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.