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Dragon Notes

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  UNDER CONSTRUCTION

Nonlinear Dynamics & Chaos:
Stability




Fixed Point Stability [Linear Sys]

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


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon