MATH3027 ODE
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 runaway
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 $Re(\lambda)<0$  $e^{tA}\vec{x}=e^{t(\lambda_i I)}e^{t(A\lambda_i I)}\vec{x}=e^{t(\lambda_i I)}\sum\limits_{k=0}^{\nu_i1}\frac{t^k}{k!}(A\lambda_i I)^k \vec{x}$  $\lim\limits_{t \to\infty} \vec{x}_0e^{tA} \to 0$  
$\lambda$ 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  CauchyLipschitz 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: $\vec{x_k}(t)=\vec{x}0+\int\limits_0^t \vec{F}(\vec{x}{k1})dt$  
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 $\vert \vert \vec{x}m\vec{x}_l\vert \vert \le \sum\limits{k=l+1}^m \vert \vert \vec{x}{k}\vec{x}{k1}\vert \vert \le \sum\limits_{k=l+1}^m 2^{k+1}r \le 2^{l+1}r$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 $\lim\limits{k\to\infty}\vec{x}k=\lim\limits{k\to\infty}(\vec{x}0+\int\limits_0^tF(x{l1}(s))ds)=\vec{x}_0+\int\limits_0^tF(x(s))ds)=\vec{x}$  
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 
$\vert \vert \Phi(x_0, t)\Phi(x_0, t_0)\vert \vert \le M\vert tt_0\vert$  $\vert \vert \Phi(x_0, t_0)\Phi(y_0, t)\vert \vert \le 2e^{Lt}\vert \vert x_0  y_0\vert \vert +M\vert t_0t\vert$ 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 $\lim\limits_{y \to x_0} \frac{\vert \varphi_{t_0}(x_0)\varphi_{t_0}(y)M(t_0)(x_0y)\vert }{\vert x_0y\vert } = 0$,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 $\vert M_k(t)M_{k1}(t)\vert {\text{op}} \leq \frac{(Lt)^k}{k!}$, hence convergence of $\vert M(t)M_l(t)\vert _{\text{op}} \leq \sum\limits{k=l+1}^\infty \vert M_k(t)M_{k1}(t)\vert {\text{op}} \leq \sum\limits{k=l+1}^\infty \frac{(Lt)^k}{k!}$  
28  True in general  make $A$ into an uppertriangle 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 nondiagonalizable $A$, you can use $A=L+D$, where $L$ is diagonalizable.
For example, in HW4:
 Question:
 Trick: