Phase portraits of coupled systems
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Flip to reveal answersHow do you write a coupled linear system in matrix form, and where is its equilibrium?
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All 8 Flashcards — Phase portraits of coupled systems
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Question
How do you write a coupled linear system in matrix form, and where is its equilibrium?
Answer
d/dt(x, y) = M(x, y) with M = (a, b; c, d); the only equilibrium is the origin (0, 0) (when M is invertible).
Question
What do the eigenvalues of M tell you about the equilibrium?
Answer
Their nature classifies it: same-sign real → node, opposite-sign real → saddle, complex → spiral, purely imaginary → centre.
Question
Two real eigenvalues, both NEGATIVE — what type of equilibrium?
Answer
A stable node (sink): every trajectory decays to the origin because each e^(λt) → 0.
Question
Two real eigenvalues of OPPOSITE sign — what type?
Answer
A saddle point — always unstable (pulled in one direction, flung out the other).
Question
Complex eigenvalues a ± bi — stable or unstable spiral?
Answer
Spiral; stable (inward) if a < 0, unstable (outward) if a > 0. If a = 0 it is a centre (closed loops).
Question
What is special about a real eigenvector's direction in a phase portrait?
Answer
It is a straight-line trajectory through the origin; the state moves IN if its eigenvalue is negative, OUT if positive.
Question
Write the general solution of a coupled system with real eigenvalues λ₁, λ₂.
Answer
(x, y) = A·e^(λ₁t)·v₁ + B·e^(λ₂t)·v₂, with v₁, v₂ the eigenvectors and A, B set by the starting state.
Question
On the GDC, what is the calculator route for this topic?
Answer
Enter M as a matrix and use the eigenvalue/eigenvector tools; use it for Euler's method too (GDC allowed on every AI paper).
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Topic 5.17 hub
Phase portraits (HL only)
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