“Recent Development of Normal Form Theory
Normal form theory is one of the most useful and important tools in the study of nonlinear dynamical systems. Although some theories and methodologies have been developed in this area, better approaches are still needed in analyzing the complex behaviour such as instability and bifurcations. Efficiently combining analytical and computational (numerical/symbolic) methods is one of such approaches.
The conventional normal form (CNF) can be further simplified, leading to the simplest normal form (SNF), which is more powerful in attacking higher dimensional and/or higher order nonlinear dynamical systems. We have developed an efficient, recursive method for computing the SNF of differential equations. In addition, several specific techniques have been established for computing the SNF associated with various singularities. Lie algebra is used for theoretical proofs, while algorithms and Maple programs are developed to greatly facilitate applications in solving real problems.
This talk will also present some results of the applications of normal form theory which we have been involved, including epidemiology models in finite dimensional system, neuron network, high speed aircraft and nonlinear oscillators in infinite dimensional systems with time delay.