Here we'll talk about both linear and nonlinear planar systems of the form
$$ \begin{aligned} \dot{x} & = f(x,y) \\ \dot{y} & = g(x,y). \end{aligned} $$
A separatrix is the boundary that separating two modes of behavior in a dynamical system.
Trajectories that start and end at the same fixed point are called homoclinic orbits.
Trajectories that connect two fixed point are called heteroclinic orbits.
A hyperbolic fixed point is a fixed point for which the real part of both eigenvalues is non-zero.
Nullclines
The nullclines of a planar system are the curves where either $\dot{x} = 0$ or $\dot{y} = 0$ and indicate where the flow is either purely horizontal or purely vertical. fixed points occur at intersections of nullclines.
Classification of Linear Systems
We can write a linear planar system as
$$ \dot{\vec{x}} = A \vec{x}, \quad \vec{x} \in \mathbb{R}^2 $$
where
$$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}. $$
The characteristic equation tells us how to find the @eigenvalues and is given by:
$$ \lambda^2 - \tau \lambda + \Delta = 0. $$
Rewriting that using the quadratic equation gives:
$$ \lambda_{1,2} = \frac{1}{2} \left (\tau \pm \sqrt{\tau^2 - 4 \Delta} \right ) $$
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.