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**Examples **

**Example 1**

Consider linear dynamical system

There is only one critical point at the origin,

x_{C} = 0, y_{C} = 0.

Since the system is linear, we do not have to linearize it and can write the matrix of linearized system immediately:

Its eigenvalues are calculated from (8.27):

Eigenvectors can be found easily by hand. Recall that eigenvectors are solutions to equation

(Notice that we suppress the Einstein summation convention). For i = 1 we obtain homogeneous system of linear equations

where e_{1} = (a, b) is unknown eigenvector. Since the rows (or columns) of the matrix above are linearly dependent, this system has infinitely many non-trivial solutions satisfying condition a = -b. Hence, all eigenvectors corresponding to eigenvalue Î»_{1} = 1 have the form

We choose the eigenvector to be

By similar consideration we nd that the eigenvector associated with eigenvalue Î»_{2} = 3 is

To summarize, we have found the eigensystem of matrix J :

(8.28)

Now we can classify the critical point (0, 0). Since the eigenvalues are real and nonzero, critical point is hyperbolic. They are both positive and hence the critical point is unstable node. Finally, eigenvectors are real and so the system has two unstable manifolds given by e_{1} and e_{2}. Implementation in Mathematica is shown in figure 8.11.

**Fig. 8.11. Example 1.**

**Example 2 **

Linear dynamical system has the form

The matrix of this system is clearly

Eigenvalues are

and so it is a degenerate node. Since both eigenvalues are negative, it is a stable degenerate node. Implementation in Mathematica is shown in figure 8.12.

**Example 3. Volterra-Lotka equations **

Volterra-Lotka equations belong to the class of predator-prey models which describe interaction between two populations. The population of preys has a tendency to grow and the population of predators tends to die. It is due to their mutual interaction that also the population of predators can grow and the population of preys can die, in other words, predators are eating preys.

Let x = x(t) be the number of preys, say, rabbits, let y = y(t) be the number of predators, say, foxes. We can construct a plausible model of interaction between foxes and rabbits by following simple considerations. Suppose that y = 0, i.e. there are only rabbits present. As a rst approximation we can assume that the population of rabbits will grow, the number of rabbits x will increase because of "interaction" between rabbits and the higher is the number of rabbits, the higher is the rate of growth. Hence, we can postulate that isolated population of rabbits will be governed by equation

Roughly speaking, constant Î± can be interpreted as a probability of the birth of a new rabbit when there are no foxes. This equation has solution

which means that isolated population of rabbits will grow exponentially.

Similar consideration applies to isolated population of foxes. If Î³ is the probability of death of the fox, isolated population of foxes will be governed by equation

which has the solution

i.e. the population of foxes will die exponentially.

Now we add an interaction to our equations. The number of rabbits eaten by foxes is proportional to number of rabbits and to number of foxes. Conversely, the number of new-born foxes is proportional to number of foxes and to number of rabbits. If we introduce constants Î² and Î´ for both processes, equations for interacting populations of rabbits and foxes read

(8.29)

These are Volterra-Lotka equations. Obviously, they are non-linear and the nonlinearity represents the interaction between two populations. All constants are assumed to be positive.

Critical points can be found by

Hence, the critical points are

In order to linearize equations (8.29) we introduce the Jacobi matrix J

The Jacobi matrix can be found in Mathematica by

which shows

Next we evaluate the Jacobian at both critical points:

i.e. we have

Finally we nd eigenvalues and eigenvectors by

** Flow of the vector field **

In this section we introduce some useful notions related to the concept of dynamical system. We consider general autonomous dynamical system (8.1)

(8.30)

We know that the solution exists and is unique if prescribe initial conditions

(8.31)

where x_{a0} are constants with the meaning of initial value of coordinates x_{a}. The solution of dynamical system is then a set of functions x_{a} as functions of time,

(8.32)

where we have explicitly emphasized that particular solution depends on initial values

Hence, in the following, by symbol x(t; x_{0}) we mean the set of functions

such that

(8.33)

In other words, x(t, x_{0}) is a solution of dynamical system (8.30) with initial conditions (8.31).

It is useful to introduce slightly more formal notation for x(t, x_{0}). We de ned the phase space M as an abstract space with coordinates x_{a}. For n-dimensional dynamical system, the phase space is

The flow of dynamical system (8.30) is a mapping

defined by

Obviously,

so that we have

Vector field f_{a} can be plotted by

This dynamical system for initial conditions

x(0) = x_{0}, y(0) = y_{0,} can be solved explicitly by

which shows, in the notation introduced above,

Similarly, we de ne positive semi-orbit and negative semi-orbit by

(8.35)

Fig. 8.13. Illustration of the flow.

**Lyapunov stability**

Recall that we have de ned the critical point or fixed point x_{C }of dynamical system (8.30) as such point x_{C} for which

and hence System with initial conditions x_{0} = x_{C} is in equilibrium in the sense that it remains in the critical point at all times, i.e.

âˆ§(x_{C} ) = {x_{C }}

In other words, critical point x_{C} satis es relation

We have classi ed critical points according to behaviour of the orbits (phase trajectories) in the vicinity of the critical point. If the orbit remained in the vicinity of critical point, we have said that the critical point is stable. If the orbit was attracted to critical point, it was called stable node or stable focus, depending on the character of the system. If the orbit was circular, critical point was called centre. Finally, if the orbit escaped from the critical point to in nity, we called the critical point the unstable node or unstable focus. However, this analysis was performed for linearized dynamical system. Now we can formulate the stability for general non-linear system in terms of the

ow.

Let be standard norm de ned on the phase space M , i.e. for any x âˆˆ M its norm is

In general, the norm is a measure of distance of point x from the origin. In some situations, it is useful to introduce different notion of the norm, for example the so-called p norm (p is positive integer) de ned by

In the following we will use standard norm which is a standard Euclidean distance, as follows from the Pythagorean theorem. In general, the norm must satisfy three relations.

- Positive definiteness

- Linearity

for arbitrary real

In some contexts the first condition is relaxed, i.e. we admit there are vectors is called semi-norm. In this textbook we consider only positive definite norms satisfying the first property. Notice that positive definiteness implies that whenever

vectors x and y are equal, x = y.