The dynamics of a system (of any kind such as physical, chemical, or biological) is fully described by a differential equation, or a set of differential equations (of course, with an appropriate set of initial conditions) involving dynamical variables which together determine the state of the system uniquely at any instant of time.

For a classical (physical) system, the dynamics is studied using Newton’s law (a set of 6N second order coupled ordinary differential equations) with a set of initial conditions, called state variables (r(O), v(O)), to predict a future state of the system to any desired accuracy. Of course, a certain amount of uncertainty lies in the prediction because of some fluctuations in force-terms (inherent in all physical systems), which is usually ascribed to as a statistical phenomenon. But if the system is a nonlinear one, the long term behaviour thereof is often unpredictable.

As usual, the behaviour of a physical system is blueprinted on its phase space. A phase space is constructed with the components of x, and these at each point in this space uniquely determine a state of the system. A phase trajectory indicates the evolution of a system into a point attractor, a limit cycle (ref: figure), or a chaotic regime, asymptotically. A slight variation in the initial conditions can take two nearby trajectories in the phase space away from each other (known as exponential divergence of phase trajectories) contrary to the well-found belief in classical dynamics (that two nearby trajectories will maintain a constant separation throughout the time evolution of the system). This phenomenon of unpredictability of long-term behaviour ofa nonlinear system, which is very sensitive to its initial conditions, is called chaos.

Although chaos is seen in most nonlinear systems (linear systems never exhibit chaos), it is possible for a complex nonlinear system to behave in a well-predictable manner.