Dynamical Systems

Dynamical systems theory is the mathematics of change. It can be traced back at least to the ancient Greeks, but it is first recognizable to modern eyes in the work of Isaac Newton. Early achievements in this area were mainly in physics. However, the need to analyze change is pervasive throughout science. Thus modern dynamical systems theory has grown from Newton's time to find an important place in a much wider range of disciplines extending from mathematics, and information theory, to biology, ecology, and even financial forecasting.

Not only has its scope sharply grown, but the theory itself has undergone a revolution since the 1960's. In particular, the discovery of chaos has led to a new era in how we think about change and predictability in our world. What was a sedate 300-year-old field has erupted into an exciting new research frontier that has shaken up all of modern science to its foundations. It is an exciting time to be a part of the new discoveries being made in this field both in basic theory and in terms of applications.

The current research of applied dynamical systems in our department includes both theoretical study and practical applications. Dynamical systems may be represented by ordinary differential equations, partial differential equations, delay differential equations, or combination of differential equations and algebraic equations. They can be discrete, continuous or impulsive systems, or combinations of these.

Representative Publications

  • Y. Chen, Y. Huang and X. Zou, Chaotic invariant sets of a delayed discrete neural network with two non-identical neurons, SCIENCE CHINA: Mathematics, 56(2013), 1869–1878.
  • Q. Hu, J. Wu and X. Zou, Global continuation of periodic solutions of differential equations with state-dependent delay, SIAM J. Math. Anal., 44(2012), 2401-2427.
  • T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain (with T. Yi), J. Diff. Eqns., 251(2011), 2598-2611.
  • T. Yi and X. Zou, Map dynamics versus dynamics of associated delay R-D equation with Neumann boundary condition, Proc. Royal. Soc. London, A. 466 (2010), 2955-2973
  • B. Chan and P. Yu, Bifurcation, stability, and cluster formation of multi-strain infection models. J. Math. Biol. 67 (2013), 1507–1532
  • M. Gazor, P. Yu, Spectral sequences and parametric normal forms. J. Differential Equations 252 (2012), 1003–1031
  • J. Yang, M. Han, J. Li and P. Yu, Existence conditions of thirteen limit cycles in a cubic system. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2569–2577
  • Robert M. Corless, Pseudospectra of exponential matrix polynomials, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 70-80, ISSN 0304-3975, http://dx.doi.org/10.1016/j.tcs.2012.10.021.
  • A. Amiraslani, R. M. Corless, and P. Lancaster, Linearization of matrix polynomials expressed in polynomial bases IMA J Numer Anal (2009) 29 (1): 141-157 first published online February 27, 2008 doi:10.1093/imanum/drm051