Question 1

What is phase plane analysis and why is it important?

Accepted Answer

Phase plane analysis is a graphical method for understanding solution trajectories of 2D autonomous systems dx/dt=f(x,y), dy/dt=g(x,y) in the (x,y) plane. Even for nonlinear systems that cannot be solved analytically, direction fields, nullclines, and equilibrium stability allow qualitative understanding of system behavior (periodic orbits, limit cycles, chaos, convergence). It is a fundamental analysis tool in control engineering, ecology, epidemiology, neuroscience, and structural nonlinear vibration.

Question 2

How is equilibrium stability classified?

Accepted Answer

Classified by eigenvalues λ₁, λ₂ of the Jacobian matrix at equilibrium (x*, y*): (1) both real parts < 0 → stable (node: non-oscillatory, or focus: oscillatory); (2) both real parts > 0 → unstable (node or focus); (3) one > 0, one < 0 → saddle point (always unstable); (4) purely imaginary → center (neutrally stable in linearization, requires additional analysis for nonlinear case); (5) tr < 0, det > 0 → stable focus/node.

Question 3

What is the Lotka-Volterra model?

Accepted Answer

The Lotka-Volterra equations model predator-prey population dynamics. dx/dt = αx − βxy (prey: natural growth minus predation), dy/dt = δxy − γy (predator: growth from feeding minus natural death). The non-zero equilibrium is a center and solutions trace closed curves (conserved quantity exists). It is a foundational model in ecology, and in CAE has analogies with oscillatory systems and chemical kinetics.

Question 4

What are the characteristics of the Van der Pol oscillator?

Accepted Answer

The Van der Pol oscillator: dx/dt = y, dy/dt = μ(1−x²)y − x is a nonlinear oscillator with a limit cycle for μ>0. The origin is unstable and trajectories converge to the stable limit cycle (stable closed orbit) regardless of initial conditions. It originated as a model for vacuum-tube oscillators and is also used in neuron and cardiac rhythm models. Large μ produces relaxation oscillation.

Ordinary Differential Equations — Phase Plane Analysis Tool

Stability Classification (Jacobian Eigenvalues)