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6.2 Linear and Almost Linear Systems

revisit a system from last time:

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

cp: \((0, 0)\)

nodal sink

a related system:

\[\begin{aligned} x' &= -2x - y + 5 \\ y' &= -x - 2y + 4 \end{aligned}\]

cp: \((2, 1)\)

nodal sink

the phase diagrams are identical but centered at different cp's.

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why?

\[\begin{aligned} x' &= -2x - y \\ y' &= -x - 2y \end{aligned} \rightarrow \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} -2 & -1 \\ -1 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]

\[\begin{aligned} x' &= -2x - y + 5 \\ y' &= -x - 2y + 4 \end{aligned}\]

cp: \((2, 1)\)

define \(u = x - 2\), \(v = y - 1\)

then \(u' = x'\), \(v' = y'\)

and \(x = u + 2\), \(y = v + 1\)

\[\begin{aligned} u' &= -2(u + 2) - (v + 1) + 5 \\ v' &= -(u + 2) - 2(v + 1) + 4 \end{aligned}\]

simplify

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Linear Systems and Stability

\[\begin{cases} u' = -2u - v \\ v' = -u - 2v \end{cases} \rightarrow \begin{bmatrix} u' \\ v' \end{bmatrix} = \begin{bmatrix} -2 & -1 \\ -1 & -2 \end{bmatrix} \begin{bmatrix} u \end{bmatrix}\]

the same system as the homogeneous case each centered the their respective origin.

\[\vec{x}' = A\vec{x}\]

\(A\) has eigenvalues \(r_1, r_2\)

Critical point is asymptotically stable

if \(r_1 < 0, r_2 < 0\) or \(r_1, r_2\) have negative real part

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stable if real part of \(r_1, r_2\) is zero

unstable if one of \(r_1\) or \(r_2\) (or both) is positive or if \(r_1, r_2\) have positive real part

How sensitive are these to small perturbations to the elements of \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)

It turns out that if the cp is asympt. stable or unstable, then

Small perturbations do NOT affect stability

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and the trajectories (node/spiral/etc) do not change.

There are two cases where the stability and trajectories can change.

Case 1: purely imaginary eigenvalues

Perturbations can introduce pos. or neg. real part.

Complex plane with Im(\lambda) and Re(\lambda) axes. Green dots on the imaginary axis represent a center. Red arrows show shifts to the left (stable spiral) or right (unstable spiral). — Figure: Complex plane with \(Im(\lambda)\) and \(Re(\lambda)\) axes. Green dots on the imaginary axis represent a center. Red arrows show shifts to the left (stable spiral) or right (unstable spiral).

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Case 2: repeated eigenvalues

Stability does NOT change but the trajectories might.

Figure: Complex plane with a green dot on the positive real axis. Red arrows indicate splitting into two real eigenvalues or a complex conjugate pair.

Either split into two distinct but of the same sign as original or introduce imaginary part while keeping the real part with the same sign.

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Nonlinear Systems and Phase Diagrams

Nonlinear systems have phase diagrams that look like those of linear systems near each critical pt.

this means

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

\(\downarrow\) near each cp

\[ \vec{x}' = A\vec{x} + \vec{g}(\vec{x}) \]

if \((x_0, y_0)\) is a cp

\[ \lim_{(x,y) \to (x_0, y_0)} \frac{|\vec{g}(\vec{x})|}{|\vec{x}|} = 0 \]

then the system is almost linear or locally linear

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Example: Almost Linear Systems

for example,

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

cp: \((0,0), (\frac{1}{4}, 1)\)

\[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \underbrace{\begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}}_{A} \begin{bmatrix} x \\ y \end{bmatrix} + \underbrace{\begin{bmatrix} xy \\ 8xy \end{bmatrix}}_{\vec{g}(\vec{x})} \]

notice \((x, y) \to (0, 0)\)

\[ \lim_{(x,y) \to (0,0)} \frac{|\vec{g}|}{|\vec{x}|} = \lim_{(x,y) \to (0,0)} \frac{\sqrt{65} xy}{\sqrt{x^2 + y^2}} \]

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Analysis of Almost Linear Systems near (0,0)

\[ \lim_{(x,y) \to (0,0)} \frac{\sqrt{65} xy}{\sqrt{x^2+y^2}} \]

\(\downarrow\)

\[ \lim_{r \to 0} \frac{\sqrt{65} r^2 \cos\theta \sin\theta}{r} = 0 \]

Let \( r = \sqrt{x^2+y^2} \)

then \( (x,y) \to (0,0) \) is the same as \( r \to 0 \)

\( x = r \cos\theta, \; y = r \sin\theta \)

So, this is an almost linear system near \( (0,0) \).

It will behave like

\[ \vec{x}' = \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix} \vec{x} \]

near \( (0,0) \)

\( \lambda = -1, -2 \)

asymp. stable node

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Analysis near Critical Point \( (\frac{1}{4}, 1) \)

Now look at near \( (\frac{1}{4}, 1) \). What is \( A \)?

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

\(\vdots\)

\(\downarrow\)

\[ \begin{aligned} u' &= \frac{1}{4}v + uv \\ v' &= 8u + 8uv \end{aligned} \]

cp: \( (\frac{1}{4}, 1) \)

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

\[ \begin{bmatrix} u' \\ v' \end{bmatrix} = \underbrace{\begin{bmatrix} 0 & \frac{1}{4} \\ 8 & 0 \end{bmatrix}}_{A} \begin{bmatrix} u \\ v \end{bmatrix} + \begin{bmatrix} uv \\ 8uv \end{bmatrix} \]

\( \lambda = \pm \sqrt{2} \)

saddle pt

Neither saddle pt or asymp. stable node is sensitive to perturbation.

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Phase Portrait Analysis

This phase portrait illustrates the vector field and trajectories for a system of differential equations, highlighting two critical points with distinct stability characteristics.

Saddle Point: Located in the upper-right quadrant, where trajectories approach along one axis and diverge along another.
Asymptotically Stable Node: Located in the lower-left quadrant, where all nearby trajectories converge toward the equilibrium point.