Homoclinic bifurcations

Thesis

Previously obtained results from the study of homoclinic bifurcations in ordinary differential equations are presented. The standard technique of analysis involves the construction of a Poincaré map on a surface near to the homoclinic point. This map is the composition of an inside map, with behaviour linearized about the homoclinic point, together with an outside map, with behaviour linearized about the homoclinic orbit. The Poincaré map is then reduced to a one-dimensional map, involving the return time between successive visits to the Poincaré surface.

These standard techniques in the contemplation of homoclinic systems are then extended to a class of partial differential equations, on unbounded domains. This follows a method introduced by Fowler [Stud. Appl. Math. 83 (1990), pp. 329–353]. This extension involves more technicalities than in the case of ordinary differential equations.

The method of Fowler is extended to cover the case of vector-valued partial differential equations, and to consider the consequences of symmetry invariances. A Poincaré map is derived, and then is reduced to a finite-dimensional map. This map has dimension equal to the number of symmetry invariances of the system.

Some simple examples of this finite-dimensional map are studied, in isolation. A number of interesting bifurcation pictures are produced for these simple examples, involving considerable variation with the values of coefficients of the map.

Partial differential equations on finite domains are then considered, yielding similar results to the ordinary differential equation case. The limit as domain size tends to infinity is examined, yielding a criterion for distinction between the applicability of finite and infinite domain results.

Finally, these methods are applied to the Ginzburg-Landau system. This involves the numerical calculation of coefficients for the finite-dimensional map. The finite-dimensional map thus derived supports an interesting interlocked isola structure, and moreover correlates with numerical integration data.

Authors: Drysdale, D; David Drysdale

• Publication date: 1994
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Partial differential equations