Differential Equations Systems of Differential Equations and Dynamical Systems Syllabus

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.

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 $\vec{x}_p=\Psi(t)\int\Psi^{-1}\vec{g}\,dt$.

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.

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

What is the main advantage of variation of parameters over undetermined coefficients for systems?

Variation of parameters works for ANY continuous forcing $\vec{g}(t)$ (not just polynomial/exponential/sinusoidal forms), and requires no guessing — at the cost of computing $\Psi^{-1}$ and an integral.

What is a critical (equilibrium) point of the autonomous system $\vec{x}\,'=\vec{f}(\vec{x})$?

A point $\vec{x}^*$ where $\vec{f}(\vec{x}^*)=\vec{0}$, so all derivatives vanish and the system stays there for all time (a constant solution).

For $\vec{x}\,'=A\vec{x}$ with $\det A \neq 0$, list the linear critical-point types by eigenvalue configuration.

Real distinct same sign: node (stable if both $<0$). Real opposite signs: saddle (unstable). Complex $\alpha\pm i\beta$, $\alpha\neq 0$: spiral (stable if $\alpha<0$). Pure imaginary: center. Repeated real: proper/improper node.

Using trace $T=\operatorname{tr}A$ and determinant $D=\det A$, how do you classify the critical point of a $2\times 2$ linear system?

Discriminant $\Delta = T^2 - 4D$. If $D<0$: saddle. If $D>0$ and $\Delta>0$: node; $\Delta<0$: spiral; $T=0$: center. Stability: stable when $T<0$, unstable when $T>0$.

State the definitions of stable and asymptotically stable critical points.

Stable: solutions starting near $\vec{x}^*$ stay near it for all $t\ge 0$ ($\forall\varepsilon\,\exists\delta$). Asymptotically stable: stable AND nearby solutions satisfy $\vec{x}(t)\to\vec{x}^*$ as $t\to\infty$.

For the linear system $\vec{x}\,'=A\vec{x}$, what eigenvalue condition guarantees asymptotic stability of the origin?

All eigenvalues of $A$ have strictly negative real parts: $\operatorname{Re}(\lambda_i) < 0$ for all $i$. Then every solution decays to $\vec{0}$.

When is the origin of $\vec{x}\,'=A\vec{x}$ stable but NOT asymptotically stable?

When all eigenvalues satisfy $\operatorname{Re}(\lambda_i)\le 0$ and every eigenvalue with $\operatorname{Re}=0$ is simple (or has full geometric multiplicity). The canonical case is a center (pure imaginary eigenvalues) with bounded, non-decaying orbits.

How do you linearize the nonlinear autonomous system $\vec{x}\,'=\vec{f}(\vec{x})$ near a critical point $\vec{x}^*$?

Compute the Jacobian $J = \frac{\partial \vec{f}}{\partial \vec{x}}$ evaluated at $\vec{x}^*$. Near $\vec{x}^*$, with $\vec{u}=\vec{x}-\vec{x}^*$, the dynamics approximate $\vec{u}\,' \approx J(\vec{x}^*)\vec{u}$.

Write the Jacobian matrix for the 2D system $x'=f(x,y),\ y'=g(x,y)$.

$$J = \begin{pmatrix} \dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial y} \\[6pt] \dfrac{\partial g}{\partial x} & \dfrac{\partial g}{\partial y} \end{pmatrix}.$$

According to the Hartman-Grobman idea, when does the linearization reliably predict the nonlinear behavior near an equilibrium?

When the equilibrium is hyperbolic — no eigenvalue of the Jacobian has zero real part. Then the nonlinear phase portrait is topologically equivalent to the linear one (type and stability carry over).

Which linearized cases are 'borderline' and may NOT reflect the nonlinear behavior?

Centers (pure imaginary eigenvalues) and any case with $\operatorname{Re}(\lambda)=0$, plus borderline nodes/repeated eigenvalues. A linear center can become a stable or unstable spiral once nonlinear terms are included.

What is a Lyapunov function $V(\vec{x})$ for an equilibrium at the origin?

A continuously differentiable scalar function that is positive definite ($V(\vec{0})=0$, $V(\vec{x})>0$ nearby) whose orbital derivative $\dot V = \nabla V\cdot \vec{f}$ is used to test stability without solving the ODE.

State Lyapunov's stability theorem (direct method) conditions for stability vs. asymptotic stability.

