Differential Equations Systems of Differential Equations and Dynamical Systems Syllabus

Systems of Differential Equations and Dynamical Systems syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Systems of Differential Equations and Dynamical Systems in Differential Equations, not a summary of it.

1. 1

   Linear Systems Theory

   2 topics

   • Matrix Formulation of Systems
   • Homogeneous Linear Systems
2. 2

   Eigenvalue Methods

   4 topics

   • Distinct Real Eigenvalues
   • Complex Eigenvalues
   • Repeated Eigenvalues and Generalized Eigenvectors
   • Matrix Exponential
3. 3

   Nonhomogeneous Systems

   2 topics

   • Undetermined Coefficients for Systems
   • Variation of Parameters for Systems
4. 4

   Phase Plane and Stability Analysis

   4 topics

   • Critical Points and Classification
   • Stability of Linear Systems
   • Linearization of Nonlinear Systems
   • Lyapunov Stability Methods
5. 5

   Nonlinear Dynamics

   4 topics

   • Predator-Prey Models
   • Competing Species Models
   • Limit Cycles and the Poincare-Bendixson Theorem
   • Introduction to Bifurcation and Chaos

Systems of Differential Equations and Dynamical Systems flashcards for Differential Equations

23 of 50 cards from the Systems of Differential Equations and Dynamical Systems deck — real questions with worked answers.

1. How is a system of $n$ first-order linear ODEs written in matrix form?

   As $\vec{x}\,' = A(t)\vec{x} + \vec{g}(t)$, where $\vec{x}$ is the $n\times 1$ vector of unknowns, $A(t)$ is the $n\times n$ coefficient matrix, and $\vec{g}(t)$ is the forcing vector. The system is homogeneous when $\vec{g}(t)=\vec{0}$.

2. How do you convert the higher-order ODE $y'' + p\,y' + q\,y = 0$ into a first-order system?

   Let $x_1 = y$ and $x_2 = y'$. Then $$\begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -q & -p \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}.$$

3. What does the Wronskian tell you about solutions $\vec{x}^{(1)},\dots,\vec{x}^{(n)}$ of a linear system?

   $W = \det[\vec{x}^{(1)}\ \cdots\ \vec{x}^{(n)}]$. If $W(t_0)\neq 0$ at any point, the solutions are linearly independent and form a fundamental set; by Abel's theorem $W$ is either always zero or never zero on the interval.

4. What is the general solution of a homogeneous linear system $\vec{x}\,'=A\vec{x}$ given $n$ linearly independent solutions?

   $\vec{x}(t) = c_1\vec{x}^{(1)}(t) + c_2\vec{x}^{(2)}(t) + \cdots + c_n\vec{x}^{(n)}(t)$, a linear combination of the fundamental set of solutions with arbitrary constants $c_i$.

5. For the constant system $\vec{x}\,'=A\vec{x}$, what form of solution is sought, and what equation must its parameters satisfy?

   Seek $\vec{x} = \vec{v}\,e^{\lambda t}$. Substituting gives $A\vec{v} = \lambda\vec{v}$, so $\lambda$ must be an eigenvalue of $A$ and $\vec{v}$ a corresponding eigenvector: $(A-\lambda I)\vec{v}=\vec{0}$.

6. How do you find eigenvalues of a matrix $A$?

   Solve the characteristic equation $\det(A - \lambda I) = 0$, a polynomial of degree $n$ in $\lambda$.

7. Give the general solution of $\vec{x}\,'=A\vec{x}$ when $A$ ($2\times 2$) has distinct real eigenvalues $\lambda_1,\lambda_2$ with eigenvectors $\vec{v}_1,\vec{v}_2$.

   $$\vec{x}(t) = c_1\vec{v}_1 e^{\lambda_1 t} + c_2\vec{v}_2 e^{\lambda_2 t}.$$

8. For distinct real eigenvalues, when is the origin a saddle point versus a node?

   Saddle point when eigenvalues have opposite signs ($\lambda_1 < 0 < \lambda_2$); node when they share a sign — an unstable node if both positive, a stable (asymptotically stable) node if both negative.

9. If $A$ is a real matrix with a complex eigenvalue $\lambda = \alpha + i\beta$ and eigenvector $\vec{v}=\vec{a}+i\vec{b}$, how do you get two real solutions?

   Take real and imaginary parts of $\vec{v}\,e^{\lambda t}$: $$\vec{x}_1 = e^{\alpha t}(\vec{a}\cos\beta t - \vec{b}\sin\beta t),\quad \vec{x}_2 = e^{\alpha t}(\vec{a}\sin\beta t + \vec{b}\cos\beta t).$$

10. For complex eigenvalues $\alpha \pm i\beta$, how does the sign of $\alpha$ classify the critical point?

    It is a spiral (focus): stable spiral if $\alpha < 0$ (trajectories spiral inward), unstable spiral if $\alpha > 0$. If $\alpha = 0$ (purely imaginary), it is a center with closed periodic orbits.

