The left fixed point is a stable point, while the right FP is unstable.
Linear Stability Analysis
In order to know the behavior around fixed point linearization is a great method to analyze.
Let be a fixed point, and let be a small perturbation away from . To see whether the perturbation grows or decays, we derive a differential equation for . Differentiation yields
since is constant. Thus . Now using Taylor’s expansion we obtain
where denotes quadratically small terms in . Finally, note that since is a fixed point. Hence
Now if , the terms are negligible and we may write the approximation
This is a linear equation in , and is called the linearization about x*. It shows that the perturbation:
Potentials
There’s another way to visualize the dynamics of the first-order system: potential is defined by
Consider the relation between potential and time , by using chain rule we have
since , we obtain
The equilibrium point happens at , the remains constant. Since implies .
Example
Graph the potential for the system , and identify all equilibrium points.
Solving yields
Once again we set . Figure shows the graph of . The local minima at correspond to stable equilibria, and the local maximum at corresponds to an unstable equilibrium.
Some people say that an exchange of stabilities has taken place between the two fixed points.
Pitchfork Bifurcation
This bifurcation is common in physical problems that have a symmetry.
Supercritical Pitchfork Bifurcation
Normal Form:
Note that this equation is invariant under the change of variables .
One stable splitting two stable and one unstable.
Subcritical Pitchfork Bifurcation
Normal Form:
One stable and two unstable to one stable.
Dimensional Analysis
Non-dimensionalization
Differential equations that show up in modeling “real world situations” usually have many constants in them. Often one can reduce the number of constants in a problem by choosing the right units for the various quantities in the problem. In the other words, we can convert the variables to ratio that has no unit to analyze multiple different coefficient problems.
We define a dimensionless time by
where is dimensionless time, is dimensional time, is characteristic time scale. We need to choose very carefully to do the non-dimensionalization.
Timescale
is the rate of growth. is timescales of growth.
Example 1
Suppose that a quantity changes in time according to the ODE
The coefficients , , and must all have different units, otherwise we could not add the terms , , and .
To simplify the equation we choose a constant value for , let's say , and we let this value be our unit. The ratio
has no units. In the same way we can pick a unit of time and introduce the quantity
which also has no units.
The quantities and are nondimensionalized versions of our original variables and . The point of nondimensionalization is that we can now derive a differential equation for and , and then afterwards figure out which choice of the units and simplifies things most.
In this example we substitute
which leads to
by the chain rule, and
by direct substitution. The differential equation (1) for and is therefore equivalent with
and thus
At this point we choose and . We can try to make the constant term and the coefficient of both equal to 1. If and then this is possible provided we choose
The coefficient of then becomes
and we get the following differential equation for as a function of :
This is a nondimensionalized version of equation (1). Note that instead of three undetermined parameters it only has one parameter, namely .
Example 2
The Logistic equation:
where is population, is time, growth rate, carrying capacity.
The non-dimensionalize term:
Thus, we have
when and .
Two-Dimensional System
General form
system:
Example
Let , , then we have
Linear form
where are . Let , , so we have .
The fixed points: when or .
Uncoupled System
Consider the system
Notice = the trajectory is different and depends on :
Created by potrace 1.10, written by Peter Selinger 2001-2011
Classified these cases: Stable node, Stable star, stable node, line of fixed points, saddle point
Stability Technology
When is a fixed point:
is attracting if all trajectories near approach it as .
is globally attracting if all trajectories approach it as .
is Lyapunov stable if all trajectories that start sufficiently close to remain close for all time
There are three cases to help understand:
Attracting but not Lyapunov Stable
Created by potrace 1.10, written by Peter Selinger 2001-2011
Lyapunov Stable but not attracting
Created by potrace 1.10, written by Peter Selinger 2001-2011
Attracting and Lyapunov is Stable Start case.
General Linear System
Consider following form
Note: The straight-line trajectories is if then which is stays on -axis.
Now let's generalize this idea. Let's guess , hence is the growth rate and is the direction of growth (the vector).
So, implies and
So, is an eigenvector and is an eigenvalue. is an eigensolution.
2D Eigenvalues
Definition
Then the this singular matrix has determinant zero:
Thus, , , and .
The eigenvectors are when you plug the eigenvalues back to the original form.
Note: ,
System form
When the general form of the system is
Example
For linear system:
we have and . Then the , , so gives and .
So, by plugging backs to determent equation, we have