5.3 Systems, Stability, Fourier, PDEs

Systems of DEs

First solution:

$x^{\left(1\right)}=e^{\lambda t}{\xi }_{\lambda }$

Second solution:

$x^{\left(2\right)}=t{ℇ}^{\lambda t}{\xi }_{}+{ℇ}^{\lambda t}\eta$

where ${\xi }_{}=(A-\lambda 𝕀)\eta$. Third solution:

$x^{\left(3\right)}=\frac{t^2}{2}{e}^{\lambda t}{\xi }_{\lambda }+t{ℇ}^{\lambda t}\eta +e^{\lambda t}\zeta$

where $\eta =(A-\lambda 𝕀)\zeta$.

Critical Points

Jacobian Matrix:

$J=\left[\begin{array}{cc}{\partial }_x({{{{F_{}}_{}}_{}^{}}^{}}^{}(x_0,y_0))& {\partial }_x({{{{F_{}}_{}}_{}^{}}^{}}^{}(x_0,y_0))\\ {\partial }_x(G(x_0,y_0))& {\partial }_y(G(x_0,y_0))\end{array}\right]$

Sometimes a nonlinear ODE has an exact phase portrait given by:

$\frac{d\left(y\right)}{d\left(x\right)}=\frac{G(x,y)}{{{{{F_{}}_{}}_{}^{}}^{}}^{}(x,y)}\to$ integrate $H(x,y)=c$

Almost Linear Systems: The Proper/Improper bits become Node/Spiral Point, and the Centre becomes Centre or Spiral (Indeterminate).

Lyapunov Functions

Let $V(x,y)$ be defined on a domain $D$ containing $(0,0)$.

• $V(x,y)$is positive (negative) definite if$V(0,0)=0$and$V(x,y)>0(\forall (xy)\in D)(V(x,y)<0\forall (x,y)\in D)$

• $V(x,y)$is positive (negative) semi-definite if$V(0,0)=0$and$V(x,y)\ge 0(\forall (xy)\in D)(V(x,y)\le 0\forall (x,y)\in D)$

Theorem: Given an autonomous system with critical point $(0,0)$, if $\exists V(x,y)$ continuous with continuous first partial derivatives, is positive definite, then $(0,0)$ is:

• Asymptotically Stable: If $\frac{d\left(V\right)}{d\left(t\right)}$ is negative definite on some domain $D$ containing $(0,0)$.

• Stable (at non-linear level): If $\frac{d\left(V\right)}{d\left(t\right)}$ is negative semi-definite.

Theorem: Given an autonomous system with critical point $(0,0)$, assume $\exists V(x,y)$ continuous with continuous first partial derivatives, such that $V(0,0)=0$ and that in every neighbourhood of $(0,0)$ there exists at least one point $(x_1,y_1)$ where $V(x_1,y_1)$ is positive negative. If there exists some domain $D$ containing $(0,0)$ where $\frac{d\left(V\right)}{d\left(t\right)}$ is positive definite negative definite on $D$, then $(0,0)$ is an unstable critical point.

$\frac{d\left(V\right)}{d\left(t\right)}=x^{\prime }{\partial }_x\left(V\right)+y^{\prime }{\partial }_y\left(V\right)$

Fourier Analysis

Fourier Series of $ƒ\left(x\right)$:

$ƒ\left(x\right)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }(a_n\cos \left(\frac{n\pi x}{L}\right)-b_n\sin \left(\frac{n\pi x}{L}\right))$

where $T=2L$ or $ƒ(x+2L)=ƒ\left(x\right)$.

$a_0=\frac{1}{L}{\int }_{-L}^L(ƒ\left(x\right)d\left(x\right))$

$a_n=\frac{1}{L}{\int }_{-L}^L(ƒ\left(x\right)\cos \left(\frac{n\pi x}{L}\right)d\left(x\right))$

$b_n=\frac{1}{L}{\int }_{-L}^L(ƒ\left(x\right)\sin \left(\frac{n\pi x}{L}\right)d\left(x\right))$

Function is even if $ƒ(-x)=ƒ\left(x\right)$ only cosine terms.

Function is odd if $ƒ(-x)=-ƒ\left(x\right)$ only sine terms.

Partial Differential Equations

We have:

$a^2{\partial }_x^2\left(u\right)+\mathrm{\beta }u={\partial }_t\left(u\right)$

With non-homogeneous boundary conditions $u(0,t)=T_1$, $u(L,t)=T_2$, look for $v\left(x\right)$ to solve:

$a^2{v}^{″}\left(x\right)+\beta v\left(x\right)=0$, $v\left(0\right)=T_1,$ $v\left(L\right)=T_2$

Then determine solution using:

$u(x,t)=v\left(x\right)+w(x,t)$

Identities

• $\sinh \left(x\right)=\frac{e^x-e^{-x}}{2}$

• $\cosh \left(x\right)=\frac{e^x+e^{-x}}{2}$

• $\cos \left(x\right)=\frac{e^{ix}+e^{-ix}}{2}$

• $\sin \left(x\right)=\frac{e^{ix}-e^{-ix}}{2i}$

• $\cos \left(\mu \right)=0\to \mu =(2n-1)\frac{\pi }{2}$

• 1-D Heat Eq: ${\partial }_t\left(u\right)=a^2{\partial }_x^2\left(u\right)$

• $\int ({\sin }^2(ax)d\left(x\right))=\frac{x}{2}-\frac{\sin (2ax)}{4a}$

• $\int ({\cos }^2(ax)d\left(x\right))=\frac{x}{2}+\frac{\sin (2ax)}{4a}$

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