Dynamical systems and Normal forms


A dynamical system is a description of the deterministic evolution of a state variable, whihc may be a finite or an infinite dimensional object. Typically, a dynamical system is given as the flow of a vector field, or the solution of a differential or a difference equation. Many examples of dynamical systems arise in physical or engineering sciences and in biological or medical applications. Specific systems under study by members of the AIMS Laboratory include waves in nonlinear optics, atmospheric vortices, and surface and internal waves in the ocean, among many others. Modeling of physical systems by a dynamical system can be either over finite dimensional phase space, the continuum, or on discrete lattices. Due to the energy conservative nature of many of the problems of interest to AIMS Lab members, it is quite common to work with Hamiltonian dynamical systems in particular.

Among the numerous analytical and numerical techniques available in the field of dynamical systems, a classical point of view advocates an initial normal forms analysis. Under transformation, a mathematician seeks to reduce a nonlinear dynamical system to its simplest form, eliminating inessential nonlinearities or degrees of freedom. In the context of Hamiltonian systems this technique is in particular very valuable in perturbation theory of invariant tori, in Nekhoroshev stability theory, in a Hamiltonian version of multiple scales analysis for partial differential equations, and in the renormalization process by which nonlinear wave turbulence can be understood.

Bifurcations occur in a dynamical system which depends upon a parameter when the behavior of the system undergoes sudden changes at particular parameter values. The range of bifurcation problems includes symmetry breaking and pattern formation in partial differential equations, resonance phenomena in nonlinear oscillations, persistence and break-up of invariant tori, and transition to chaos involked by nonlinear mode-interactions. These phenomena are investigated using diverse mathematical techniques, including the analysis of normal forms of the governing differential equations, the Lyapunov-Schmidt reduction to isolate the relevant degrees of freedom implicated in the phenomena under investigation, singularity theory to resolve the possibly complex parameter dependent multiplicity of solutions, group theoretic considerations as an organizing principle in the bifurcation and the normal forms analysis, and perturbation techniques to classify and analyze the local behaviour of bifurcation branches. Numerical computation combined with workstation computer graphics for visualizing quantitative/global information are available to AIMS Lab researchers to provide further understanding of the physical phenomena that lie as the ultimate goal of ones research.



  Stephanella Boatto  
  Walter Craig  

Dmitri Pelinovsky


Andrei Biryuk


Philippe Guyenne


Vladislav Panferov



  • Hamiltonian long wave expansions for water waves over a rough bottom (Craig, Guyenne, Nicholls and Sulem, 2004)
  • Hamiltonian long wave expansions for free surfaces and interfaces (Craig, Guyenne and Kalisch, 2004)
  • Normal forms for discrete breathers