Download original notes
PAGE 1

6.2 Linear and Almost Linear Sys (continued)

last time:

\[\begin{aligned} x' &= F(x, y) \\ y' &= G(x, y) \end{aligned}\]

is almost/locally linear if near critical pt \((x_0, y_0)\) it behaves like

\[\vec{x}' = A\vec{x} + \vec{g}(\vec{x}) \quad \text{such that} \quad \lim_{(x,y) \to (x_0, y_0)} \frac{|\vec{g}|}{|\vec{x}|} = 0\]

Generally, there is a different \(A\) near each critical pt

example from last time:

\[\begin{aligned} x' &= -x + xy \\ y' &= -2y + 8xy \end{aligned}\]

has a critical pt at \((\frac{1}{4}, 1)\)

PAGE 2

define \(u = x - \frac{1}{4}\)

\(v = y - 1\)

\(x = u + \frac{1}{4}\), \(y = v + 1\)

\(u' = x'\)

\(v' = y'\)

rewrite system

\[\begin{aligned} u' &= -(u + \frac{1}{4}) + (u + \frac{1}{4})(v + 1) = \frac{1}{4}v + uv \\ v' &= \dots = 8u + 8uv \end{aligned}\]
\[\begin{bmatrix} u' \\ v' \end{bmatrix} = \underbrace{\begin{bmatrix} 0 & \frac{1}{4} \\ 8 & 0 \end{bmatrix}}_{A \text{ near } (\frac{1}{4}, 1)} \begin{bmatrix} u \\ v \end{bmatrix} + \begin{bmatrix} uv \\ 8uv \end{bmatrix}\]

there is a different (more efficient) way to find the \(A\) matrix

\(\rightarrow\) linearize the nonlinear system

\[\begin{aligned} x' &= F(x, y) \\ y' &= G(x, y) \end{aligned}\]

do Taylor series expansion for \(F\) and \(G\) near critical pt \((x_0, y_0)\)

PAGE 3

Linearization of Nonlinear Systems

The Taylor series expansion for a system of two variables near a critical point \((x_0, y_0)\) is given by:

\[F(x,y) = F(x_0, y_0) + \frac{\partial F}{\partial x}(x_0, y_0)(x-x_0) + \frac{\partial F}{\partial y}(x_0, y_0)(y-y_0) + \dots\]
\[G(x,y) = G(x_0, y_0) + \frac{\partial G}{\partial x}(x_0, y_0)(x-x_0) + \frac{\partial G}{\partial y}(x_0, y_0)(y-y_0) + \dots\]

Note: The terms denoted by \(\dots\) are higher order and not important for local linearization. At a critical point, \(F(x_0, y_0) = 0\) and \(G(x_0, y_0) = 0\).

Near the critical point \((x_0, y_0)\), the system:

\[\begin{aligned} x' &= F(x,y) \\ y' &= G(x,y) \end{aligned}\]

becomes, by letting \(u = x - x_0\) and \(v = y - y_0\):

\[\begin{aligned} u' &= \frac{\partial F}{\partial x}(x_0, y_0) u + \frac{\partial F}{\partial y}(x_0, y_0) v \\ v' &= \frac{\partial G}{\partial x}(x_0, y_0) u + \frac{\partial G}{\partial y}(x_0, y_0) v \end{aligned}\]
\[\begin{bmatrix} u' \\ v' \end{bmatrix} = \begin{bmatrix} F_x(x_0, y_0) & F_y(x_0, y_0) \\ G_x(x_0, y_0) & G_y(x_0, y_0) \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix}\]

Jacobian matrix \(\rightarrow\) our A matrix near \((x_0, y_0)\)

PAGE 4

Example: Linearization Analysis

\(x' = x^2 + y^2 - 6 = F\)

\(y' = x^2 - y = G\)

Critical points (cp): \((\sqrt{2}, 2)\), \((-\sqrt{2}, 2)\)

Jacobian matrix:

\[J(x,y) = \begin{bmatrix} 2x & 2y \\ 2x & -1 \end{bmatrix}\]

Linearized near \((\sqrt{2}, 2)\)

\[J = \begin{bmatrix} 2\sqrt{2} & 4 \\ 2\sqrt{2} & -1 \end{bmatrix}\]

System acts like \(\vec{x}' = \begin{bmatrix} 2\sqrt{2} & 4 \\ 2\sqrt{2} & -1 \end{bmatrix} \vec{x}\)

  • \(\lambda \approx 4.8, -3\)
  • Saddle point
  • Unstable
  • (not sensitive to perturbation)
Phase portrait sketch on x-y axes showing a spiral sink in the upper left and a saddle in the upper right.

