Unit 12: Nonlinear Systems and Phase-Plane Analysis
Trajectories flowing in the phase plane, equilibria read straight off eigenvalues, and what linearization can and can't tell you about a nonlinear system.
9 topics · 32 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
9.1 The Phase Plane
- How it worksHow does a system's velocity field assign an arrow to every point in the phase plane, sketching out trajectories?
- How it worksHow do you tell a system's equilibrium points apart from everywhere else just by looking at the phase plane?
- Going furtherWhat happens to the phase portrait's arrows near a point where the velocity field is exactly zero?
- Worked exampleFind the equilibrium points of the system x' = y, y' = -x + x3 by setting both rates to zero.
Your turn1 of 2
Find the nonzero equilibrium x-value(s) of the system x' = y, y' = -7x + x3 (other than the origin), by setting both rates to zero.9.1 Classifying Equilibria from Eigenvalues
- How it worksHow do the sign and type of a linear system's eigenvalues sort every equilibrium into a node, saddle, spiral, or center?
- Worked exampleFind the equilibrium type at the origin of x' = Ax for A = [[-2, 1], [1, -2]] using its eigenvalues.
- Going furtherWhat happens to an equilibrium's classification as a system's eigenvalues are continuously varied from negative real parts to positive?
- Ties togetherHow does classifying an equilibrium by its eigenvalues here connect to solving the same system's equations explicitly?
Your turn1 of 2
For A = [[-2, 2.12], [2.12, -2]] (symmetric, off-diagonal entries equal), find the two eigenvalues, a+b and a-b. Report the larger one.9.1 Nodes
- How it worksHow do two real eigenvalues of the same sign force every nearby trajectory toward or away from a node?
- Worked exampleSketch the phase portrait near the origin for x' = Ax with A = [[-1, 0], [0, -4]], a node.
- Common mistakeWhy don't trajectories near a node cross the eigenvector lines even though they curve to approach them?
- Going furtherWhat happens to a node's trajectories when the two real eigenvalues are equal instead of merely the same sign?
Your turn1 of 2
For the diagonal system x' = Ax with A = [[-3, 0], [0, -3]] (a node, both eigenvalues the same sign), find the ratio of eigenvalues d/a, which controls how sharply trajectories bend toward the slower eigendirection.9.1 Saddle Points
- How it worksHow do one positive and one negative eigenvalue create the incoming and outgoing directions that define a saddle?
- Worked exampleSketch the phase portrait near the origin for x' = Ax with A = [[2, 0], [0, -3]], a saddle point.
- Common mistakeWhy isn't a saddle point considered stable even though one special direction leads straight into it?
Your turn1 of 2
For the diagonal system x' = Ax with A = [[4, 0], [0, -2]] where a > 0 and d < 0 (a saddle point), find the product of the eigenvalues, a·d (always negative for a saddle).9.1 Spirals and Centers
- How it worksHow does the sign of the real part of a complex eigenvalue pair decide whether trajectories spiral in, spiral out, or form closed loops?
- Worked exampleFind the equilibrium type of x' = Ax for A = [[-1, 2], [-2, -1]] using its complex eigenvalues.
- Going furtherWhat happens to a spiral equilibrium's trajectories as the real part of its eigenvalues shrinks toward exactly zero?
Your turn1 of 2
For A = [[-2, 2], [-2, -2]], find the modulus |lambda| = √(p2+q2) of its complex eigenvalues p plus or minus qi.9.1 Why a Center Is the Fragile, Borderline Case
- Common mistakeWhy doesn't linearizing a nonlinear system near an equilibrium reliably tell you whether that equilibrium is truly a center?
- Going furtherWhat happens to a linear system's closed orbits around a center the instant a tiny bit of damping is added?
- Ties togetherHow does a center's fragility here connect back to why undamped free vibration sits at the boundary of the three damping cases?
Your turn1 of 2
A linear system has a center with pure imaginary eigenvalues plus or minus 4i. A tiny damping term is added, shifting the eigenvalues to -0.224 plus or minus 4i. Find the new real part's magnitude, epsilon = 0.224, as a fraction of the frequency w: epsilon/w.9.2 Linearizing a Nonlinear System Near an Equilibrium
- How it worksHow does the Jacobian matrix of partial derivatives at an equilibrium build the best linear approximation to a nonlinear system there?
- Worked exampleFind the Jacobian and classify the equilibrium at (0,0) for the nonlinear system x' = x - x2 - xy, y' = 2y - xy - y2.
- Going furtherWhat happens to the accuracy of a linearization as you move farther away from the equilibrium point it was built at?
Your turn1 of 2
For the nonlinear system x' = 3x - x2 - xy, y' = 3y - xy - y2, find the Jacobian's trace at the equilibrium (0,0): a+b (since the Jacobian there is [[a,0],[0,b]]).9.5 The Predator-Prey System
- How it worksHow do the Lotka-Volterra equations turn a predator's need to eat and a prey's need to reproduce into a coupled system?
- Worked exampleFind the nonzero equilibrium of the predator-prey system dx/dt = x(3 - y), dy/dt = -y(2 - x).
- Worked exampleFind the linearization of the predator-prey system dx/dt = x(3 - y), dy/dt = -y(2 - x) at its nonzero equilibrium.
- Ties togetherHow does the predator-prey system's closed orbits around its equilibrium connect back to the center case in linear systems?
Your turn1 of 2
Find the sum of the coordinates (x+y) at the nonzero equilibrium of the predator-prey system dx/dt = x(4 - y), dy/dt = -y(4 - x).9.4 The Competing Species System
- How it worksHow do the nullclines of a competing species system carve the phase plane into regions with different long-term outcomes?
- Worked exampleFind all four equilibria of the competing species system dx/dt = x(1 - x - y), dy/dt = y(0.75 - y - 0.5x).
- Going furtherWhat happens to the competing species system's long-term outcome when the competition coefficients are lowered enough?
- Common mistakeWhy doesn't a competing species system always end with one species driving the other completely extinct?
Your turn1 of 2
For the competing species system dx/dt = x(1 - x - y), dy/dt = y(0.779 - y - 0.56x) with c not equal to 1, find the x-coordinate of the coexistence equilibrium: x = (K-1)/(c-1).