1. ECE-7000: Nonlinear Dynamical Systems 3. Phase Space Methods 3.1 Determinism: Uniqueness in phase space We Assume that the system is linear stochastic or non linear deterministic. For a purely deterministic system, it is important to establish a vector space such that specifying a point in the space specifies the state of the system and vice versa. The dynamical systems are usually defined by the set of differential equations. The concept of state is very powerful in the case of nondeterministic systems. They are usually described by a set of states and transition rules ( in the form of transition probabilities). EX : Markov Process Main Feature: The essential feature of these features processes is their strictly finite memory : the transition probabilities depend on the present state and not on the past states. A purely deterministic process is the limiting case of Markov Process.

2. ECE-7000: Nonlinear Dynamical Systems 3. Phase Space Methods If phase space is finite dimensional vector space R^m, the dynamics are described by am m dimensional map or by m first – order differential equations. Xn+1 = F(Xn) For continuous case : d/dt X(t) = f (X(t)) This situation is usually referred to as a flow. The vector field f is defined not to depend on time and thus is called ‘autonomous’. If f contains explicit time dependence, the system is no more dynamical since time invariance is broken. The system may be made autonomous by introducing additional degrees of freedom.

3. ECE-7000: Nonlinear Dynamical Systems 3. Phase Space Methods In autonomous case the solution to the initial value is known to exist and unique if the vector field is Lipshitz Continuous. Lipshitz Continuous: A function is Lipschitz continuous if it statisfoes Lipshitz condition for a finite constant C. Lipshitz Condition: A mapping f: X  Y between metric spaces is said to satisfy Lipshitz condition if there exists a real constant C d y ( f(p), f(q) ) <= C d x (p, q) Trajectory: A sequence of points X n solving the above equation is called ‘trajectory of the dynamical system’. Typical trajectories will run away to infinite or stay bounded as time proceeds. The observed behavior depends on form of F and initial conditions. Basin of attraction: The set of initial conditions leading to the same asymptotic behavior of the trajectory is called ‘basin of attraction’ for particular motion.

4. ECE-7000: Nonlinear Dynamical Systems 3. Phase Space Methods For an autonomous system with two degrees of freedom and continuous time, there are only 2 possibilities: a) Fixed point and b) limit cycle. For flows in at least three dimensions, the method of stretching and folding can produce chaos. 3.2 : DELAY RECONSTRUCTION The attractor formed by S n is equivalent to the attractor of unknown space in the original system if dimension m of delay coordinate space is twice the box counting dimension D F of the attractor. 3.3: FINIDING A GOOD EMBEDDING: m * tau is the most important embedding parameter. False Neighbours: A method for determining the smallest dimension for embedding. mutual information is used to find appropriate time lag.