Week 1 - Introduction to ODE

Differential Equation Def:

A differential equation is any relation involving ht derivatives of an unknown function, the function itself, and known quantities

Application

Applications of ODE include

Chemistry:
- models for chemical reactions
Physics:
- naturally comes with Newton’s second law, planetary motion, and much more

Applications of PDE include:

Physics:
- electricity and magnetism, heat dissipation

For example:

Question:

Solution:
\[p(t) = Ae^{rt+b}\]

Another example would be:

For example:

Question:
Solution:

Goal of Studying ODE

The main goal is of course finding all solutions of a give ODE. Ideally, we would want to

closed form expression
- solution containing standard functions such as $cosine$.
possible cases include:
- linear equations
- separable equations

However, what if

analyze its existence and uniqueness of solutions
study its quantitative behavior, such as steady state or run-away

Linear ordinary differential equations and the method

First consider the example of:

\[x^\prime (t) = ax(t)\]

Now consider the more general

where:

primitive means antiderivative

ODE Equations

ID	Title	Condition/Definitions	Key Steps	Theorem/Solution Equation
1	Variable coefficient and inhomogeneous:	$x'(t)=a(t)x(t)+f(t)$	RHS does not have $x(t)$:$\frac{d}{dt}(e^{-\phi(t)}x(t))=e^{-\phi(t)}(x'(t)-\phi'(t)x(t))=e^{-\phi(t)}f(t)$	$x(t)=e^{\phi(t)}(\int e^{-\phi(t)}f(t) dt + c)$
			$\phi=\int a(t)dt$
2	Separable	$x’(t)=f(x_{(t)})g(t)$	RHS does not have $x(t)$: $\frac{d}{dt}\phi (x) = \phi'(x)x'(t)=\frac{x'(t)}{f(x)}=g(t)$	$x(t)=\phi^{-1}(\int g(t)dt +c)$
			$\phi'(x) = \frac{1}{f(x)}$
3	Linear System	$\vec{x}'(t) = A \vec{x}(t)$	$e^{At} = \sum\limits_{k=0}^{\infty}\frac{t^k}{k!}A^k$	$\vec{x}=\vec{x}_0 e^{At}$

