# Textbook Recommendation: Topological Dynamics

I need to take credits satisfying a topology requirement, and can structure it myself. My field of study is dynamical systems, can someone recommend a textbook that handles differential equations/dynamical systems from a topological point of view? Or is there another recommended field of study that would fill this?

Not completely sure what you are looking for....The books are ordered w.r.t. difficulty. First one the easiest.

Classical bifurcation theory If you want to classify dynamical systems it is important to see if the solutions curves can be continuously' deformed into each other. In the first order ODE this can be very easily done by introducing continuous parameter dependencies in the vector field which let you deform the solution curves.

ARNOLD Geometrical Methods in the Theory of ODEs

GUCKENH. HOLMES Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Chaos theory The chaos generating mechanism for an ODE/map can typically be brought down to an abstract underlying topological phenomenon. You see this for example when the tangles of the unstable and stable manifold can intersect each other in such a way that you get a mechanism which is equivalent to that in the Smale-horseshoe.

ALLIGOOD, SAUER, YORKE Chaos: an Introduction to dynamical systems

PALIS TAKENS Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Advanced)

Index theory Index theory for dynamical systems associates an index, which is a discrete function on the dynamical system. Furthermore, the index is constructed such that it does not change when the system is continuously' deformed. Consequently giving a classification of the dynamical system. Milnor's his topology from the differentiable point of view is quite nice to start with in this topic. His main focus is to introduce the Poincare-Hopf theorem for vector fields.

MILNOR topology from the differentiable point of view

If you are interested in more advanced readings in the topics of connecting index theory to dynamical systems then you might look into Morse or Conley index theory. For Morse index theory there is a nice online lecture set by Georgia U on youtube and for Conley index theory you might want to start with any of the introductory papers by Mischaikow (just google his name).

You could do a study of "Thurston-Nielsen Classification theorem", which classifies diffeomorphisms of surfaces.

You could also take a look at V.I. Arnold's "Ordinary Differential Equations" and S. Lefshetz's "Differential Equations: Geometric Theory". Also, there's a pair of iconic books; first is by J.Palis and W. de Melo called "Geometric theory of dynamical systems". Second is A. Katok and B. Hasselblatt's "Introduction to the Modern Theory of Dynamical Systems".