Local properties of solutions.

1) If ф(t) is a solution, then  ф (t+c) is a solution for any c€R.

2) Existence: For any t₀€R , x⁰ €G, a solution x(t, t_0, x⁰)  exists in a certain interval 

3) Smoothness: If f € C^p(G), p≥1 , then ф(t) € C^p+1 .

4) Dependence on parameters: Let f =f(x,α),α€G_a, , where G_a is a domain, then x(t, t_0, x⁰,α)€ C^p( x G_a)  

5) Let x⁰ be a non-equilibrium point; then there exist neighbourhoods V,W of the points , x⁰ ,f(x⁰) respectively, and a diffeomorphism y=h(x):V W  such that the autonomous system has the form y=const

Global properties of solutions.

1. Any solution x= ф(t)  of the autonomous system (1) may be extended to an interval  (t _–t₊). If =R the solution is said to be unboundedly extendable;

2. The extension is unique in the sense that any two solutions with common initial data are identical through out their range of definition

3. Any solution of an autonomous system belongs to one of the following three types: a) a periodic, with ф(t ₁) ≠ ф(t ₂)  for all t _1≠ t₂

b) periodic, non-constant;

25) Phase pattern of trajectory of system of the second order in the neighborhood rest point.

The second-order differential equation of general type

¨x = f (x, ˙x , t)

with initial conditions, say x(t0) and ˙x(t0), is an example of a dynamical system. The evolution or future states of the system are then given by x(t) and ˙ x(t). Generally, dynamical systems are initial-value problems governed by ordinary or partial differential equations, or by difference equations.

The equation above can be interpreted as an equation of motion for a mechanical system,in which x represents displacement of a particle of unit mass, ˙x its velocity, ¨x its acceleration,and f the applied force, so that this general equation expresses Newton’s law of motion for

the particle:

acceleration = force per unit mass

A mechanical system is in equilibrium if its state does not change with time. This implies that an equilibrium state corresponds to a constant solution of the differential equation, and conversely. A constant solution implies in particular that ˙x and ¨x must be simultaneously zero.

Note that ˙x = 0 is not alone sufficient for equilibrium: a swinging pendulum is instantaneously at rest at its maximum angular displacement, but this is obviously not a state of equilibrium.

Such constant solutions are therefore the constant solutions (if any) of the equation

f (x, 0, t) = 0.

We distinguish between two types of differential equation:

(i) the autonomous type in which f does not depend explicitly on t ;

(ii) the non-autonomous or forced equation where t appears explicitly in the function f .

A typical non-autonomous equation models the damped linear oscillator with a harmonic

forcing term

¨x+k˙x + ω20x = F cos ωt,

in which f (x, ˙x, t) = −k˙x −ω20x +F cos ωt.

There are no equilibrium states. Equilibrium states

are not usually associated with non-autonomous equations although they can occur as, for

example, in the equation

x + (α + β cos t)x = 0.

which has an equilibrium state at x = 0, ˙x = 0.

We shall consider only autonomous systems, given by the differential

equation

¨x = f (x, ˙ x), (1)

in which t is absent on the right-hand side. To obtain the representation on the phase plane, put

˙x= y, (2.1)

so that

˙y= f (x, y). (2.2)

This is a pair of simultaneous first-order equations, equivalent to ¨x = f (x, ˙ x),.

The state of the system at a time t0 consists of the pair of numbers (x(t0), ˙ x(t0)), which can

be regarded as a pair of initial conditions for the original differential equation. The initial

state therefore determines all the subsequent (and preceding) states in a particular free motion.

In the phase plane with axes x and y, the state at time t0 consists of the pair of values

(x(t0), y(t0)). These values of x and y, represented by a point P in the phase plane, serve

as initial conditions for the simultaneous first-order differential equations (2.1) & (2.2), and

therefore determine all the states through which the system passes in a particular motion. The

succession of states given parametrically by x = x(t), y = y(t),

traces out a curve through the initial point P: (x(t0), y(t0)), called a phase path, a trajectory or

an orbit.

The direction to be assigned to a phase path is obtained from the relation ˙x = y (eqn 1.7a).

When y > 0, then ˙x > 0, so that x is increasing with time, and when y < 0, x is decreasing

with time. Therefore the directions are from left to right in the upper half-plane, and from right

to left in the lower half-plane.

To obtain a relation between x and y that defines the phase paths, eliminate the parameter t

between (2.1) and (2.2) by using the identity

˙y/˙x= dy/dx.

Then the differential equation for the phase paths becomes

dy/dx= f (x, y)/y