4	Euclidean and Operator Norm of Matrix	Euclidean Norm: $$		A		= \sqrt{\sum\limits_{i,j=1}^{n}a_{ij}}$$		$$		A		_{op} \le		A		\le \sqrt{n}	A	_{op}$$
		Operator Norm: $$		A		_{op} = max{		A\hat{x}		,		\hat{x}		=1}$$
5	Existence of $e^A$	Using a Cauchy Sequence, so that $$		S_m - S_l		\le \epsilon$for large$l$and$m \ge l$$.	$\left\|\sum\limits_{k=l+1}^{\infty}\frac{1}{k!}A^k\right\| \le \sum\limits_{k=l+1}^{\infty}\frac{1}{k!}\left\| A^k\right\|_{op} \le \sum\limits_{k=l+1}^{\infty}\frac{1}{k!}\left\| A\right\|_{op}^k$	Limit exists for $e^A$
6	Property of $e^{A+B}$	If $A$ commutes with $B$		$e^{A+B}=e^A e^B$
7	Alternative formula for $e^A$			$e^A=\lim_{m\to \infty} (I+\frac{1}{m}A)^m$
8	Derivative of $e^{tA}$		$e^{hA}-I=hA+\sum\limits_{k=0}^{\infty}\frac{h^{k+2}}{(k+2)!}A^{k+2}$	$\frac{d}{dt}e^{tA}=Ae^{tA}$
9	Inhomogeneous Linear System	$\vec{x}'(t)=A\vec{x}(t)+\vec{f}(t)$		$\vec{x}(t)=e^{tA}\left(\int\limits_0^t e^{-sA}f(s) ds + \vec{x}_0\right)$
10	Diagonal Matrix Exponentials	$e^{tD}$, $D$ is diagonal		$e^{tD}= \begin{bmatrix}e^{t\lambda_1} & 0 & 0 &... \\ 0 & e^{t\lambda_2} & 0 & ...\\... & ... & ... & ...\\... & ... &... &e^{t\lambda_1}\end{bmatrix}$
11	Diagonalizable Matrix Exponential	$e^{tA}$, $A=SDS^{-1}$		$e^{tA}=Se^{tB}S^{-1}$
12	Nilpotent Matrix	$N^m=0$, $N$ is nilpotent	$A^m\vec{x}=\lambda^m x = 0$	All eigenvalues of $N$ is 0
13	LN Decomposition	$A=L+N$ for any $A$, and $L,N$ commute	$L\vec{x}=\lambda_i \vec{x}$, for $\vec{x}\in ker(A-\lambda_iI)^{\nu_i}$
			$N^{max(\nu_i)}\vec{x}=(A-\lambda_i I)^{max(\nu_i)}\vec{x}=0$, for $\vec{x}\in ker(A-\lambda_iI)^{\nu_i}$
14	Asymptotic Behavior of Linear System	$\vec{x}'=A\vec{x}$, all $Re(\lambda)<0$	$e^{tA}\vec{x}=e^{t(\lambda_i I)}e^{t(A-\lambda_i I)}\vec{x}=e^{t(\lambda_i I)}\sum\limits_{k=0}^{\nu_i-1}\frac{t^k}{k!}(A-\lambda_i I)^k \vec{x}$	$\lim\limits_{t \to\infty} \vec{x}_0e^{tA} \to 0$
		$\lambda$ are the generalized eigenvalues, $\vec{x}$ in generalized eigenspace	and $e^{t(\lambda_i I)}\to 0$ much faster as all $Re(\lambda)<0$
15	Asymptotic Behavior of Inhomogeneous System	$\vec{x}'=A\vec{x}+f(t)$, all $Re(\lambda)<0$, and $f(t)$ has period $\tau$	Start with $\bar{x}(t_\tau = \tau)=\bar{x}(t_\tau = 0)$, get $(e^{\tau A}-I)\bar{x}_0 = \int\limits_0^\tau e^{-sA}f(s)ds$	There is a unique solution $\bar{x}(t_{\tau})$that is also period $\tau$
		$\lambda$ are the generalized eigenvalues, $\vec{x}$ in generalized eigenspace	$\vec{x}’(t)-\bar{x}’(t_{\tau})=A(\vec{x}(t)-\bar{x}(t_{\tau}))$. Therefore: $\vec{x}(t)-\bar{x}(t_{\tau}) = e^{At}(\vec{x}(t)-\bar{x}(t_{\tau}))$, and $e^{At}\to 0$ if all $Re(\lambda)<0$	There can be other solutions $\vec{x}(t)$, but they will tend to $\bar{x}(t_{\tau})$ as $t \to \infty$
16	Asymptotic Behavior of Inhomogeneous System	This applies for both homogenous and inhomogeneous, as you can write them all in the form of $\vec{x}=e^{At}\vec{x}_0$	same as (14), but take case 1 being $\vec{x}\in ker(p_{-}A)$, and the second case the opposite	$e^{tA}\vec{x}0 \to 0$ if $\vec{x}_0 \in ker(p{-}A)$, and $t\to \infty$ and $e^{tA}\vec{x}0 \to 0$ if $\vec{x}_0 \in ker(p{+}A)$ and $t \to -\infty$
		Assumes no purely imaginary eigenvalues	$ker(p_{-}A)$ is the generalized eigenspace for negative (real part) generalized eigenvalues, and vice versa
17	Asymptotic Behavior of Inhomogeneous System	Accounting $Re(\lambda)=0$ case	Split the total space into three spaces, one for each $Re(\lambda<0)$, $Re(\lambda=0)$, $Re(\lambda>0)$	$e^{tA}\vec{x}\to 0$ if $\vec{x}\in ker(p_{-}A)=ker(A-\lambda_i I)^{\nu_i}$, for all $Re(\nu_i)<0$
			Do the same analysis as (16) and (14)	$e^{tA}\vec{x}$ is bounded if $\vec{x}\in ker(p_{-}A)\bigoplus ker(A-\lambda_jI)$, for all $Re(\nu_j)=0$
