Nonlinear Dynamics & Chaos

A real dynamic system is described by differential equations ( that describe the evolution of systems in continuous time), but it is often convenient to consider impler models, called difference equations, where the system evolves through a sed of discrete time steps. The simplest models of difference equations are the iterated maps because the future value of some variable at time t= (n+1) depends only on its value at t=n. Thus x_{n+1}=f(x_n)

A very general framework for ordinary differential equations is provided by the system:

\dot{x_1}=f_1(x_1,...,x_n)

\dot{x_n}=f_n(x_1,...,x_n)

System is said to be linear, if all x_i on the right hand side appear to the first power only. Otherwise the system would be nonlinear.

If we construct an abstract space with coordinates (x_1,x_2), then the solution (x_1(t),x_2(t)) corresponds to a point moving along a curve in this space:

This curve is called a trajectory, and the space is called the phase space for the system. The phase space is completely filled with trajectories, since each point can serve as an initial condition. The phase space for the general system is the space with coordinates x_1,...,x_n. Because this space is n- dimensional, we will refer to the system as an n-dimensional system or an nth-order system. Thus n represents the dimension of the phase space.

ONE-DIMENSIONAL FLOWS:

Flows on the line

We introduced the general system:

\dot{x_1}=f_1(x_1,...,x_n)

\dot{x_n}=f_n(x_1,...,x_n)

and mentioned that its solutions could be visualized as trajectories flowing through an n-dimensional phase space with coordinates (x_1,...x_n). With the simple case n=1, we get a single equation of the form:

\dot{x}=f(x)

We’ll call such equations one dimensional or first order systems.

Interpreting a differential equation as a vector field, we consider the following nonlinear differential equation:

\dot{x}=sinx (1)

We separate the variables and then integrate:

dt= \frac{dx}{\sin x}

which implies

t = \int\csc x dx = -ln|\csc x +\cot x|+c To evaluate the constant c, suppose that x=x_0 at t=0. Then C=ln|\csc x_0 +\cot x_0|. Hence the solution is:

t =| \frac{ln|\csc x_0 +\cot x_0|}{ln|\csc x +\cot x|}|
A graphical analysis of (1) is clear and simple. We think of t as time, x as position of an imaginary particle moving along the real line, and \dot{x} as the velocity of that particle. Then the differential equation \dot{x}= \sin x represents a vector field on the line. It’s convenient to plot \dot{x} versus x.

The arrows point to the right when \dot{x}>0 and to the left when \dot{x}0 and to the left when \dot{x}<0. At points when \dot{x}=0, there is no flow;such points are therefore called fixed points.There are two kinds of fixed points: solid black dots represent stable fixed points ( often called attractors or sinks) and open circles represent unstable fixed points (also known as repellers or sources).
A particle tarting at x_0= \frac{\pi}{4} moves to the right faster until it crosses x= \frac{\pi}{2}. Then the particle starts slowing down and eventually approaches the stable fixed point x= \pi from the left.
To find the solution to \dot{x}=f(x) starting from an arbitrary initial condition x_0, we place an imaginary particle (known as a phase point) at x_0 and watch how it is carried along by the flow. The phase point moves along the x-axis according to some function x(t). This function is called the trajectory based at x_0, and it represents the solution of the differential equation starting from the initial condition x_0. The given picture is called a phase portrait. The fixed points x* are defined by f(x*)=0. In terms of the original differential equation, fixed points represent equilibrium solutions ( called steady, constant).

EXAMPLE:
Find all fixed points for \dot{x}=x^2-1 and classify their stability.
Solution : Here f(x)=x^2-1 . To find the fixed points, we set f(x*)=0 and solve for x*. Thus x*=+-1 . To determine stability, we plot x^2-1 and then sketch the vector field. The flow is to the right where x^2-1>0 and to the left where x^2-1<0. Thus x*=-1 is stable and x*=1 is unstable.

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