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Global Analysis and Synthesis of Oscillations

Final Activity Report Summary - GASO (Global Analysis and Synthesis of Oscillations)

The main theme of this project concerns the global analysis and synthesis of nonlinear oscillations in networks of interconnected oscillators. Oscillators are ubiquitous in physical, biological, biochemical, and electromechanical systems. Detailed models of oscillators abound in the literature, most frequently in the form of a set of nonlinear differential equations whose solutions robustly converge to a limit cycle oscillation. Local stability analysis of the oscillations is possible by means of Floquet theory but global stability is usually restricted to second order models. For these models, global analysis is performed by using specific low dimensional tools (phase plane methods, Poincaré-Bendixson theorem, etc.) which do not easily generalise to higher dimensional (complex) models.

As a consequence, global analysis of complex oscillator models is quite hard since currently no general analysis method exists. This lack of general analysis methods typically forces complex models of oscillators to be studied only through numerical simulation methods. Although the latter may give a first insight into the behaviour of the model, a more in-depth understanding of its fundamental properties is generally impeded by the complexity of the model and the challenge of rigorous global stability, robustness, and sensitivity analysis. Moreover, even in the case of simple two-dimensional models, the low dimensional methods used for their analysis do not generalise to the analysis of their network interconnection. These considerations show the need for developing general methods that allow the global analysis of oscillators, either isolated or interconnected into a network.

This project is concerned with the development of such general methods for the existence, robustness and global stability analysis of oscillators. The development of such general methods represent the first step towards the development of a unified mathematical theory for oscillator analysis and synthesis. This theory will then be useful for answering various open questions in fundamental research such as:
- validation of models of oscillators proposed in the literature (biochemical models for circadian rhythms, or models of cardiac cells, oscillatory enzymes, epidemiological processes or neural networks are just a few examples);
- characterisation and classification of fundamental oscillations mechanisms;
- synthesis of artificial networks of oscillators for the control of rhythmic tasks robots (such as walking robots or general dexterous robots).

The objective of this project has been reached by the development of a new global analysis methodology based on a generalization of dissipativity theory (introduced by Jan Willems in 1972). This new methodology allows to characterise the existence, uniqueness and global asymptotic stability of limit-cycle oscillations in high-dimensional models of passive oscillators and of interconnected passive oscillators. All these new results are presented in the forthcoming IEEE TAC journal paper 'Analysis of interconnected oscillators by dissipativity theory'. This paper includes results that, for the first time, allow to globally analyse arbitrarily complex models of oscillators (belonging to a particular class of systems called 'passive oscillators') and to extends their global analysis to networks of identical passive oscillators.

In parallel to this work, three other line of research have been opened. The first one focuses on better understanding the synchronization mechanisms responsible for the generation of stable synchronised oscillations in particular biochemical networks of oscillators. The second one develops a new numerical analysis method for the global stability analysis of limit-cycle oscillations in high-dimensional piecewise-linear oscillators. The third one is not directly related to the analysis of networks of oscillators but to the creation of algorithmic optimisation methods for solving the drug-scheduling problem intervening in the development of 'intelligent' treatments for chronic-like diseases like cancers and HIV.