Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour

Part 1 Introduction: preliminary ideas autonomous equations autonomous systems in the plane construction of phase portraits in the plane flows and evolution. Part 2 Linear systems: linear changes of variable similarity types for 2x2 real matrices phase portraits for canonical systems in the plane classification of simple linear phase portraits in the plane the evolution operator affine systems linear systems of dimension greater than two. Part 3 Non-linear systems in the plane: local and global behaviour linearization at a fixed point the linearization theorem non-simple fixed points stability of fixed points ordinary points and global behaviour first integrals limit points and limit cycles Poincare-Bendixson theory. Part 4 Flows on non-planar phase spaces: fixed points closed orbits attracting sets and attractors further integrals. Part 5 Applications I - planar phase spaces: linear models affine models non-linear models relaxation oscillations piecewise modelling. Part 6 Applications II - non-planar phase spaces, families of systems and bifurcations: the Zeeman models of heart beat and nerve impulse a model of animal conflict families of differential equations and bifurcations a mathematical model of tumor growth some bifurcations in families of one-dimensional maps some bifurcations in families of two-dimensional maps area-preserving maps, homoclinic tangles and strange attractors symbolic dynamics new directions.