Introduction to Dynamical Systems

In this tutorial we are concerned with dynamical systems. A simple definition of dynamical system adequate for our purposes is as follows. A dynamical system consists of a state space along with an operation which maps the state at the current time to the state at some later time. Thus the current state contains all the needed information to completely describe the system for all times in the future. In other words, to determine the state of the system at a future time, only the state at the current time is needed; information about the past states is not necessary. The state at some initial time t0 can be considered an initial condition. We consider two classes of dynamical systems; discrete-time and continuous time dynamical systems. In discrete-time dynamical systems, time can only take on discrete values, usually the integers, while in continuous-time dynamical systems, the time can be any real number. Given an initial conditions x0 at t0, we can find the state at all future time t, denoted x(t). The collection x(t) for all (future time) t is called the trajectory. Sometimes x(t) is written as x(t,t0,x0) to indicates its dependence on t0 and x0. If x(t,t0,x0) does not depend on t0, then the system is called autonomous. Otherwise, the system is called nonautonomous.

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