PROGRAM
 
Wednesday, July 9
Thursday, July 10
Friday, July 11
9.00 - welcome address
A. Vicino, CSC director

9.15 - 12.00
CONTROL OF CHAOS AND BIFURCATIONS (i)

Chaos Control: Theory and Applications
M. Ogorzalek

Model Reduction for Systems with Low-Dimensional Chaos
C. Piccardi

9.00 - 12.00
NEURAL NETWORKS (i)

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science
L.O. Chua

On the Application of Spectral Techniques for the Analysis of Nonlinear Dynamic Arrays
M. Gilli

9.00 - 12.00
NEW TRENDS IN STABILITY THEORY FOR COMPLEX SYSTEMS

Normal Form, Bifurcation, and Stabilization of Highly Nonlinear Systems
W. Kang

The Input-to-State Stability Approach for Global Analysis of Nonlinear Systems
D. Angeli

14.30 - 17.30
CONTROL OF CHAOS AND BIFURCATIONS (ii)

Bifurcation Control and Applications
E.H. Abed

Controller Synthesis for Stabilizing Periodic Orbits in Chaotic Systems
M. Basso

 
14.30 - 17.30
NEURAL NETWORKS (ii)

Programmable Complex Systems, Visual Microprocessors and Universal Machines on Flows - with Bioinspired Aspects
T. Roska

14.30 - 17.30
CHAOS IN COMMUNICATION SYSTEMS

Statistical Approach to Chaos: Some Theoretical Results and Applications
G. Setti

Bifurcation Phenomena in the Behavior of Access Protocols for Wireless Multimedia Communication Systems
G. Giambene
 

17.30 - conclusion of the school


 

ABSTRACTS

CONTROL OF CHAOS AND BIFURCATIONS

Chaos Control: Theory and Applications
M. Ogorzalek
This lecture will present basic techniques available for controlling chaotic dynamical systems. We will present first special features of chaotic systems which make the approaches to chaos control different from standard methods available in the control engineers toolbox. Sensitive dependence on initial conditions, ergodic properties of trajectories, bifurcation behavior can help in building special control strategies. Applications in engineering, biology and medicine will be presented.

Model Reduction for Systems with Low-Dimensional Chaos
C. Piccardi
The lecture will discuss a method for deriving a reduced model of a continuous-time dynamical system with low-dimensional chaos. The method relies on the identification of peak-to-peak dynamics, i.e. the possibility of approximately (but accurately) predicting the next peak amplitude of a variable from the knowledge of its two previous peaks. The reduced model is a simple one-dimensional map or, in the most complex case, a set of one-dimensional maps. Its use in control system design will also be discussed by means of some examples.

Bifurcation Control and Applications
E.H. Abed
The topic of bifurcation control will be reviewed with a view toward recent results and applications of current interest. Bifurcation control relates to the design of control systems that improve the operating characteristics of a nonlinear system in the vicinity of a bifurcation in its dynamics. Issues such as delay and stabilization of bifurcating solutions by feedback will be discussed. System monitoring for the automatic detection of impending bifurcations will also be considered. Recent developments related to bifurcation control of nonsmooth systems will be introduced. Applications will be used throughout the lecture to motivate the theory. This includes control problems for aircraft, computer networks and heart arrhythmia.

Controller Synthesis for Stabilizing Periodic Orbits in Chaotic Systems
M. Basso
The lecture deals with the use of finite-dimensional linear time-invariant controllers for the stabilization of periodic solutions in sinusoidally forced chaotic systems. Such controllers can be interpreted as rational approximations of the well-known Delayed Feedback Controllers (DFC). By exploiting results concerning absolute stability of nonlinear systems and robustness of linear systems, it is shown that controller synthesis techniques based on Linear Matrix Inequalities (LMI) can be devised. In particular, a synthesis algorithm to maximize the amplitude of the forcing input ensuring stable periodic solutions is discussed, together with some application examples.

<top of page>


NEURAL NETWORKS

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science
L.O. Chua
This lecture provides a nonlinear dynamics perspective to Wolfram’s monumental work on “A New Kind of Science”. By mapping a Boolean local rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, it is possible to characterize the complexity of the dynamics of each Boolean local rule. In particular, Wolfram’s seductive idea of a  “threshold of complexity”  can be rigorously defined via a “complexity index”.