11. What does the imaginary part $\beta$ of a complex eigenvalue control in the solution?

    $\beta$ is the angular frequency of rotation; the trajectories oscillate/rotate with period $\frac{2\pi}{\beta}$ while the factor $e^{\alpha t}$ controls growth or decay of amplitude.

12. What is the algebraic vs. geometric multiplicity of an eigenvalue, and when do repeated eigenvalues cause trouble?

    Algebraic multiplicity is the root order in the characteristic polynomial; geometric multiplicity is the number of independent eigenvectors ($\dim\ker(A-\lambda I)$). Trouble (defective/deficient case) occurs when geometric $<$ algebraic multiplicity.

13. For a $2\times 2$ matrix with a repeated eigenvalue $\lambda$ having only one eigenvector $\vec{v}$, what is the general solution?

    $$\vec{x}(t) = c_1 \vec{v}\,e^{\lambda t} + c_2\big(\vec{v}\,t\,e^{\lambda t} + \vec{w}\,e^{\lambda t}\big),$$ where $\vec{w}$ is a generalized eigenvector solving $(A-\lambda I)\vec{w} = \vec{v}$.

14. How is a generalized eigenvector $\vec{w}$ (rank 2) for eigenvalue $\lambda$ defined?

    By $(A-\lambda I)\vec{w} = \vec{v}$ where $\vec{v}$ is an ordinary eigenvector, equivalently $(A-\lambda I)^2\vec{w}=\vec{0}$ but $(A-\lambda I)\vec{w}\neq\vec{0}$.

15. Define the matrix exponential $e^{At}$ as a power series.

    $$e^{At} = \sum_{k=0}^{\infty} \frac{(At)^k}{k!} = I + At + \frac{A^2 t^2}{2!} + \frac{A^3 t^3}{3!} + \cdots$$

16. What key properties make $e^{At}$ useful for solving $\vec{x}\,'=A\vec{x}$?

    $\frac{d}{dt}e^{At} = A e^{At}$, $e^{A\cdot 0}=I$, and $(e^{At})^{-1}=e^{-At}$. Thus the solution to the IVP with $\vec{x}(0)=\vec{x}_0$ is $\vec{x}(t)=e^{At}\vec{x}_0$.

17. How can $e^{At}$ be recovered from any fundamental matrix $\Psi(t)$?

    $e^{At} = \Psi(t)\,\Psi(0)^{-1}$. This is the special fundamental matrix normalized to equal $I$ at $t=0$.

18. When can $e^{At}$ be computed via diagonalization, and what is the formula?

    When $A = PDP^{-1}$ is diagonalizable ($D$ diagonal of eigenvalues), $e^{At} = P e^{Dt} P^{-1}$, where $e^{Dt}=\operatorname{diag}(e^{\lambda_1 t},\dots,e^{\lambda_n t})$.

19. For a nilpotent part, if $(A-\lambda I)$ is nilpotent, how does the series for $e^{(A-\lambda I)t}$ behave?

    It terminates. Writing $A = \lambda I + N$ with $N^k=0$, $e^{At}=e^{\lambda t}\big(I + Nt + \frac{N^2 t^2}{2!}+\cdots+\frac{N^{k-1}t^{k-1}}{(k-1)!}\big)$ — a finite polynomial in $t$.

20. What form of particular solution does the method of undetermined coefficients assume for $\vec{x}\,'=A\vec{x}+\vec{g}(t)$ when $\vec{g}(t)$ is a constant vector $\vec{b}$?

    Assume $\vec{x}_p = \vec{a}$ (constant). Substituting gives $\vec{0}=A\vec{a}+\vec{b}$, so $\vec{a}=-A^{-1}\vec{b}$ (provided $A$ is invertible).

21. In undetermined coefficients for systems, how do you handle $\vec{g}(t)=\vec{c}\,e^{\lambda t}$ when $\lambda$ IS an eigenvalue of $A$?

    The naive guess $\vec{a}e^{\lambda t}$ fails (resonance). Use $\vec{x}_p = \vec{a}\,t\,e^{\lambda t} + \vec{b}\,e^{\lambda t}$ — include both the $t e^{\lambda t}$ term and a pure $e^{\lambda t}$ term, then match coefficients.

22. State the variation of parameters formula for $\vec{x}\,'=A\vec{x}+\vec{g}(t)$ using a fundamental matrix $\Psi(t)$.

    $$\vec{x}(t) = \Psi(t)\vec{c} + \Psi(t)\int \Psi(t)^{-1}\vec{g}(t)\,dt.$$ The particular part is $\vec{x}_p=\Psi(t)\int\Psi^{-1}\vec{g}\,dt$.

23. Write the variation-of-parameters solution to the IVP $\vec{x}\,'=A\vec{x}+\vec{g}(t)$, $\vec{x}(t_0)=\vec{x}_0$, using the matrix exponential.

    $$\vec{x}(t) = e^{A(t-t_0)}\vec{x}_0 + \int_{t_0}^{t} e^{A(t-s)}\vec{g}(s)\,ds.$$

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