Phase plane
In the study of dynamical systems, a phase plane is a two-dimensional graphical representation that depicts the behavior of solutions to a system of two ordinary differential equations, with the axes corresponding to the system's state variables and trajectories showing how solutions evolve over time without explicit dependence on the independent variable.[1] This tool is particularly useful for analyzing autonomous systems of the form \dot{x} = f(x, y), \dot{y} = g(x, y), where trajectories form curves in the plane that do not intersect due to the uniqueness of solutions guaranteed by theorems like Picard's.[2] Phase plane analysis reveals qualitative behaviors such as equilibrium points—where f(x_0, y_0) = 0 and g(x_0, y_0) = 0—which can be stable (attracting nearby trajectories), unstable (repelling them), or centers (indicating periodic motion).[1] For linear systems \vec{x}' = A \vec{x}, the phase portrait near the origin (the typical equilibrium) is classified based on the eigenvalues of A, yielding types like nodes, saddles, spirals, or centers, with stability determined by the real parts of the eigenvalues.[3] In nonlinear systems, the method extends to identify more complex features, including limit cycles—isolated closed trajectories that attract or repel nearby paths—and attractors like spirals where trajectories converge as time approaches infinity.[3] By sketching vector fields or using computational tools to plot these portraits, researchers can infer long-term system dynamics, stability, and bifurcations without solving the equations analytically, making it essential in fields like physics, engineering, and biology.[2]Fundamentals
Definition and Purpose
A phase plane is a two-dimensional graphical construct employed in the study of second-order autonomous dynamical systems, with each axis representing one of the system's state variables—typically position and velocity—and trajectories depicted as parametric curves that illustrate the system's temporal evolution without explicit time parameterization.[2] These trajectories arise from solutions to the system's governing equations, providing a visual map of how states transition from initial conditions.[1] Such systems are inherently autonomous, meaning their dynamics are governed by ordinary differential equations (ODEs) where the rates of change depend solely on the current state and not on time explicitly: \begin{align*} \frac{dx}{dt} &= f(x, y), \\ \frac{dy}{dt} &= g(x, y), \end{align*} with f and g as smooth functions of the state variables x and y.[4] This autonomy ensures that the phase plane captures invariant behavioral patterns, independent of the starting time.[5] The purpose of the phase plane lies in facilitating qualitative analysis of dynamical behavior, bypassing the often intractable task of obtaining closed-form solutions to the ODEs; instead, it reveals structural properties like the location and stability of equilibrium points (where \frac{dx}{dt} = 0 and \frac{dy}{dt} = 0), the presence of attractors drawing nearby trajectories inward, repellors pushing them outward, and periodic orbits forming closed loops indicative of sustained oscillations.[6] This approach is particularly valuable for understanding long-term system tendencies and global flow patterns in the state space.[7] The phase plane concept originated in the late 19th century within Henri Poincaré's foundational work on the qualitative theory of differential equations, where he introduced geometric methods to describe trajectory arrangements in planar systems without relying on explicit integration.[8]Representation of Trajectories
In the phase plane, trajectories for a general two-dimensional autonomous system \dot{x} = f(x, y), \dot{y} = g(x, y) are represented as solution curves (x(t), y(t)) parameterized by time t, tracing the evolution of the state variables from initial conditions.[2] These curves illustrate the qualitative behavior of the system, with the direction of motion at each point given by the vector field (f(x, y), g(x, y)), which indicates both the tangent direction and the instantaneous velocity.[9] The magnitude of this velocity vector, \sqrt{f(x, y)^2 + g(x, y)^2}, represents the speed along the trajectory, allowing time to be inferred by integrating the arc length divided by this speed, though phase portraits often emphasize geometric flow over explicit temporal scaling.[9] To construct trajectories graphically, a slope field (or direction field) is commonly used, where a grid of points in the phase plane is overlaid with short line segments whose slopes approximate \frac{dy}{dx} = \frac{g(x, y)}{f(x, y)}, provided f(x, y) \neq 0.[6] These segments visualize the local direction of the vector field, enabling trajectories to be sketched by connecting aligned segments while following the indicated directions; numerical integration or software tools can refine this for precise curves.[2] Equilibrium points, where \dot{x} = 0 and \dot{y} = 0 (i.e., f(x, y) = 0 and g(x, y) = 0), appear as singular points in the field with no defined direction, serving as fixed locations where trajectories may converge or diverge.[9] Trajectories exhibit key properties rooted in the theory of ordinary differential equations: by the existence and uniqueness theorem for Lipschitz-continuous right-hand sides, solutions are unique for given initial conditions, ensuring that trajectories do not intersect or cross each other in the phase plane.[9] This non-intersection preserves the topological structure of the flow, with trajectories forming a foliation of the plane except at equilibria.[2] Isoclines provide an auxiliary tool for sketching, consisting of curves where the slope \frac{g(x, y)}{f(x, y)} = k is constant for some fixed k, partitioning the plane into regions of uniform direction.[6] Along each isocline, parallel line segments are drawn with slope k, facilitating the approximation of the slope field and the connection of trajectories across these loci.[6]Linear Dynamical Systems
Eigenvalue Decomposition
