 # Dragon Notes UNDER CONSTRUCTION
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# Nonlinear Dynamics & Chaos:Stability

Fixed Point Stability [Linear Sys]

Fixed point stability in a linear system is mediated by its eigenvalues - given by
$$\ds \vplLup \lambda_{1,2}=\sfrac{1}{2}(\tau\pm\sqrt{\tau^2-4\Delta}),\quad \Delta = \lambda_1\lambda_2,\quad \tau=\lambda_1+\lambda_2,$$
$$\ds \bn{\dot{x}}=A\bn{x}\Rightarrow \Delta = \t{det}(A),\ \tau = \t{tr}(A)$$ $$\Delta < 0$$: real and opposite in signs $$\rightarrow$$ saddle point

$$\vplLup\Delta > 0$$: real with same signs (nodes), or complex conjugates (spirals)
 Nodes: $$\hspace{3px}\tau^2 - 4\Delta > 0$$ Spirals: $$\vplup\tau^2 - 4\Delta < 0$$ Stars & degen. nodes$$\vphantom{\frac{A}{A}}$$: $$\hspace{58px}\tau^2 - 4\Delta = 0$$ $$\ds \left.\vphantom{\begin{matrix}0\\0\\0\\0\end{matrix}}\right\}$$ stable: $$\hspace{17px} \tau < 0$$ unstable: $$\tau > 0$$
$$\hspace{59px}$$Centers: $$\tau = 0$$, neutrally stable
$$\vplLup\Delta = 0$$: one or both are zero
$$\hspace{59px}\ds\vphantom{A^A}$$- Line of fixed points, or plane ($$A=0$$)
$$\hspace{59px}\vphantom{\int}$$- Origin $$\neq$$ isolated fixed point ($$=$$ no adjacent fixed points)

Stable/Unstable Manifold

Stable: IC's $$\bn{x}_0$$ such that $$\bn{x}(t)\rightarrow\bn{x}^*$$ as $$t\rightarrow\infty$$
Unstable: IC's $$\bn{x}_0$$ such that $$\bn{x}(t)\rightarrow\bn{x}^*$$ as $$t\rightarrow - \infty$$
\ds \begin{align} \vplup \dot{x} &= ax \\ \dot{y} &= -y;\ \ 0 < a < 1 \end{align} \begin{align}y\t{-axis}&=\t{stable manifold}\\x\t{-axis}&=\t{unstable manifold}\end{align}
Attracting & Globally Attracting $$\bn{x}^*$$

Attracting: trajectories starting near $$\bn{x}^*$$ approach it as $$t\rightarrow\infty$$
Globally Att: all phase plane trajectories approach $$\bn{x}^*$$ as $$t\rightarrow\infty$$
\ds \begin{align} \vplup \dot{x} &= ax \\ \dot{y} &= -y;\ \ -1 < a < 0 \end{align} $$\bn{x}^*=0$$ is globally attracting

Liapunov/Neutrally Stable $$\bn{x}^*$$

Liapunov: all trajectories starting near $$\bn{x}^*$$ remain near it for all time
Neutrally: Liapunov-stable but not attracting

\ds \begin{align} \vplup \dot{x} &= ax \\ \dot{y} &= -y;\ \ a = 0 \end{align}
$$\ds \dot{\theta}=1-\cos{\theta}\hspace{20px}$$  LHS: All $$\bn{x}^*$$ on $$x$$-axis are neutrally stable
RHS: $$\theta^*=0$$ is globally-attracting, but not Liapunov stable
Stable/Unstable $$\bn{x}^*$$

Stable: Attracting + Liapunov-stable
Unstable: Not Attracting nor Liapunov-stable
\ds \begin{align} \vplup \dot{x} &= ax \\ \dot{y} &= -y;\ \ a = -1,\ >0 \end{align}  LHS: $$\hspace{3px}\bn{x}^*=0$$ is stable
RHS: $$\bn{x}^*=0$$ is unstable

Fast/Slow Eigendirections

When $$\lambda_2<\lambda_1<0$$ in a linear sys, both eigensolutions decay exponentially
Fast:$$\vplup\hspace{4px}$$ eigenvector w/ the larger $$|\lambda |$$;$$\hspace{11px}$$ trajectories flow in parallel as $$t\rightarrow -\infty$$
Slow: eigenvector w/ the smaller $$|\lambda |$$; trajectories flow in parallel as $$t\rightarrow \infty$$ Stable/Unstable Spiral

In a linear system, complex $$\lambda=\alpha+j\omega$$ yields spirals $$(\alpha\neq 0)$$
$$\hspace{19px}$$Stable: $$\vplup\Re e(\lambda)<0$$
Unstable: $$\Re e(\lambda)>0$$
Stable
Unstable

Hyperbolicity, Topological equivalence, Structural stability

A fixed point is hyperbolic if $$\Re (\lambda)\neq 0$$ for all linearization eigenvalues (all lie off the imaginary axis).
-  Stability is unaffected by small nonlinear terms$$\vplup$$
- Local phase portrait is topologically equivalent to the phase portrait of the linearization (lineariz. accurately predicts fp stability)

$$\vplup$$Topological equivalence implies a homeomorphism (a continuous deformation with a continuous inverse
) that maps one local phase portrait onto the other, where trajectories are mapped onto trajectories such that their sense of time (orientation) is preserved.
$$\vplup$$A phase portrait is structurally stable if its topology cannot be changed by an arbitrarily small perturbation to the vector field:
The phase portrait of a saddle point is structurally stable, but that of a center isn't; an arbitrarily small amount of damping converts the center to a spiral (example of hyperbolicity)