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Phase space is the space of all possible states of a dynamical system, with each axis representing one state variable (position, velocity, concentration, etc.). A system of first-order ODEs defines a vector field on phase space, and the evolution of the system traces out trajectories called orbits. The collection of all orbits constitutes the flow — a continuous map that advances every initial condition forward (or backward) in time, giving a global portrait of all possible behaviors.
In your earlier work on autonomous equations and phase portraits for linear systems, you learned to visualize how solutions evolve by plotting trajectories in the plane of state variables rather than against time. Nonlinear dynamics takes this idea and makes it the central organizing principle: the phase space is the arena where all dynamics play out, and understanding the geometry of trajectories in this space is the primary goal.
The formal setup is straightforward. Given a system ẋ = f(x) where x is a vector of n state variables, the function f defines a vector field — at every point in n-dimensional phase space, there is an arrow telling you the direction and speed the system moves. A trajectory starting from initial condition x₀ follows the vector field forward in time, tracing out an orbit. The existence and uniqueness theorem (assuming f is smooth enough) guarantees that exactly one trajectory passes through each point, which means orbits can never cross. This no-crossing property is profoundly constraining: in two dimensions, it implies that trajectories can only approach fixed points, closed orbits, or infinity — there are no other options.
The flow φ_t is the function that maps every initial condition to where it ends up after time t. It satisfies φ_0(x) = x (do nothing at time zero) and the group property φ_{s+t} = φ_s ∘ φ_t (evolving for time s + t is the same as evolving for t then for s). This group structure is a consequence of the system being autonomous — the rules don't change with time. The flow provides a complete description of the dynamics: if you know φ_t for all t and all initial conditions, you know everything the system can do.
What makes phase space powerful is that it converts analytical questions into geometric ones. Instead of asking "what is x(t)?", you ask "what does the flow look like?" Fixed points become dots where the vector field vanishes. Periodic orbits become closed curves. The stability of these objects becomes visible in whether nearby trajectories approach or recede. Basins of attraction become regions of phase space. Separatrices — special trajectories that form boundaries between qualitatively different behaviors — become curves or surfaces. This geometric language is what allows nonlinear dynamics to make qualitative predictions even when exact solutions are impossible, which is almost always the case for nonlinear systems.