18	Cauchy-Lipschitz Theorem	$\vec{x}’(t)=\vec{F}(\vec{x})=\begin{bmatrix}f_1(\vec{x})\f_2(\vec{x})\…\f_n(\vec{x})\end{bmatrix}$	If we take a small time interval $\delta$ away from the initial condition at $t_0$, there $\exists$ a solution.	That solution will be the convergence of Picard Iterates: $\vec{x_k}(t)=\vec{x}0+\int\limits_0^t \vec{F}(\vec{x}{k-1})dt$
			Then we can bound: $\vert \vert \vec{F}(\vec{x})\vert \vert \le M$ and $\vert \vert D\vec{F}(\vec{x})\vert \vert _{op}\le L$, where $D\vec{F}$ would be the Jacobian on $\vec{F}$
19	Convergence of Picard Iterates	$\vert \vert \vec{x}_{k+1}-\vec{x}_k\vert \vert \le 2^{-k}r$	Then define $\vec{x}=\lim\limits_{k\to\infty}\vec{x}_k$	Solution can be obtained from Picard Iterates: $\vec{x}=\lim\limits_{k\to\infty}\vec{x}_k$
			Calculate $\vert \vert \vec{x}m-\vec{x}_l\vert \vert \le \sum\limits{k=l+1}^m \vert \vert \vec{x}{k}-\vec{x}{k-1}\vert \vert \le \sum\limits_{k=l+1}^m 2^{-k+1}r \le 2^{-l+1}r$Now send $m\to \infty$, still bounded by $2^{-l+1}r$
			Therefore, $\vec{F}(\vec{x})=\lim\limits_{k\to\infty}\vec{F}(\vec{x}k)$, hence the solution $\lim\limits{k\to\infty}\vec{x}k=\lim\limits{k\to\infty}(\vec{x}0+\int\limits_0^tF(x{l-1}(s))ds)=\vec{x}_0+\int\limits_0^tF(x(s))ds)=\vec{x}$
20	Uniqueness of Solution	$\vec{x}’(t)=\vec{F}(\vec{x})$	Suppose there is another solution $\vec{y}$. Then bound $$		\vec{F}(\vec{x})-\vec{F}(\vec{y})		\le L		\vec{x}-\vec{y}		$$, $\vec{F}$ is Lipschitz continuous	The solution $\vec{x}$ is unique if $\vec{F}$ is Lipschitz continuous, and $\vec{x}$ is continuously differentiable.
			i.e. Lipschitz continuous function is limited in how fast it can change
			Compute $$\frac{d}{dt}		\vec{x}-\vec{y}		^2 \le 2L		\vec{x}-\vec{y}		^2$$, hence $\frac{d}{dt}e^{-2Lt}		\vec{x}-\vec{y}		^2\le0$
			Therefore, if $\vec{x}(0)-\vec{y}(0)=0$, for all $t$, $\vec{x}-\vec{y}=0$
21	Unique Maximal Domain for Solution	$\begin{cases}\vec{x}’(t)=\vec{F}(\vec{x}) \ \vec{x}(0)=\vec{x}_0\end{cases}$	This complements the previous theorem (20), saying the solution will uniquely solve up to some time $\beta$	If $\beta < \infty$, then it must be that $\vert \vert \vec{x}\vert \vert \to \infty$ as $t\to \beta$
			Proof by contradiction that , if $\beta < \infty$, yet your solution is still bounded in a set $U =\R^n$. This cannot be because then you can extend a time $\delta$ by theorem (18).
22	Definition of Flow	For each point $(x_0,t_0)\in \Omega$, we have:
		$\Phi(x_0,t_0)=\phi_{t_0}(x_0)=x(t_0)$ meaning a solution of the ODE starting from point $x_0$, and is defined up to time $t_0$.
23	Bounded Neighborhood Solution (Part 1)	If we have another initial condition $y_0$ s.t. $\vert \vert x_0 - y_0\vert \vert \le e^{-Lt_1}r$, where $t_1$ is the closed upper bound for maximal defined time interval		$\vert \vert \Phi(x_0, t)-\Phi(y_0, t)\vert \vert \le 2e^{Lt}\vert \vert x_0 - y_0\vert \vert$where $0\le t\le t_1$.
		$\Phi(x_0, t)$ and $\Phi(y_0,t)$, namely two solutions up to the same time yet starting from a different initial condition		Also hints at possible chaotic behavior as $t\to \infty$ exploded $e^{Lt}$
24	Interval for Neighborhood Solution	If we have another initial condition $y_0$ s.t. $\vert \vert x_0 - y_0\vert \vert \le e^{-Lt_1}r$, where $t_1$ is the closed upper bound for maximal defined time interval		$\Phi(y_0,t)$ is *defined at least* in the same interval** of $\Phi(x_0,t)$, where $0 \le t \le t_1$
25	Bounded Neighborhood Solution (Part 2)	If we have another initial condition $y_0$ s.t. $\vert \vert x_0 - y_0\vert \vert \le e^{-Lt_1}r$, where $t_1$ is the closed upper bound for maximal defined time interval	$\vert \vert \Phi(x_0, t)-\Phi(x_0, t_0)\vert \vert \le M\vert t-t_0\vert$	$\vert \vert \Phi(x_0, t_0)-\Phi(y_0, t)\vert \vert \le 2e^{Lt}\vert \vert x_0 - y_0\vert \vert +M\vert t_0-t\vert$ where we only have an initial point $(x_0,t_0)$, and we bound another solution up to time $t\in[0,t_1]$
