I would make this a comment, but it seems to be too long for that. As I have suggested making the question a CW, I would like this answer to be treated as an extensive comment expressing personal opinion.
Great answers have already been provided, so I just want to make it clear that the theory of dynamical systems in general, and what the reader seems to be looking for, are somewhat different things. First, the theory of dynamical systems is incredibly vast (see the book series suggestions by Ian Morris in one of the answers). There are a few reasons for this vastness, the most obvious of which are probably the following.
1) Dynamical systems is a field that draws from many different fields (topology, geometry, complex, real and functional analysis, algebra and number theory, and more). To understand why this is so, consider:
2) A dynamical system, in its broadest sense, is an action of a semigroup or a group on a set. For example, consider $f: \mathbb{R}\rightarrow\mathbb{R}$ given by, say, $f(x) = x/2$. Taking the semigroup $G = \mathbb{Z}_{\geq 0}$ of non-negative integers, we can define an action on $\mathbb{R}$ as follows: $k\in G$ acts on $x$ via $kx = f^k(x)$ where $k$ denotes $k$-fold composition of $f$ with itself. Since $f$ is invertible, we can consider the action of the group $\mathbb{Z}$ on $\mathbb{R}$ via $kx = f^k(x)$, where if $k < 0$, then $f^k$ denotes $k$-fold composition of the inverse of $f$ with itself. The interesting questions begin to appear when the set on which the dynamics is defined (action of a group) carries more than just a set structure; say a topological space, or some kind of an algebraic structure, such as a module or a ring or a field, or a smooth structure, such a smooth manifold, or an analytic structure, such as a complex-analytic manifold, or a measurable. In this case the actions that are most interesting to consider are those that preserve the structure (for example, $f$ above is a homeomorphism, an (analytic) diffeomorphism and an isomorphism when $\mathbb{R}$ is viewed as a topological space with the usual Euclidean topology, an (analytic) manifold, or an algebraic field, respectively. Then one beigns to raise the following questions: what is the structure (in terms of the structure carried by the space) of orbits? For example, are there dense orbits? Periodic ones? How many periodic orbits of a given period are there? How does the number of periodic orbits grow as a function of the period?
The modern theory of dynamical systems can thus be roughly broken into areas according to the "structure" of the space on which the dynamics is considered: topological dynamics, algebraic/arithmetic dynamics, smooth dynamics, holomorphic dynamics and ergodic theory (considered, respectively, on topological spaces, algebraic spaces such as number fields, smooth manifolds, analytic manifolds where most of the time the action is given by a rational--i.e. quotient of polynomials--map, and measured spaces where the action is typically measure-preserving and satisfies a few other technical conditions). Of course, all these may intersect. The modern theory is concerned with the following (related) problems:
(1) Investigation of specific dynamical systems as a source of interesting examples, or looking for examples of certain dynamical phenomena;
(2) Classification: classifying dynamical systems up to a certain quite natural equivalence (here is an example for topological systems: http://en.wikipedia.org/wiki/Topological_conjugacy). This program is huge and the problems here are very difficult;
(3) Related to (2), search for invariants (properties that are preserved by the equivalence from (2)). There are two sets of invariants: complete and incomplete. A complete invariant is a property such that if two dynamical systems have this property, they they are necessarily equivalent, and an incomplete invariant is a property that is shared by two equivalent dynamical systems, but is not a complete invariant.
(4) Study of invariant objects. As an example, let us return back to the example $f$ acting on $\mathbb{R}$ from above. Notice that $f(0) = 0$. This in particular shows that the set $\{0\}$ is invariant under $f$. Also notice that for each $x\in\mathbb{R}$, $f^n(x)\rightarrow 0$ as $n\rightarrow\infty$. In this case the point $0$ (or the set $\{0\}$ is called an attractor. In general, given a dynamical system and a subset of the space on which the system acts which is invariant, one wants to study the structure of this set in terms of the structure of the ambiant space (e.g. the topology of invariant sets in topological dynamics) as well as the dynamics of the dynamical system when restricted to the invariant set. Notice that this also includes dimension theory (that has been mentioned above).
I presume the dynamical systems that you were curious about are those that arise as ordinary and partial differential equations, in which case one is working with an action of a continuous group on some space in the smooth (or analytic) category (manifolds). Certainly the field here is huge (in fact, systematic development of dynamical systems started as study of certain differential equations which model planetary orbits). Some of the biggest questions in this area are the $N$-body problems (http://en.wikipedia.org/wiki/N-body_problem), the Navier-Stokes equations (http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations) and in general, as has already been mentioned above, different types of ODEs and PDEs that arise as models in biology, physics, computer science, etc (in physics in particular, wave equations are abundant (http://en.wikipedia.org/wiki/Wave_equation), such as the Schroedinger equation from quantum mechanics).
Also arising in quantum and statistical mechanics are dynamical systems on infinite-dimensional spaces (Hilbert spaces). In this case the dynamics may be given by the action of a unitary operator on a Hilbert space, such as the solution of the Schrodinger equation. Techniques here are usually principally different than those in the finite dimensional case (e.g. on a finite dimensional manifold).
In general, the field of mathematical physics is huge, and dynamical systems are certainly not foreign to mathematical physics. Some of the areas of math physics where dynamical systems arise naturally are: statistical mechanics (here measurable dynamics, ergodic theory are prevalent); classical mechanics and general relativity (Newtonian dynamics modeled by a system of ODEs and systems of PDEs from relativity); fluid mechanics (the main problem here is the Navier-Stokes equations) which studies dynamics of compressible and incompressible fluids (or gasses) in a given environment, modeled by (a system of) PDEs; quantum mechanics and spectral theory (infinite dimensional dynamics given by unitary operators acting on Hilbert spaces); wave mechanics (wave equations, given by (a system of) PDEs).
I hope this helps.