If $V$ is positive definite and $\dot V \le 0$ (negative semidefinite), the origin is stable. If additionally $\dot V < 0$ (negative definite) for $\vec{x}\neq\vec{0}$, the origin is asymptotically stable.

What does a Lyapunov function establish if $V$ is positive definite but $\dot V$ is positive definite?

By Chetaev/Lyapunov instability theorem, the origin is unstable — the 'energy' $V$ increases along trajectories, so they move away from the equilibrium.

Write the classic Lotka-Volterra predator-prey model equations and name the variables.

$$\frac{dx}{dt} = ax - bxy,\qquad \frac{dy}{dt} = -cy + dxy,$$ where $x$ is prey population, $y$ is predator population, and $a,b,c,d>0$.

Find the nonzero (coexistence) equilibrium of the Lotka-Volterra system $x'=ax-bxy,\ y'=-cy+dxy$.

Set both rates to zero (with $x,y\neq 0$): $x^* = \dfrac{c}{d}$, $y^* = \dfrac{a}{b}$.

What is the qualitative behavior of trajectories in the basic Lotka-Volterra model?

The coexistence equilibrium is a center: populations cycle in closed periodic orbits (predator peaks lag prey peaks). There is a conserved quantity, and the time-averages equal the equilibrium values.

Write a competing-species model (Lotka-Volterra competition) for populations $x$ and $y$.

$$\frac{dx}{dt} = x(a_1 - b_1 x - c_1 y),\qquad \frac{dy}{dt} = y(a_2 - b_2 y - c_2 x),$$ where each species inhibits its own and the other's growth ($c_1,c_2>0$ competition terms).

In competing-species models, what determines 'competitive exclusion' versus 'stable coexistence'?

If interspecific competition is strong relative to intraspecific, the coexistence equilibrium is a saddle (unstable) and one species drives the other extinct (exclusion). If intraspecific competition dominates, the interior equilibrium is a stable node and both coexist.

What is a limit cycle?

An isolated closed (periodic) trajectory in the phase plane toward which (or away from which) neighboring trajectories spiral. Stable if nearby orbits approach it, unstable if they diverge — a genuinely nonlinear phenomenon.

How does a limit cycle differ from the closed orbits of a linear center?

A limit cycle is isolated (no closed orbits immediately beside it) and attracts/repels neighbors, giving self-sustained oscillation of fixed amplitude. Center orbits form a continuous family — perturbing the amplitude just moves you to another nearby closed orbit.

State the Poincaré-Bendixson theorem.

For a $C^1$ planar autonomous system, if a trajectory is confined to a closed, bounded region containing no equilibria, then its $\omega$-limit set is a periodic orbit (a limit cycle). In 2D, bounded non-equilibrium orbits must approach a cycle.

State Bendixson's (Dulac's) negative criterion for ruling out limit cycles.

If $\nabla\cdot\vec{f} = \frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}$ has constant sign (never zero) on a simply connected region $D$, then $D$ contains no closed orbits. (Dulac: same with a multiplier $B(x,y)$ on $\nabla\cdot(B\vec{f})$.)

What is a bifurcation in a parametrized dynamical system?

A qualitative change in the phase portrait (number, type, or stability of equilibria/orbits) as a parameter passes through a critical value $\mu_c$ — e.g. equilibria appearing, disappearing, or changing stability.

Name and describe the Hopf bifurcation.

As a parameter crosses a critical value, a pair of complex-conjugate eigenvalues of the Jacobian crosses the imaginary axis ($\operatorname{Re}=0$), so a stable equilibrium loses stability and a small-amplitude limit cycle is born (or absorbed).

Describe the saddle-node and pitchfork bifurcations.

Saddle-node (fold): two equilibria (a saddle and a node) collide and annihilate, e.g. $x'=\mu - x^2$. Pitchfork: one equilibrium loses stability while two new symmetric equilibria appear, e.g. $x'=\mu x - x^3$ (supercritical).

What characterizes deterministic chaos in a dynamical system, and what is the minimum dimension for chaos in a continuous autonomous system?

Chaos: bounded aperiodic dynamics with sensitive dependence on initial conditions (positive Lyapunov exponent), often on a strange attractor (e.g. the Lorenz system). A continuous autonomous system needs at least $3$ dimensions; the Poincaré-Bendixson theorem forbids chaos in the plane.