Near \((-\sqrt{2}, 2)\)

\[J = \begin{bmatrix} -2\sqrt{2} & 4 \\ -2\sqrt{2} & -1 \end{bmatrix}\]
  • \(\lambda \approx -2 \pm 3.2i\)
  • Spiral sink
  • Asymptotically stable
  • (not sensitive to perturbation)
PAGE 5

Phase Portrait Analysis

The following figure displays a vector field and phase portrait for a system of differential equations. The plot shows trajectories (red curves) and direction vectors (black arrows) on a coordinate plane ranging from -4 to 4 on both axes.

Vector field and phase portrait on a Cartesian plane with red trajectories and black direction arrows.

The phase portrait reveals critical points and the qualitative behavior of solutions, including spiral patterns and saddle-like regions.

PAGE 6

example

Consider the nonlinear system:

\[ x' = 2xy = F \]\[ y' = 1 - x^2 + y^2 = G \]

Critical points (cp): \( (1, 0), (-1, 0) \)

The Jacobian matrix is given by:

\[ J = \begin{bmatrix} F_x & F_y \\ G_x & G_y \end{bmatrix} = \begin{bmatrix} 2y & 2x \\ -2x & 2y \end{bmatrix} \]

Evaluating at the first critical point \( (1, 0) \):

\[ J(1, 0) = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} \quad \lambda = \pm 2i \]

Linearized sys. says center (stable). Sensitive to perturbation; true behavior could still be a center but we can't tell from linearization.

Complex plane showing eigenvalues as red dots on the imaginary axis at plus and minus 2i.

Evaluating at the second critical point \( (-1, 0) \):

\[ J(-1, 0) = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} \quad \lambda = \pm 2i \]

Same story as the other cp.

We can't be sure what the true picture is like w/o either plotting the nonlinear phase diagram or solving the system (for this sys, the sub \( u = y^2/x \) can solve the system).

PAGE 7

In this case, the linearized sys tell us the "truth"

Vector field plot with red streamlines showing two symmetric spiral or center-like patterns around points on the x-axis.
PAGE 8

Example

\[ \begin{aligned} x' &= -3y + ay(x^2 + y^2) \\ y' &= 3x + ay(x^2 + y^2) \end{aligned} \]

\( a \) is some constant

cp: \( (0,0) \)

\[ J = \begin{bmatrix} 2xya & -3 + ax^2 + 3y^2 \\ 3 + 2axy & 3ay^2 + ax^2 \end{bmatrix} \]
\[ J(0,0) = \begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix} \]

no \( a \) here.

clearly "\( a \)" is doing something

but linearization completely lost it

PAGE 9

Phase Portraits of Linearized and Nonlinear Systems

Linearized System

The phase portrait below represents a linearized system exhibiting a stable center. The trajectories form closed concentric loops around the equilibrium point at the origin.

Phase portrait of a linearized system showing concentric circular trajectories around a stable center at the origin.

Nonlinear System: Unstable Spiral

For the nonlinear case where \( a = 1 \), the system exhibits an unstable spiral point. Trajectories spiral outward away from the equilibrium point.

Phase portrait of a nonlinear system with a=1 showing trajectories spiraling outward from an unstable spiral point.

Nonlinear System: Asymptotically Stable Spiral

For the nonlinear case where \( a = -1 \), the system exhibits an asymptotically stable spiral point. Trajectories spiral inward toward the equilibrium point at the origin.

Phase portrait of a nonlinear system with a=-1 showing trajectories spiraling inward toward a stable spiral point.