26	Differentiability of Flow and Linearized Solution	Given an ODE: $\vec{x}’(t)=\vec{F}(\vec{x})$, and its Flow: $\vec{x}(t)=\Phi(x_0,t)$	The final goal is to show that $\lim\limits_{y \to x_0} \frac{\vert \varphi_{t_0}(x_0)-\varphi_{t_0}(y)-M(t_0)(x_0-y)\vert }{\vert x_0-y\vert } = 0$,where we get $M(t_0)$ being the derivative on the “initial condition $x_0$” (the only other variable is time, whose derivative would be the trivial).	$\vec{x}(t)=\Phi(x_0,t)$ is differentiable at $\vec{x}0$ which is given by $D\phi{t_0}(x_0)=M(t_0)$
		Let $A(t)\equiv D\vec{F}(\vec{x})$		$M$ is defined by the solution to the ODE: $M’(t)=A(t)M(t)$, $M(0)=I$
27	Existence of $M$	Let $A(t)\equiv D\vec{F}(\vec{x}):[0,T]\to \R^{n\times n}$ be a continuous function	Proven using Picard Iterates, constructing $M(t) := \lim\limits_{k \to \infty} M_k(t)$	There exists a $M$ that solves the ODE $M’(t)=A(t)M(t)$, and $M(0)=I$
			Induction proof that, if $M_0(t)=I$, $\vert A(t)\vert {op}\le L$ for all $t\in[0,T]$, and $M{k+1}(t) = I + \int\limits_0^t A(s) M_k(s) \, ds$
			Then $\vert M_k(t)-M_{k-1}(t)\vert {\text{op}} \leq \frac{(Lt)^k}{k!}$, hence convergence* of $\vert M(t)-M_l(t)\vert _{\text{op}} \leq \sum\limits{k=l+1}^\infty \vert M_k(t)-M_{k-1}(t)\vert {\text{op}} \leq \sum\limits*{k=l+1}^\infty \frac{(Lt)^k}{k!}$
28		True in general	make $A$ into an upper-triangle matrix $B$ by $A=SBS$	$det(e^{tA}) = e^{t\cdot tr(A)}$
			Expand the exponent, and show $det(A)^k = det(B)^k = tr(B)^k = tr(A)^k$
29				$\frac{d}{dt}det(M(t))\vert _{t_0} = tr(A(t_0))det(M(t_0))$
30		If $M$ is differentiable	Follows from above (29)	$det(M(t))=det(M(0)\cdot e^{\int_0^ttr(A(s))ds})$
31	Liouville Theorem	If $F$ is a divergence free vector field (for the nonlinear differential system)	$detD\varphi_{t_0}(x_0)=detM(t_0)=detM(0)=1$	The flow $\varphi_{t_0}$ preserves volume (i.e. two solution $\varphi_{t_0}(x_0),\varphi_{t_0}(y_0)$ occupies the same volume in space)
32		If given an equilibrium point $\bar{x}$	For that equilibrium point, $M’(t)=AM(t)$, $A$ independent of $t$	For the solution $\bar{x}$, $M(t)=e^{At}$
33	Stability of Equilibrium Point	For an equilibrium point $\bar{x}$, if $DF(\bar{x})$ has eigenvalues with only negative real parts		$\bar{x}$ is asymptotically stable
34	Stability of Equilibrium Point	For an equilibrium point $\bar{x}$, if $DF(\bar{x})$ has eigenvalues with at least one positive real parts		$\bar{x}$ is not stable
35	Stability of Equilibrium Point	For an equilibrium point $\bar{x}$, if $DF(\bar{x})$ has eigenvalues with at least one purely imaginary		inconclusive
36	Lyapunov’s criterion for stability	If we can find a function $L$, such that: $L(\bar{x})$ is a strict local minimum, and $\nabla L(x) \cdot F(x) \le 0$ for all $x$ near $\bar{x}$	Show that $L(\varphi_t(x_0))$ is monotonously decreasing by showing its derivative being $\le 0$	$\bar{x}$ is stable
37		If we can find a function $L$, such that: $L(\bar{x})$ is a strict local minimum, and $\nabla L(x) \cdot F(x) < 0$ for all $x$ near $\bar{x}$		$\bar{x}$ is asymptotically stable
38	Stability of Gradient System	If $F=-\nabla V$, and $\bar{x}$ is a local min for $V$	Show that $V$ is a Lyapunov Function, hence $<\nabla V(x), F(x)> \le 0$	$\bar{x}$ is a stable equilibrium point
			If the function $V$ is found to be $\nabla V(x)\cdot F(x) < 0$	$\bar{x}$ is an asymptotically stable equilibrium point
39	Stability of Hamiltonian System	If $F=J \nabla H$, and $\bar{x}$ is a local min for $H$	Same as above, but have $<\nabla H(x), F(x)> = 0$	$\bar{x}$ is only a stable equilibrium point (not asymptotically stable)
		$J=\begin{bmatrix}0 & 1& \ -1 & 0 & \ & & 0 & 1\ & & -1 & 0\&&&&… & \&&&&&0&1\&&&&&-1&0\end{bmatrix}$
40	Property of Hamiltonian System		Show that $\frac{d}{dt}H(x)=0$	The quantity $H$ is preserved
			If stuck, first show that $\vec{w} \cdot J\vec{w}=0$

