Multi-Parameter Dynamical Systems: Bifurcations and Applications

Samenvatting

This project together with Dr. V. Gaiko is a continuation of the current Dutch research on Global Analysis, Bifurcation and Catastrophe Theory, and Applications of Multi-Parameter Dynamical Systems as this is carried out at the Bernoulli Institute of Mathematics, Computer Science and Artificial Intelligence of Groningen University. Many dynamical systems, e.g., in population dynamics, at the end are generated and approximated from planar polynomial systems and their precise description therefore is of great importance. Here the notorious Hilbert’s sixteenth problem comes into play, concerning the number and distribution of limit cycles. With help of global methods, the dynamics of given multi-parameter dynamical systems can be completely described. We apply this to three-dimensional Lorenz-like dynamical systems.

Processes in the natural environment or in living organisms differ intrinsically from those studied in the “hard sciences”. Classically, a physical or chemical process is reduced into one or more idealized building blocks which are subsequently studied in isolation. Modern developments in life sciences have exposed the limitations of this approach. Collective cell behavior or the behavior of a biomedical or an ecological system is inherently governed by the simultaneous interaction and competition between various mechanisms. Since the coupling between these mechanisms is typically nonlinear, the system cannot be understood as a combination of one or more of its subsystems.

The mathematics of nonlinear systems is firmly rooted in the classical hard sciences, especially since the reduced models, e.g., in hydrodynamics and classical mechanics are highly nonlinear. However, at present, the challenges posed by processes in the brain, in the cell, or in the ecological system, to the theory of dynamical systems surpass those coming from more classical fields in richness, in magnitude and in difficulty. This leads to a deep cross-fertilization, because the mathematics of nonlinear systems itself is at a critical phase in its development as well. Especially in idealized settings, many fundamental aspects of the theory are well-understood. It has only very recently become possible to delve beneath the surface of more complex phenomena by combining these aspects and subsequently extending these into new theoretical insights. Hence, mathematics is ready, and well-prepared, to meet the challenges posed by, for instance, biological networks and predator-prey systems. Since the interaction between mathematics and life sciences has traditionally not been very intense, the mathematical approach has not yet been exploited to its full strength and depth within this context.

The project will be of great interest for the NWO-cluster Nonlinear Dynamics of Natural Systems (NDNS+). The recent theoretical results on limit cycle bifurcations and new original bifurcational geometric methods which will be developed in this project will contribute to completing the global qualitative analysis of some well-known mathematical models and their modifications which are used in ecology, biology, and medicine. Besides, the development of global bifurcation theory of polynomial dynamical systems will give good perspectives for completing Hilbert’s sixteenth problem on the maximum number and distribution of limit cycles for planar polynomial dynamical systems and understanding the strange attractor bifurcation scenario in the three-dimensional case.

Kenmerken

Projectnummer

040.11.719

Hoofdaanvrager

Prof. dr. H.W. Broer

Verbonden aan

Rijksuniversiteit Groningen, Faculty of Science and Engineering (FSE), Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence

Looptijd

01/09/2019 tot 31/12/2019