Questions: Phase Space and Flows

Question 1

A system has three state variables (x, y, z). Its phase space is three-dimensional. A student claims that two different trajectories in phase space can cross at a point. Under what condition is this possible?

• A. It is always possible — trajectories in higher dimensions routinely cross
• B. It is never possible for an autonomous system with a unique solution, because the crossing point would have two different velocity vectors
• C. It is possible only if the system is non-autonomous, so the vector field changes with time
• D. It is possible only at a fixed point, where the velocity is zero

Answer: (B)

For an autonomous ODE system with unique solutions (guaranteed by Lipschitz continuity), the vector field assigns exactly one velocity vector to each point in phase space. If two trajectories crossed at a point, that point would need two different futures — contradicting uniqueness. This is the existence and uniqueness theorem at work. The only apparent 'crossing' happens at fixed points, but those are single trajectories (the constant solution), not two trajectories meeting. Non-autonomous systems can have crossing projected trajectories, but in the extended phase space (including time) they still don't cross.

Question 2

The flow map φ_t satisfies φ_0(x) = x and φ_{s+t}(x) = φ_s(φ_t(x)). This means the flow forms a group under composition.

• T. True
• F. False

Answer: True

The flow of an autonomous system forms a one-parameter group: the identity is φ_0, composition satisfies φ_{s+t} = φ_s ∘ φ_t, and the inverse is φ_{-t} (running time backward). This group property reflects the time-translation symmetry of autonomous systems — the dynamics don't care when you start the clock. For non-autonomous systems, you lose this group structure and must work with a two-parameter family φ_{t,t_0} instead.

Question 3

Consider the system ẋ = y, ẏ = -x (simple harmonic oscillator). What do the orbits look like in the (x, y) phase plane?

• A. Spirals converging to the origin
• B. Closed ellipses (in this case circles) centered at the origin
• C. Straight lines through the origin
• D. Hyperbolas opening along the axes

Answer: (B)

The system ẋ = y, ẏ = -x has x² + y² = constant along every trajectory (differentiate: 2xẋ + 2yẏ = 2xy + 2y(-x) = 0). So orbits are circles centered at the origin. The origin is a center — a fixed point surrounded by closed orbits. There is no energy dissipation, so trajectories neither spiral in nor spiral out. This is the phase portrait of an undamped harmonic oscillator.

Question 4

Why is the phase space formulation more powerful than simply plotting x(t) versus t for understanding a dynamical system?

Model answer: The phase space portrait shows all possible trajectories simultaneously, reveals the geometry of the dynamics (fixed points, limit cycles, basins of attraction, separatrices), and makes qualitative behavior visible without solving the equations. A time series x(t) shows only one trajectory and hides the global structure. Phase space also makes the non-crossing property of autonomous systems geometrically apparent, which constrains what behaviors are topologically possible.

Phase space converts a dynamical question ('what does this system do?') into a geometric question ('what does the flow look like?'). You can see stability, periodicity, chaos, and bifurcations as geometric features — basins of attraction become regions, separatrices become curves, and limit cycles become closed curves that attract nearby orbits. This geometric viewpoint is the foundation of everything in nonlinear dynamics.