In the analysis of linear dynamical systems within the phase plane, the fundamental equation is the autonomous system \dot{x} = Ax, where x \in \mathbb{R}^2 represents the state vector, A is a constant 2×2 matrix, and the origin serves as the equilibrium point.[10] The general solution to this system is given by x(t) = e^{At} x(0), where e^{At} is the matrix exponential, which encapsulates the evolution of initial conditions x(0) over time.[11] To compute this solution explicitly, eigenvalue decomposition is employed by solving the characteristic equation \det(A - \lambda I) = 0, which yields the eigenvalues \lambda_1 and \lambda_2.[10] For distinct real eigenvalues, the corresponding eigenvectors v_1 and v_2 are found by solving (A - \lambda_i I)v_i = 0 for i=1,2, allowing the matrix A to be diagonalized as A = P \Lambda P^{-1}, with \Lambda = \diag(\lambda_1, \lambda_2) and P = [v_1 \, v_2].[11] The general solution then takes the form x(t) = c_1 v_1 e^{\lambda_1 t} + c_2 v_2 e^{\lambda_2 t}, where c_1 and c_2 are constants determined by initial conditions.[10] In the phase plane, this decomposition reveals the structure of trajectories: solutions along the eigenvector directions v_1 and v_2 are straight lines emanating from or approaching the origin, depending on the signs of \lambda_1 and \lambda_2.[11] General trajectories, as linear combinations of these eigensolutions, curve in the plane, with their qualitative behavior governed by the eigenvalue signs—negative values draw paths toward the origin, while positive values repel them away.[10] Degenerate cases arise when eigenvalues are repeated or complex, requiring modifications such as generalized eigenvectors for repeated roots or real and imaginary parts for complex conjugates, though full details on these are addressed in subsequent classifications.[11]Classification of Equilibrium Points
In linear two-dimensional dynamical systems of the form \dot{\mathbf{x}} = A \mathbf{x}, where A is a constant $2 \times 2 matrix, the origin is the sole equilibrium point, as homogeneous linear systems possess no other fixed points.[12] The qualitative behavior of trajectories in the phase plane near this equilibrium is determined by the eigenvalues of A, which dictate the type and stability of the fixed point.[1] Stability is assessed by the real parts of the eigenvalues: the equilibrium is asymptotically stable if all real parts are negative, unstable if any are positive, and neutrally stable (but not asymptotically) if all real parts are zero. Linear systems exhibit no limit cycles, as trajectories either converge to, diverge from, or orbit the origin without periodic attractors disconnected from it. For real and distinct eigenvalues \lambda_1 < \lambda_2, the phase portrait depends on their signs. If both are positive (\lambda_1 > 0), the equilibrium is an unstable node (source), where trajectories radiate outward from the origin along the directions of the corresponding eigenvectors, diverging to infinity.[13] Conversely, if both are negative (\lambda_2 < 0), it forms a stable node (sink), with trajectories converging toward the origin, typically approaching faster along the eigenvector for the more negative eigenvalue.[1] When the eigenvalues have opposite signs, the equilibrium is a saddle point, which is unstable; trajectories along the eigenvector for the negative eigenvalue approach the origin (stable manifold), while those along the positive one diverge (unstable manifold), creating hyperbolic paths that separate regions of inflow and outflow.[12] Repeated real eigenvalues \lambda occur when the characteristic polynomial has a double root, leading to two subcases based on the geometric multiplicity. If the eigenspace is two-dimensional (i.e., A is a scalar multiple of the identity matrix), the equilibrium is a proper node (or star node); trajectories approach or recede radially from the origin along all directions if \lambda < 0 (stable) or \lambda > 0 (unstable), respectively.[14] In the defective case, with only one independent eigenvector (Jordan canonical form), it is an improper node; stability follows the sign of \lambda (stable if \lambda < 0, unstable if \lambda > 0), but trajectories curve toward or away from the origin, aligning asymptotically with the single eigenvector direction rather than radiating uniformly.[15] Complex conjugate eigenvalues \alpha \pm i\beta with \beta \neq 0 produce rotational behavior in the phase plane. If \alpha < 0, the equilibrium is a stable spiral (spiral sink), where trajectories spiral inward toward the origin, combining decay with oscillation.[12] For \alpha > 0, it is an unstable spiral (spiral source), with trajectories spiraling outward.[1] When \alpha = 0 (purely imaginary eigenvalues), the equilibrium is a center, neutrally stable, featuring closed elliptical orbits around the origin that neither grow nor decay, corresponding to periodic solutions. The following table summarizes the classifications for the origin in linear phase planes:| Eigenvalue Type | Subtype/Behavior | Stability | Phase Plane Description |
|---|---|---|---|
| Real, distinct, both >0 | Unstable node (source) | Unstable | Trajectories diverge along eigenvectors. |
| Real, distinct, both <0 | Stable node (sink) | Asymptotically stable | Trajectories converge along eigenvectors. |
| Real, distinct, opposite signs | Saddle | Unstable | Hyperbolic trajectories; inflow on one axis, outflow on the other. |
| Real, repeated, full eigenspace | Proper node (star) | Stable if \lambda < 0; unstable if \lambda > 0 | Radial approach/recession from all directions. |
| Real, repeated, defective | Improper node | Stable if \lambda < 0; unstable if \lambda > 0 | Curved trajectories aligning with single eigenvector. |
| Complex, \alpha < 0, \beta \neq 0 | Stable spiral (sink) | Asymptotically stable | Inward spiraling orbits. |
| Complex, \alpha > 0, \beta \neq 0 | Unstable spiral (source) | Unstable | Outward spiraling orbits. |
| Purely imaginary (\alpha = 0) | Center | Neutrally stable | Closed elliptical orbits. |