On the Application of Spectral Techniques for the Analysis of Nonlinear Dynamic Arrays
M. Gilli
Regular arrays of nonlinear circuits present interesting applications in image processing and pattern recognition; they are also useful for modeling stationary and wave phenomena in many disciplines, ranging from physics to biology. Such arrays are described by large systems of  locally coupled nonlinear differential equations and they can exhibit a very complex behaviour. A complete study of their dynamics would require to classify all the attractors and possibly to estimate the domains of attraction. This is a formidable task that cannot be faced neither through computer simulation, nor through classical time-domain techniques, that are suitable for low-order dynamical systems. The lecture will briefly summarize the properties of stable arrays, i.e dynamical systems, where each trajectory (with the exception of a set of measure zero) converges towards an equilibrium point. Then it will focus on the application of spectral techniques to nonlinear arrays that exhibits either a periodic or a non-periodic behavior. It will be shown that: a) the whole set of periodic attractors can be characterized through a suitable extension of the describing function technique; b) the main properties and characteristics of such attractors can be determined through a harmonic balance (HB) based method; c) the limit cycle bifurcation processes, leading to a complex behavior, can be studied through a suitable combination of HB based and time-domain techniques.

Programmable Complex Systems, Visual Microprocessors and Universal Machines on Flows - with Bioinspired Aspects
T. Roska
Following the formal computing machine models on integers (Turing Machine and Von-Neumann computers) and on reals (Blum-Schub-Smale), recently, a new computing paradigm on flows (Roska and Chua) has been introduced. The basic idea of the latter, the CNN Universal Machine, will be introduced first with key notions and practical aspects. The possibility to algorithmically program and implement complex systems will be explored. Next, the various physical implementations will be reviewed, including visual microprocessors, and the computational and computer complexity issues will be described. Application case studies will follow for super-high-speed sensory-computing tasks (Xkframe per sec). Finally, the biological relevance will be highlighted culminating in a programmable mammalian retinal model implemented on a single chip.

<top of page>


NEW TRENDS IN STABILITY THEORY FOR COMPLEX SYSTEMS

Normal Form, Bifurcation, and Stabilization of Highly Nonlinear Systems
W. Kang
The first part of the talk is an introduction to the normal forms and invariants of nonlinear control systems. In the second part, bifurcations of control systems are defined. Classification of the bifurcations for systems with uncontrollable modes is derived using normal forms. Furthermore, feedbacks for the qualitative control of classical bifurcations will be introduced. The third part of the talk is on the feedback stabilization of nonlinear systems with a positive uncontrollable mode. Its feedback design takes the advantage of the bifurcation on controllability. In the last part, the speaker will announce some open problems and talk about possible directions for future research.

The Input-to-State Stability Approach for Global Analysis of Nonlinear Systems
D. Angeli
The notion of Input to State Stability allows in a unified framework to deal with a variety of systems theoretic properties. The focus of this talk are techniques which are in our opinion most relevant in order to analyze complex behaviours: incremental ISS and almost global ISS.

<top of page>


CHAOS IN COMMUNICATION SYSTEMS

Statistical Approach to Chaos: Some Theoretical Results and Applications
G. Setti (joint research with R. Rovatti and G. Mazzini)
Recent developments have highlighted that a statistical approach may greatly benefit the study of discrete-time chaotic systems (maps). The key idea is to consider a set of trajectories of non-vanishing measure of a chaotic map, whose study allows to track the mechanism causing the complex behavior and to characterize it quantitatively despite the well-known critical dependence on initial condition. A set of tools will be introduced which permits to assess the statistical features of the quantized process generated by a chaotic map in terms of the exact computation of high-order moments. Such a well-developed theoretical framework will be then applied to three hot information engineering topics, namely DS-CDMA communication system performance optimization, electromagnetic interference reduction and artificial multimedia traffic generation.

Bifurcation Phenomena in the Behavior of Access Protocols for Wireless Multimedia Communication Systems
G. Giambene
The steady increase of mobile traffic requires new solutions to coordinate the transmissions of wireless terminals. The first part of this talk will survey Medium Access Control (MAC) protocols, focusing on the techniques used in third-generation (3G) mobile communication systems and in Low Earth Orbit Systems. The second part of this talk will focus on MAC protocols based on uncoordinated access attempts from mobile terminals. A novel protocol will be proposed that adopts suitable control parameters to regulate the transmission of terminals belonging to different traffic classes. Such scheme will be modeled through non-linear equations by means of the Equilibrium Point Analysis (EPA) that will permit to highlight a bifurcation behavior.

<top of page>


(May 30, 2003)