Planar Systems - Mathnotes

Here we'll talk about both linear and nonlinear planar systems of the form

$\begin{align*} \dot{x} & = f(x,y) \\ \dot{y} & = g(x,y). \end{align*}$

Definition: Separatrix \@{separatrix} A separatrix is the boundary that separating two modes of behavior in a dynamical system.

Definition: Homoclinic Orbit \@{homoclinic-orbit} Trajectories that start and end at the same fixed point are called homoclinic orbits. Referenced by (1 direct)Direct references:Limit Cycles

Definition: Heteroclinc Trajectory \@{heteroclinc-trajectory} Trajectories that connect two fixed point are called heteroclinic orbits.

Definition: Hyperbolic Fixed Point \@{hyperbolic-fixed-point} A hyperbolic fixed point is a fixed point for which the real part of both eigenvalues is non-zero.

Nullclines

\dot{x} = 0

Classification of Linear Systems

We can write a linear planar system as

$\dot{\vec{x}} = A \vec{x}, \quad \vec{x} \in \reals^2$

where

$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.$

\lambda^2 - \tau \lambda + \Delta = 0.

Rewriting that using the quadratic equation gives:

$\lambda_{1,2} = \frac{1}{2} \parens{\tau \pm \sqrt{\tau^2 - 4 \Delta}}$

where

$\tau = \tr{(A)} = a + d = \lambda_1 + \lambda_2, \quad \Delta = \det{(A)} = ad - bc = \lambda_1 \lambda_2.$

Type	Condition
Stable Node	$\lambda_1,\lambda_2 < 0,$ real
Unstable Node	$\lambda_1, \lambda_2 > 0,$ real
Saddle	$\Delta < 0$ , equiv $\lambda_1 < 0 < \lambda_2$
Stable Spiral	$\Re{\lambda} < 0,$ complex
Unstable Spiral	$\Re{\lambda} > 0,$ complex
Center	$\Re{\lambda} = 0,$ complex (purely imaginary)

Linear systems only have one fixed point, the origin!

Jacobian Linearization

Given general (can be nonlinear planar system:

$\dot{x} = f(x,y), \quad \dot{y} = g(x,y),$

if we have a fixed point $(x^*, y^*),$

then

$f(x^*, y^*) = 0, \quad g(x^*, y^*) = 0.$

We can do linear stability analysis on this system to understand the local behavior near fixed points. We use the Jacobian matrix of the system, then plug in $(x^*, y^*).$ For the nonlinear case, we have the same geometric interpretations as the linear case. Topologically, if the real part of any eigenvalue of a fixed point is positive, then the fixed point is unstable. If the real part of all eigenvalues of a fixed point is negative, then the fixed point is asymptotically stable. We call these cases (both eigenvalues have nonzero real parts) hyperbolic @fixed points

However, if the real part of any eigenvalue is zero, this linearized approached does not tell us about the stability of the fixed point.