Math notes/Differential Equations/Systems and Phase Planes

THE WHOLE UNIT · ONE REFERENCE

Systems and Phase Planes
Cheat sheet.

The key rules, formulas and reminders from all 5 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Core rule

A\mathbf v=\lambda\mathbf v\implies\mathbf x(t)=e^{\lambda t}\mathbf v.

Watch for

Eigenvectors describe state directions, which may mix quantities with different physical units. Define the state and scaling before interpreting a phase-plane angle as a physical angle.

Core rule

For planar linear systems, inspect eigenvalue real parts, then draw arrows along representative trajectories.

Watch for

A center in a linear system has closed orbits and neutral stability, not asymptotic attraction. A nonlinear system with the same linearized eigenvalues can behave differently, so do not extend the classification without checking its hypotheses.

Matrix Exponentials and Repeated Modes

Core rule

\mathbf x(t)=e^{At}\mathbf x_0,

e^{(\lambda I+N)t}=e^{\lambda t}\sum_{k\ge0}\frac{t^kN^k}{k!}.

Watch for

A negative repeated eigenvalue can produce a transient polynomial factor before eventual decay. A plot over a short interval may show growth in one component even though all states eventually approach zero.

Forced Systems and Variation of Constants

Core rule

\mathbf x=e^{At}\mathbf x_0+\int_0^t e^{A(t-\tau)}\mathbf b(\tau)d\tau.

Watch for

An equilibrium can exist even if it is unstable. Long-time convergence requires stability of the homogeneous dynamics, not only successful solution of $A\mathbf x_*+\mathbf b=0$ .

Nonlinear Systems and Local Linearization

Core rule

J=D\mathbf f(\mathbf x_*),

\mathbf u'\approx J\mathbf u.

Watch for

Local stability is not a global claim about every starting point. A stable equilibrium may have a limited basin of attraction, and a linear approximation loses accuracy far from its expansion point.

Linear Systems and Eigenmodes

2 reference blocks

Read lesson ↗

Core rule

A\mathbf v=\lambda\mathbf v\implies\mathbf x(t)=e^{\lambda t}\mathbf v.

Watch for

Eigenvectors describe state directions, which may mix quantities with different physical units. Define the state and scaling before interpreting a phase-plane angle as a physical angle.

Phase Portraits and Stability

2 reference blocks

Read lesson ↗

Core rule

For planar linear systems, inspect eigenvalue real parts, then draw arrows along representative trajectories.

Watch for

Matrix Exponentials and Repeated Modes

2 reference blocks

Read lesson ↗

Core rule

\mathbf x(t)=e^{At}\mathbf x_0,

e^{(\lambda I+N)t}=e^{\lambda t}\sum_{k\ge0}\frac{t^kN^k}{k!}.

Watch for

Forced Systems and Variation of Constants

2 reference blocks

Read lesson ↗

Core rule

\mathbf x=e^{At}\mathbf x_0+\int_0^t e^{A(t-\tau)}\mathbf b(\tau)d\tau.

Watch for

An equilibrium can exist even if it is unstable. Long-time convergence requires stability of the homogeneous dynamics, not only successful solution of $A\mathbf x_*+\mathbf b=0$ .

Nonlinear Systems and Local Linearization

2 reference blocks

Read lesson ↗

Core rule

J=D\mathbf f(\mathbf x_*),

\mathbf u'\approx J\mathbf u.

Watch for

Local stability is not a global claim about every starting point. A stable equilibrium may have a limited basin of attraction, and a linear approximation loses accuracy far from its expansion point.