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Introduction to Nonlinear Dynamics
We begin with a model of two-dimensional dynamical system
that obeys the following ordinary differential equations
(ODEs):
f(x,y) dt
dy (8.1a)
g(x,y) dt
dx (8.1b)
It is an autonomous system as it does not explicitly involve the
independent variable t. If such a system involves a nonlinear
term such as x^2 (t), x(t)y(t), sinx(t), exp( t) etc., then the system is
said to be nonlinear one.
Recall the chain rule for differentials: (dy/dx)(dx/dt) is equal to
dy/dt. It means that
g(x,y)
f(x,y) dx/dt
dy/dt dx
dy . (8.2)
When y is given as a function of x or x is represented as a
function of y , a trajectory of solution is formed for the model.
The case when both x and y are constant means that both
derivatives in Eq. 8.1 are zero. Such points are called
stationary points , equilibrium points or fixed points of the
system.
A Simple Pendulum
Let us first study a simple nonlinear system in which a mass m is attached to a rigid massless rod so the center of gravity of the mass is at distance L from the frictionless pivot. The system is constrained to move in a plane. If there is no viscous damping, then the motion of this system is governed by the equation
- Ia(t) (8.3)
where, (t), is the torque around the pivot point; I is the moment of inertia about the pivot point, and a(t) is the angular acceleration.
We use x as a angular displacement, v(t) as its time rate of change,
and a(t) as the second derivative of x(t).
In the case of the simple pendulum the displacement is an angle measured in radians, v(t) is the angular velocity in radians per s, and the acceleration a is the angular acceleration in radians/s and is mgLsinx with I = mL^2. Making these substitutions gives
dt
d x
- mgLSinx mL 2
2
^2
This equation can be simplified by dividing by mL. In addition, let us choose the pendulum with L = 9.8 m , then the equation reduces to the following form:
-sinx
dt
d x
2
2
Because this model has sinx rather than x on the right-hand side, it forms a second-order non-linear differential equation.
NONLINEAR DYNAMICS
Visualization in Phase Space
o The phase space is a coordinate space with x, vx, y, vy,.. ., where v is velocity or any other canonical state variable. It has position graphed along the horizontal axis and velocity or momentum graphed along the vertical axis.
o Phase space can have fixed points (x, v) such that they satisfy the model (say Eq. 8.1 as f(x, v) = 0 and g(x, v) = 0). This corresponds to steady state.
o The set of points in the phase space are identified as orbit or trajectory****. If the set of points in the simulation repeat itself after some time (T), then the orbit is said to be periodic , that is x(t + T) = x(t). The orbit of mass-spring system in a friction free environment is an ellipse in phase space.
o A closed curve is called a limit cycle in phase space towards which an orbit evolves as time goes to large values. It has property that all other curves move towards it or away from it.
o When all the neighboring trajectories are going towards the limit cycle it is called a stable or attracting cycle , otherwise it is an unstable or repelli ng one.
Let us observe such cycles through examples of various famous nonlinear models.
Phase Space: Undamped Motion of a Simple Pendulum
Let us construct the phase curves of the simple pendulum without any damping and driving force. It will be useful to make a phase portrait using a variety of initial conditions. To do this we will start the pendulum at x = 0 , but we will give it various velocities corresponding to various kinetic energies. The results are shown in Figure below.
Let us allow velocity range to be from -3 to 3 in steps of 0.5 and domain is given by [ -3, 3 ].
♫ We can modify the initial conditions to get a set of concentric ellipses centered on points (2n, 0). ♫ The point in phase space (0, 0) and the set of points ( 2n, 0), where n is an integer are points of stable equilibrium for the simple pendulum; the pendulum is hanging down from the pivot. ♫ The set of points [(2n - 1), 0] are points of unstable equilibrium; the pendulum is "standing" on its pivot.
Fig. 8.1 The angular velocity v (radians per second) as a function of displacement x (radians) for an undamped simple pendulum.
