Analytical, numerical, and software tools are essential for understanding of nonlinear dynamical systems

Inaugural lecture by Prof. Yuri Kuznetsov, Professor of Numerical Bifurcation Methods at the University of Twente.

Dynamical systems theory allows mathematicians to predict and describe abrupt changes and transitions (known as bifurcations) in different deterministic systems. Professor Yuri Kuznetsov will give his inaugural speech on 10 January, at the University of Twente. In his speech, Prof. Kuznetsov will state that without efficient numerical bifurcation methods and interactive software tools that go beyond conventional simulations, one could make no further progress in the analysis of nonlinear dynamical systems. Yuri Kuznetsov holds a post at the University of Twente’s MIRA research institute (in the AAMP group).

Multi-dimensional non-linear differential equations can have a complex solution structure that includes an infinite number of periodic and non-periodic solutions. Such solutions are extremely sensitive to variations in the parameters. The figure illustrates the degree of complexity involved. It is a parametric portrait of three coupled ordinary differential equations. The equations describe the behaviour of a food chain, consisting of prey (algae), predator (plankton), and super-predator (fish). The prey population’s growth rate (r) and niche capacity (K) are two control parameters. The food chain exhibits a specific dynamic for each combination of these parameters. The dark colours in this illustration correspond to the more complex periodic solutions that result from an increase in the number of predator-prey oscillations per period. The chaotic domain is bounded by several black curves on which bifurcations of periodic solutions (abrupt changes and transitions in oscillatory dynamics) occur. These curves can only be computed using special interactive bifurcation software tools. Such software must be numerically efficient, designed for interactive management, and equipped with a user-friendly graphical interface.

Hot topic
Working with an international team of scientists, Yuri Kuznetsov has already developed three generations of interactive bifurcation software: LOCBIF, CONTENT, and MATCONT. Advances in the theoretical analysis of dynamical systems require the development of new numerical methods for detection and analysis of bifurcations. Prof. Kuznetsov collaborates with biomedical scientists, in fields such as infectious diseases and clinical neurophysiology. Numerical bifurcation analysis will lead to a better understanding of dynamic phenomena in these fields and elsewhere. So the development of interactive bifurcation software is a hot topic.

Inaugural lecture
Prof. Yuri Kuznetsov will deliver his inaugural speech (which is open to public) at 16:00 on 10 January, in the Professor M.P. Breedveld room in the Waaier building on the University of Twente campus. For further details, or a digital copy of the inaugural lecture, entitled Exploring Borders of Chaos, please contact Juliette Fhij of the University of Twente’s Press Relations Department.