ODEs + dynamical systems¶

Weeks¶

Week 15¶

Spine: Jiří Lebl, Notes on Diffy Qs v6.11

Reading: Ch. 1 First-order equations: modeling, separable/linear equations, autonomous equations

Know: Translate verbal rate laws into ODEs; solve and interpret first-order models and equilibrium behavior.

Reconstruct: Derive exponential growth/decay and logistic solution structure; derive integrating factor for linear first-order ODE.

Do: Model battery self-discharge + load, microbial growth, and thermal cooling using the same state-balance pattern.

Defend: Which features of these three systems are structural analogies and which are domain-specific?

Gate: formulate an ODE from prose, solve or approximate it, and check units/limiting behavior.

Source: source

Week 16¶

Spine: Jiří Lebl, Notes on Diffy Qs

Reading: Ch. 2 Higher-order linear ODEs; oscillation, forcing, resonance

Know: Analyze oscillators, characteristic roots, transient/steady response and resonance.

Reconstruct: Derive the characteristic equation for constant-coefficient ODEs and the forced damped oscillator response structure.

Do: Simulate a suspension/robot-joint oscillator; sweep damping and forcing frequency; identify resonance and tradeoffs.

Defend: Why does resonance matter far beyond mechanical oscillators?

Gate: Closed-book: classify damping regimes and predict response before simulation.

Source: source

Week 17¶

Spine: Jiří Lebl, Notes on Diffy Qs

Reading: Ch. 3 Systems of ODEs

Know: Represent coupled state dynamics; connect eigenstructure to modes and stability.

Reconstruct: Derive first-order state-space form from an nth-order ODE; derive modal behavior for x'=Ax.

Do: Model a two-tank mixing system and a two-population interaction system in common state-space form.

Defend: When is diagonalization physically meaningful versus merely computationally convenient?

Gate: given A, predict qualitative trajectories from eigenvalues/eigenvectors before plotting.

Source: source

Week 18¶

Spine: Jiří Lebl, Notes on Diffy Qs

Reading: Ch. 8 Nonlinear systems: phase plane, equilibria, linearization, stability

Know: Analyze nonlinear equilibria, local stability, phase portraits and bifurcation-style qualitative changes.

Reconstruct: Derive Jacobian linearization of x'=f(x) about equilibrium; explain local validity.

Do: Construct a nonlinear feedback model that is locally stable but has an unsafe remote basin/attractor.

Defend: Why is local stability not global safety?

Gate: sketch a phase portrait from equations and defend each qualitative feature.

Source: source

Week 19¶

Spine: Strogatz-style synthesis using Diffy Qs + simulation

Reading: Review Ch. 1-3, 8; add numerical experiments in bifurcation/chaos

Know: Recognize timescale separation, nonlinear feedback, bifurcation and chaotic sensitivity.

Reconstruct: Regenerate fixed-point stability test and nondimensionalize one two-parameter system.

Do: Simulate logistic-map or Lorenz-style sensitivity; distinguish deterministic chaos from stochastic noise empirically.

Defend: What evidence would let you distinguish model chaos from measurement noise?

Gate: Module defense: receive an unfamiliar nonlinear system and produce state variables, equilibria, local stability, simulation and limitations.

Source: source

Exit gate¶

Closed-book: 120 min: formulate 3 ODE models; solve 2 analytic cases; linearize 1 nonlinear system; classify equilibria.

Novel problem: Analyze a previously unseen coupled system with competing positive/negative feedback.

Artifact: Simulate phase portrait, parameter sweep and perturbation recovery; compare to local analysis.

Defend: Defend state choice, equilibrium meaning, local/global stability and model omissions.

Pass criterion: Pass if qualitative predictions precede simulation and match it except where explicitly revised.

Transfer problems¶

Try these before consulting solutions or asking for the complete answer.

1. Balance model: Build an ODE from stock = in - out + generation - consumption.

2. First-order: Compare exponential and logistic growth under the same initial local growth rate.

3. Oscillator: Predict damping regime from coefficients before solving/simulating.

4. Resonance: Find forcing frequency that maximizes a damped response and explain the energy mechanism.

5. State conversion: Turn a second-order ODE into first-order state space in two different coordinate choices.

6. Eigenmodes: Predict qualitative x'=Ax behavior from eigenvalues/eigenvectors.

7. Nonlinear equilibrium: Find equilibria of a two-state nonlinear system and classify local stability.

8. Basins: Construct a locally stable system with multiple attractors and show dependence on initial condition.

9. Nondimensionalization: Reduce a dimensional ODE to minimal dimensionless groups.

10. Chaos/noise: Design a numerical test attempting to distinguish deterministic sensitivity from stochastic noise.

Textbooks¶

See the five-book resource page.