 # Dynamical Systems

Dynamical systems(1,9,10) as a field of study have been around since the time of Newton due to their great importance in the sciences. Only in rare instances can such systems be solved algebraically, with linear (time independent) systems and some Hamiltonian systems as exceptions. Usually we need computers to find the solution.

A dynamical system is usually presented in one of two ways:

1. As a set of differential equations, for example Newton's laws for the motion of planets, population dynamics, fluid mechanics, etc. With a given initial condition, such a differential equation has typically one solution which varies with time. The solution then depends on the initial condition, time, and relevant parameters (for example the masses of the various planets).
1. As a transformation which models the change of an initial condition after one unit of time. Iterating such transformations then produces the solution at a discrete set of times in the future.

The questions to be answered using dynamical systems are:  What is the behavior of the solution as time tends to infinity? Does the solution converge to a constant (equilibrium/steady state) solution, a periodic one, or a chaotic one, and what do these outcomes depend on? How stable are the dynamics? What is turbulence? Can we define some average behavior?

A typical class in dynamical systems will devote itself to a few of these topics:

1. Basic definitions. Notions of stability and instability. Becoming familiar with the types of phenomena encountered.
2. Bifurcation Theory(2), which addresses the question of how changes in parameters affect the behavior of solutions. This topic tends to have a heavily algebraic/analytic flavor. The student will learn about many different kinds of bifurcations. An important example is the Hopf bifurcation(11), in which a stable steady state solution becomes unstable, while a stable periodic solution emerges.
3. Hamiltonian systems(3,12). These are of great interest in physics. Such systems have at least one conserved quantity (energy). A similar class of transformations is one that preserves an additional quantity, such as area, volume, or a symplectic form. Such dynamical systems occur frequently and have special properties.
4. Systems with sensitive dependence on initial conditions and chaotic systems(4,13). This builds on a class of very interesting examples of hyperbolic dynamical systems that can be fully analyzed using tools such as symbolic dynamics that help elucidate their behavior. An important classical example has been Smale's Horseshoe.
5. Ergodic theory(5,14). The historical background for this emerges when one tries to relate the movement of microscopic atoms in a gas to macroscopic thermodynamic quantities, such as pressure and temperature. In modern mathematical language, one learns here about invariant measures, averages, ergodic theorems (Birkhoff, Von Neumann) and its important generalization: Oscledec's multiplicative ergodic theorem. Lyapunov exponents are defined. Consider two nearby points and follow these as time evolves. Does their distance increase or decrease at an exponential rate? If so, then these points lie along unstable or stable directions of the system, respectively.
6. Perturbation theory(6,15). When we consider the motion of planets, the sun has by far the greatest mass, and so the motion of individual planets is mostly dominated by just the sun. The corresponding solutions are the Keppler solutions which give rise to elliptical planetary orbits. In reality of course the motion of a single planet is determined not just by the sun, but also by the other planets which have small mass relative to the sun, and perhaps large distances to the planet of interest. Therefore, the deviations of planetary motion from a Keppler orbit can be analyzed in terms of power series that involve the masses and distances of the other planets. This was the beginning of perturbation theory. More modern and far more sophisticated methods are also used, for example the Kolmogorov-Arnold-Moser (KAM) theory(7,16).
7. Special systems. Some classes in dynamical systems concentrate on very specific examples, such as holomorphic (analytic) transformations of the complex plane. Already for quadratic transformations z->z^2+c this leads to wonderful graphics and theory, with concepts such as Julia sets, Mandelbrot sets(8,17), and fractals.

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