Tricks

Tricks for Computing Matrix Exponentials with Inhomogeneous Term

Basically, we you are dealing with terms such as $e^{tA}\vec{f}$, and $A$ is diagonalizable:

you should decompose the vector $\vec{f}$ into the eigenvectors
this will be always used since for non-diagonalizable $A$, you can use $A=L+D$, where $L$ is diagonalizable.

For example, in HW4:

Question:

Trick:

ID	Title	Condition/Definitions	Key Steps	Theorem/Solution Equation
1	Variable coefficient and inhomogeneous:	\(x'(t)=a(t)x(t)+f(t)\)	RHS does not have $x(t)$:\(\frac{d}{dt}(e^{-\phi(t)}x(t))=e^{-\phi(t)}(x'(t)-\phi'(t)x(t))=e^{-\phi(t)}f(t)\)	\(x(t)=e^{\phi(t)}(\int e^{-\phi(t)}f(t) dt + c)\)
			\(\phi=\int a(t)dt\)
2	Separable	$x’(t)=f(x_{(t)})g(t)$	RHS does not have $x(t)$: \(\frac{d}{dt}\phi (x) = \phi'(x)x'(t)=\frac{x'(t)}{f(x)}=g(t)\)	\(x(t)=\phi^{-1}(\int g(t)dt +c)\)
			\(\phi'(x) = \frac{1}{f(x)}\)
3	Linear System	\(\vec{x}'(t) = A \vec{x}(t)\)	\(e^{At} = \sum\limits_{k=0}^{\infty}\frac{t^k}{k!}A^k\)	\(\vec{x}=\vec{x}_0 e^{At}\)

