algebra to bifurcation theory and computational fluid dynamics. We will show how these techniques can be applied to various hydrodynamic pattern-forming systems. A system of particular current interest is plane Couette flow, which is linearly stable at all Reynolds numbers, but undergoes sudden transition to turbulence both experimentally and numerically. In an effort to understand transition, intermediate states have been stabilized by inserting a wire into the flow. We report on the bifurcation diagram computed numerically for perturbed plane Couette flow.
J Yorke (Maryland)	Transient Chaos Transient Chaos occurs in cases where there are horseshoes and chaos in some region but where the chaos does not lie within an attractor. Instead there is a chaotic saddle or chaotic repeller. Trajectories near such sets behave chaotically for awhile and then leave the region. Our group over the years has developed a number of techniques for detecting and locating the chaotic saddles, but they have worked well only in simple cases. Now we have discovered a simple robust method for computing and visualizing these sets. This is joint work with Dr. Helena E. Nusse and David Sweet.