In this section, I will talk about higher-dimensional
systems.

In higher-dimensional systems, movement of
trajectories can exhibit a wider range of

dynamical behaviour.

Fixed points still exist, but can be more
interesting depending on how trajectories

approach or repel from the equilibrium point
for example system could spiral in to a stable

point.

Other types of stability exist such as saddle-nodes,
and importantly cyclic/periodic behaviour:

limit cycles.

So More interesting, but more difficult to
analyse…

Similar to one-dimensional system I will cover
how to find fixed points and classifying fixed

points for these systems

Suppose we have the following system:

First we need to learn how we can find fixed
points and then examine stability of fixed

points and finally we will examine the phase
plane and trajectories.

To find fixed points as before 
we need to solve dx/dt = 0 and dy/dt = 0 to

get fixed points (x0, y0).

So let’s look at a Predator-prey system:

so again, (0,0) is fixed point.

Other fixed point at (80, 12).

To examine behaviour at/near fixed points
we can examine their change against time or

in phase plane.

View in phase-space (or phase-plane) where
x plotted against y rather than against time

gives more information about system.

As you see, the system can show cyclic behaviour
that is fixed but system is not at a fixed

point: complications of higher dimensions.

Before going forward we need to define Jacobian
matrix.

Jacobian matrix is the matrix of all first-order
partial derivatives of a vector- or scalar-valued

function with respect to another vector.

This matrix is frequently being marked as
J, Df or A. Jacobian matrix is used to classify

fixed points of higher order 
linear dynamical system:

Suppose x0 =(x0, y0)T is a fixed point.

Define the Jacobian:

Find eigenvalues and eigenvectors of J evaluated
at the fixed point:

If eigenvalues have negative real parts, x0
asymptotically stable

If at least one has positive real part, x0
unstable

If eigenvalues are pure imaginary, stable
or unstable

Various behaviours depending on the eigenvalues
(ei) and eigenvectors can be seen.

Here you can see some examples for 2d linear
dynamical system.

In general, points attracted along negative
eigenvalues and repelled by positive.

Axes of attraction are eigenvectors.

If we go back to our example:

Then we have fx = 0.6 – 0.05y, fy = 0.05x,
gx = 0.005y, gy = 0.005x – 0.4

For J(0,0) eigenvalues 0.6 and –0.4 and
eigenvectors (1,0) and (0,1).

Unstable (a saddle point) with main axes coordinate
axes.

For J(80,12) eigenvalues are pure imaginary.

For a 
dynamical system to be stable:

The real parts of all eigenvalues must be
negative.

All eigenvalues lie in the left half complex
plane.

A dynamical system is Underdamped if there
is spiral fixed point (some complex eigenvalue)

and we say it is Overdamped if exhibit nodal
behaviour (all eigenvalues real) and Critically

damped at the boundary between.

We can classify dynamical systems using trace
and determinant of the Jacobian matrix.

Is there any classification for linear 3D
systems using Eigen values?

Lets look at more examples for 2d systems.

Consider the following system:

The Jacobean matrix will be:

And Eigen values are λ1= -1, λ2= -4 and
Eigen vectors will be

There is a stable fixed point, the attracting
node (sink).

Now if we change the system to:

And 
Eigen values are λ1= 1 , λ2= 4 which means

it is an unstable fixed point, the repelling
node

In contrast, of attractor, a repeller is a
point of state space away from which system

will tend when in surrounding region.

The third system has another type of unstable
fixed point called saddle point.

For non-linear equations, behaviour near the
fixed points will be ‘almost like’ the

behaviour of a linear system depending how
‘almost linear’ it is.

Behaviour gets less linear-like the further
away trajectories get from the

fixed point.