4	Euclidean and Operator Norm of Matrix	Euclidean Norm: $$		A		= \sqrt{\sum\limits_{i,j=1}^{n}a_{ij}}$$		$$		A		_{op} \le		A		\le \sqrt{n}	A	_{op}$$
		Operator Norm: $$		A		_{op} = max{		A\hat{x}		,		\hat{x}		=1}$$
5	Existence of \(e^A\)	Using a Cauchy Sequence, so that $$		S_m - S_l		\le \epsilon\(for large\)l\(and\)m \ge l$$.	\(\left\|\sum\limits_{k=l+1}^{\infty}\frac{1}{k!}A^k\right\| \le \sum\limits_{k=l+1}^{\infty}\frac{1}{k!}\left\| A^k\right\|_{op} \le \sum\limits_{k=l+1}^{\infty}\frac{1}{k!}\left\| A\right\|_{op}^k\)	Limit exists for \(e^A\)
6	Property of \(e^{A+B}\)	If \(A\) commutes with \(B\)		\(e^{A+B}=e^A e^B\)
7	Alternative formula for \(e^A\)			\(e^A=\lim_{m\to \infty} (I+\frac{1}{m}A)^m\)
8	Derivative of \(e^{tA}\)		\(e^{hA}-I=hA+\sum\limits_{k=0}^{\infty}\frac{h^{k+2}}{(k+2)!}A^{k+2}\)	\(\frac{d}{dt}e^{tA}=Ae^{tA}\)
9	Inhomogeneous Linear System	\(\vec{x}'(t)=A\vec{x}(t)+\vec{f}(t)\)		\(\vec{x}(t)=e^{tA}\left(\int\limits_0^t e^{-sA}f(s) ds + \vec{x}_0\right)\)
10	Diagonal Matrix Exponentials	\(e^{tD}\), \(D\) is diagonal		\(e^{tD}= \begin{bmatrix}e^{t\lambda_1} & 0 & 0 &... \\ 0 & e^{t\lambda_2} & 0 & ...\\... & ... & ... & ...\\... & ... &... &e^{t\lambda_1}\end{bmatrix}\)
11	Diagonalizable Matrix Exponential	\(e^{tA}\), \(A=SDS^{-1}\)		\(e^{tA}=Se^{tB}S^{-1}\)
12	Nilpotent Matrix	\(N^m=0\), \(N\) is nilpotent	\(A^m\vec{x}=\lambda^m x = 0\)	All eigenvalues of \(N\) is 0
13	LN Decomposition	\(A=L+N\) for any \(A\), and \(L,N\) commute	\(L\vec{x}=\lambda_i \vec{x}\), for \(\vec{x}\in ker(A-\lambda_iI)^{\nu_i}\)
			\(N^{max(\nu_i)}\vec{x}=(A-\lambda_i I)^{max(\nu_i)}\vec{x}=0\), for \(\vec{x}\in ker(A-\lambda_iI)^{\nu_i}\)
14	Asymptotic Behavior of Linear System	\(\vec{x}'=A\vec{x}\), all $Re(\lambda)<0$	$e^{tA}\vec{x}=e^{t(\lambda_i I)}e^{t(A-\lambda_i I)}\vec{x}=e^{t(\lambda_i I)}\sum\limits_{k=0}^{\nu_i-1}\frac{t^k}{k!}(A-\lambda_i I)^k \vec{x}$	$\lim\limits_{t \to\infty} \vec{x}_0e^{tA} \to 0$
		$\lambda$ are the generalized eigenvalues, $\vec{x}$ in generalized eigenspace	and $e^{t(\lambda_i I)}\to 0$ much faster as all $Re(\lambda)<0$
15	Asymptotic Behavior of Inhomogeneous System	\(\vec{x}'=A\vec{x}+f(t)\), all $Re(\lambda)<0$, and $f(t)$ has period $\tau$	Start with $\bar{x}(t_\tau = \tau)=\bar{x}(t_\tau = 0)$, get $(e^{\tau A}-I)\bar{x}_0 = \int\limits_0^\tau e^{-sA}f(s)ds$	There is a unique solution $\bar{x}(t_{\tau})$that is also period $\tau$
		$\lambda$ are the generalized eigenvalues, $\vec{x}$ in generalized eigenspace	$\vec{x}’(t)-\bar{x}’(t_{\tau})=A(\vec{x}(t)-\bar{x}(t_{\tau}))$. Therefore: $\vec{x}(t)-\bar{x}(t_{\tau}) = e^{At}(\vec{x}(t)-\bar{x}(t_{\tau}))$, and $e^{At}\to 0$ if all $Re(\lambda)<0$	There can be other solutions $\vec{x}(t)$, but they will tend to $\bar{x}(t_{\tau})$ as $t \to \infty$
16	Asymptotic Behavior of Inhomogeneous System	This applies for both homogenous and inhomogeneous, as you can write them all in the form of $\vec{x}=e^{At}\vec{x}_0$	same as (14), but take case 1 being $\vec{x}\in ker(p_{-}A)$, and the second case the opposite	$e^{tA}\vec{x}0 \to 0$ if $\vec{x}_0 \in ker(p{-}A)$, and $t\to \infty$ and $e^{tA}\vec{x}0 \to 0$ if $\vec{x}_0 \in ker(p{+}A)$ and $t \to -\infty$
		Assumes no purely imaginary eigenvalues	$ker(p_{-}A)$ is the generalized eigenspace for negative (real part) generalized eigenvalues, and vice versa
17	Asymptotic Behavior of Inhomogeneous System	Accounting $Re(\lambda)=0$ case	Split the total space into three spaces, one for each $Re(\lambda<0)$, $Re(\lambda=0)$, $Re(\lambda>0)$	$e^{tA}\vec{x}\to 0$ if $\vec{x}\in ker(p_{-}A)=ker(A-\lambda_i I)^{\nu_i}$, for all $Re(\nu_i)<0$
			Do the same analysis as (16) and (14)	$e^{tA}\vec{x}$ is bounded if $\vec{x}\in ker(p_{-}A)\bigoplus ker(A-\lambda_jI)$, for all $Re(\nu_j)=0$
18	Cauchy-Lipschitz Theorem	$\vec{x}’(t)=\vec{F}(\vec{x})=\begin{bmatrix}f_1(\vec{x})\f_2(\vec{x})\…\f_n(\vec{x})\end{bmatrix}$	If we take a small time interval $\delta$ away from the initial condition at $t_0$, there $\exists$ a solution.	That solution will be the convergence of Picard Iterates: $\vec{x_k}(t)=\vec{x}0+\int\limits_0^t \vec{F}(\vec{x}{k-1})dt$
			Then we can bound: $\vert \vert \vec{F}(\vec{x})\vert \vert \le M$ and $\vert \vert D\vec{F}(\vec{x})\vert \vert _{op}\le L$, where $D\vec{F}$ would be the Jacobian on $\vec{F}$
19	Convergence of Picard Iterates	$\vert \vert \vec{x}_{k+1}-\vec{x}_k\vert \vert \le 2^{-k}r$	Then define $\vec{x}=\lim\limits_{k\to\infty}\vec{x}_k$	Solution can be obtained from Picard Iterates: $\vec{x}=\lim\limits_{k\to\infty}\vec{x}_k$
			Calculate $\vert \vert \vec{x}m-\vec{x}_l\vert \vert \le \sum\limits{k=l+1}^m \vert \vert \vec{x}{k}-\vec{x}{k-1}\vert \vert \le \sum\limits_{k=l+1}^m 2^{-k+1}r \le 2^{-l+1}r$Now send $m\to \infty$, still bounded by $2^{-l+1}r$
			Therefore, $\vec{F}(\vec{x})=\lim\limits_{k\to\infty}\vec{F}(\vec{x}k)$, hence the solution $\lim\limits{k\to\infty}\vec{x}k=\lim\limits{k\to\infty}(\vec{x}0+\int\limits_0^tF(x{l-1}(s))ds)=\vec{x}_0+\int\limits_0^tF(x(s))ds)=\vec{x}$
20	Uniqueness of Solution	$\vec{x}’(t)=\vec{F}(\vec{x})$	Suppose there is another solution $\vec{y}$. Then bound $$		\vec{F}(\vec{x})-\vec{F}(\vec{y})		\le L		\vec{x}-\vec{y}		$$, $\vec{F}$ is Lipschitz continuous	The solution $\vec{x}$ is unique if $\vec{F}$ is Lipschitz continuous, and $\vec{x}$ is continuously differentiable.
			i.e. Lipschitz continuous function is limited in how fast it can change
			Compute $$\frac{d}{dt}		\vec{x}-\vec{y}		^2 \le 2L		\vec{x}-\vec{y}		^2$$, hence $\frac{d}{dt}e^{-2Lt}		\vec{x}-\vec{y}		^2\le0$
			Therefore, if $\vec{x}(0)-\vec{y}(0)=0$, for all $t$, $\vec{x}-\vec{y}=0$
21	Unique Maximal Domain for Solution	$\begin{cases}\vec{x}’(t)=\vec{F}(\vec{x}) \ \vec{x}(0)=\vec{x}_0\end{cases}$	This complements the previous theorem (20), saying the solution will uniquely solve up to some time $\beta$	If $\beta < \infty$, then it must be that $\vert \vert \vec{x}\vert \vert \to \infty$ as $t\to \beta$
			Proof by contradiction that , if $\beta < \infty$, yet your solution is still bounded in a set $U =\R^n$. This cannot be because then you can extend a time $\delta$ by theorem (18).
22	Definition of Flow	For each point $(x_0,t_0)\in \Omega$, we have:
		$\Phi(x_0,t_0)=\phi_{t_0}(x_0)=x(t_0)$ meaning a solution of the ODE starting from point $x_0$, and is defined up to time $t_0$.
23	Bounded Neighborhood Solution (Part 1)	If we have another initial condition $y_0$ s.t. $\vert \vert x_0 - y_0\vert \vert \le e^{-Lt_1}r$, where $t_1$ is the closed upper bound for maximal defined time interval		$\vert \vert \Phi(x_0, t)-\Phi(y_0, t)\vert \vert \le 2e^{Lt}\vert \vert x_0 - y_0\vert \vert$where $0\le t\le t_1$.
		$\Phi(x_0, t)$ and $\Phi(y_0,t)$, namely two solutions up to the same time yet starting from a different initial condition		Also hints at possible chaotic behavior as $t\to \infty$ exploded $e^{Lt}$
24	Interval for Neighborhood Solution	If we have another initial condition $y_0$ s.t. $\vert \vert x_0 - y_0\vert \vert \le e^{-Lt_1}r$, where $t_1$ is the closed upper bound for maximal defined time interval		$\Phi(y_0,t)$ is *defined at least* in the same interval** of $\Phi(x_0,t)$, where $0 \le t \le t_1$
25	Bounded Neighborhood Solution (Part 2)	If we have another initial condition $y_0$ s.t. $\vert \vert x_0 - y_0\vert \vert \le e^{-Lt_1}r$, where $t_1$ is the closed upper bound for maximal defined time interval	$\vert \vert \Phi(x_0, t)-\Phi(x_0, t_0)\vert \vert \le M\vert t-t_0\vert$	$\vert \vert \Phi(x_0, t_0)-\Phi(y_0, t)\vert \vert \le 2e^{Lt}\vert \vert x_0 - y_0\vert \vert +M\vert t_0-t\vert$ where we only have an initial point $(x_0,t_0)$, and we bound another solution up to time $t\in[0,t_1]$
26	Differentiability of Flow and Linearized Solution	Given an ODE: $\vec{x}’(t)=\vec{F}(\vec{x})$, and its Flow: $\vec{x}(t)=\Phi(x_0,t)$	The final goal is to show that $\lim\limits_{y \to x_0} \frac{\vert \varphi_{t_0}(x_0)-\varphi_{t_0}(y)-M(t_0)(x_0-y)\vert }{\vert x_0-y\vert } = 0$,where we get $M(t_0)$ being the derivative on the “initial condition $x_0$” (the only other variable is time, whose derivative would be the trivial).	$\vec{x}(t)=\Phi(x_0,t)$ is differentiable at $\vec{x}0$ which is given by $D\phi{t_0}(x_0)=M(t_0)$
		Let $A(t)\equiv D\vec{F}(\vec{x})$		$M$ is defined by the solution to the ODE: $M’(t)=A(t)M(t)$, $M(0)=I$
27	Existence of $M$	Let $A(t)\equiv D\vec{F}(\vec{x}):[0,T]\to \R^{n\times n}$ be a continuous function	Proven using Picard Iterates, constructing $M(t) := \lim\limits_{k \to \infty} M_k(t)$	There exists a $M$ that solves the ODE $M’(t)=A(t)M(t)$, and $M(0)=I$
			Induction proof that, if $M_0(t)=I$, $\vert A(t)\vert {op}\le L$ for all $t\in[0,T]$, and $M{k+1}(t) = I + \int\limits_0^t A(s) M_k(s) \, ds$
			Then $\vert M_k(t)-M_{k-1}(t)\vert {\text{op}} \leq \frac{(Lt)^k}{k!}$, hence convergence* of $\vert M(t)-M_l(t)\vert _{\text{op}} \leq \sum\limits{k=l+1}^\infty \vert M_k(t)-M_{k-1}(t)\vert {\text{op}} \leq \sum\limits*{k=l+1}^\infty \frac{(Lt)^k}{k!}$
28		True in general	make $A$ into an upper-triangle matrix $B$ by $A=SBS$	$det(e^{tA}) = e^{t\cdot tr(A)}$
			Expand the exponent, and show $det(A)^k = det(B)^k = tr(B)^k = tr(A)^k$
29				$\frac{d}{dt}det(M(t))\vert _{t_0} = tr(A(t_0))det(M(t_0))$
30		If $M$ is differentiable	Follows from above (29)	$det(M(t))=det(M(0)\cdot e^{\int_0^ttr(A(s))ds})$
31	Liouville Theorem	If $F$ is a divergence free vector field (for the nonlinear differential system)	$detD\varphi_{t_0}(x_0)=detM(t_0)=detM(0)=1$	The flow $\varphi_{t_0}$ preserves volume (i.e. two solution $\varphi_{t_0}(x_0),\varphi_{t_0}(y_0)$ occupies the same volume in space)
32		If given an equilibrium point $\bar{x}$	For that equilibrium point, $M’(t)=AM(t)$, $A$ independent of $t$	For the solution $\bar{x}$, $M(t)=e^{At}$
33	Stability of Equilibrium Point	For an equilibrium point $\bar{x}$, if $DF(\bar{x})$ has eigenvalues with only negative real parts		$\bar{x}$ is asymptotically stable
34	Stability of Equilibrium Point	For an equilibrium point $\bar{x}$, if $DF(\bar{x})$ has eigenvalues with at least one positive real parts		$\bar{x}$ is not stable
35	Stability of Equilibrium Point	For an equilibrium point $\bar{x}$, if $DF(\bar{x})$ has eigenvalues with at least one purely imaginary		inconclusive
36	Lyapunov’s criterion for stability	If we can find a function $L$, such that: $L(\bar{x})$ is a strict local minimum, and $\nabla L(x) \cdot F(x) \le 0$ for all $x$ near $\bar{x}$	Show that $L(\varphi_t(x_0))$ is monotonously decreasing by showing its derivative being $\le 0$	$\bar{x}$ is stable
37		If we can find a function $L$, such that: $L(\bar{x})$ is a strict local minimum, and $\nabla L(x) \cdot F(x) < 0$ for all $x$ near $\bar{x}$		$\bar{x}$ is asymptotically stable
38	Stability of Gradient System	If $F=-\nabla V$, and $\bar{x}$ is a local min for $V$	Show that $V$ is a Lyapunov Function, hence $<\nabla V(x), F(x)> \le 0$	$\bar{x}$ is a stable equilibrium point
			If the function $V$ is found to be $\nabla V(x)\cdot F(x) < 0$	$\bar{x}$ is an asymptotically stable equilibrium point
39	Stability of Hamiltonian System	If $F=J \nabla H$, and $\bar{x}$ is a local min for $H$	Same as above, but have $<\nabla H(x), F(x)> = 0$	$\bar{x}$ is only a stable equilibrium point (not asymptotically stable)
		$J=\begin{bmatrix}0 & 1& \ -1 & 0 & \ & & 0 & 1\ & & -1 & 0\&&&&… & \&&&&&0&1\&&&&&-1&0\end{bmatrix}$
40	Property of Hamiltonian System		Show that $\frac{d}{dt}H(x)=0$	The quantity $H$ is preserved
			If stuck, first show that $\vec{w} \cdot J\vec{w}=0$