Phase portrait
A phase portrait is a graphical representation of the trajectories of solutions to a system of differential equations in the phase space, typically the plane for two-dimensional autonomous systems, illustrating the qualitative behavior of the dynamical system without explicit time dependence.[1] It depicts how solutions evolve from various initial conditions, highlighting key features such as equilibrium points, separatrices, and limit cycles.[2] The concept of the phase portrait originated with the work of French mathematician Henri Poincaré in the late 19th century, who introduced it as a tool for qualitative analysis in his 1885 paper "Sur les courbes définies par une équation différentielle," shifting focus from quantitative solutions to the geometric organization of all trajectories in state space.[3] Poincaré's approach, further developed in his seminal Les Méthodes Nouvelles de la Mécanique Céleste (1892–1899), emphasized visualizing the flow of vector fields to understand complex dynamics, such as those in celestial mechanics.[3] In construction, a phase portrait is built by plotting the vector field defined by the system's equations—such as \dot{x} = f(x, y) and \dot{y} = g(x, y)—on a grid in the phase plane, then sketching trajectories that follow the direction and magnitude of these vectors, often using numerical integration for accuracy.[2] Equilibrium points, where the vector field vanishes (f(x_e, y_e) = g(x_e, y_e) = 0), serve as fixed points around which trajectories converge (sinks), diverge (sources), or oscillate (centers or spirals).[1] These portraits reveal global stability and bifurcation behaviors, making them essential for analyzing nonlinear systems in fields like physics, biology, and engineering.[4]Fundamentals
Definition
A phase portrait is a graphical representation of the trajectories, or solution curves, of a dynamical system plotted in its phase space, illustrating the qualitative behavior of all possible solutions.[5] This visualization captures the flow of the system by showing how states evolve without explicitly parameterizing time, providing insight into long-term dynamics such as convergence, divergence, or periodic motion.[1] Dynamical systems, the foundation for phase portraits, are typically modeled by autonomous systems of ordinary differential equations (ODEs) of the form \dot{x} = f(x), where x is the state vector representing the system's variables and f defines the evolution rule.[6] The concept of phase portraits emerged in the late 19th century through the pioneering work of Henri Poincaré on the qualitative analysis of ODEs, particularly in his studies of periodic orbits and stability in celestial mechanics.[3] Poincaré's contributions laid the groundwork for understanding global system behavior beyond explicit solutions.[6] In contrast to time series plots, which depict state variables evolving explicitly against time, phase portraits focus solely on the interdependencies among state variables by omitting the time axis, thereby revealing invariant structures like attractors and separatrices more clearly.[7]Phase Space
Phase space is the multidimensional geometric space that encapsulates all possible states of a dynamical system, with each axis representing one of the system's state variables.[8] In mechanical systems, for instance, these state variables typically include position and velocity (or momentum), allowing the full configuration of the system to be represented as a point in this space.[9] This structure provides a coordinate framework independent of time, enabling the analysis of the system's evolution through trajectories plotted within it.[10] For a system governed by n first-order ordinary differential equations, the phase space is the n-dimensional real vector space \mathbb{R}^n, where each point uniquely specifies the values of all state variables at a given instant.[11] The dimensionality of the phase space thus matches the number of independent variables needed to fully describe the system's state, reflecting the degrees of freedom inherent to the dynamics.[8] In one-dimensional cases, the phase space reduces to the real line \mathbb{R}, often visualized using a phase line diagram, which indicates the direction of flow based on the sign of f(x).[12] Two-dimensional phase spaces are common for planar systems involving two variables, such as position and velocity in simple oscillators. For higher-dimensional systems, direct visualization becomes challenging, necessitating projections onto lower-dimensional subspaces to capture essential geometric features.[8] Invariant sets within phase space, such as attractors and separatrices, are subsets that are preserved under the flow of the dynamics, maintaining their structure as the system evolves.[13] These sets delineate regions of qualitatively distinct behavior and are fundamental to understanding long-term dynamics without relying on specific trajectories.Construction Methods
For Autonomous Systems
Autonomous systems are governed by ordinary differential equations of the form \dot{\mathbf{x}} = f(\mathbf{x}), where \mathbf{x} is the state vector and f does not explicitly depend on time t. These systems produce flows in phase space that are time-independent, allowing trajectories to be visualized without reference to specific time scales.[1] In two dimensions, the system takes the form \dot{x} = P(x,y), \dot{y} = Q(x,y), where P and Q are smooth functions.[14] The phase portrait is constructed by first evaluating the vector field (P(x,y), Q(x,y)) on a discrete grid of points spanning the region of interest in the xy-plane.[15] At each grid point (x_i, y_i), a short arrow is drawn starting from that point, with direction and magnitude proportional to the vector (P(x_i, y_i), Q(x_i, y_i)); the length of the arrows is often normalized for clarity to emphasize direction over speed.[15] This direction field provides a qualitative overview of the system's behavior, showing how solutions tend to move locally.[1] To add trajectories, numerical integration is performed by solving the initial value problem starting from multiple initial conditions distributed across the grid, typically using explicit methods such as the fourth-order Runge-Kutta scheme.[16] The resulting solution curves, parameterized by time and oriented with arrows indicating increasing t, are overlaid on the direction field to form the full phase portrait; trajectories are often sampled over a finite time interval to avoid computational overflow.[1] Finally, invariant manifolds—such as separatrices connecting fixed points—may be identified by tracing special trajectories that remain confined to lower-dimensional subsets of the phase space, often requiring higher precision in integration near equilibria. Software tools facilitate this process: in MATLAB, the Symbolic Math Toolbox or ODE solvers likeode45 (an adaptive Runge-Kutta method) can compute and plot trajectories directly.[17] Similarly, Python libraries such as Matplotlib for visualization and SciPy's solve_ivp function, which implements RK45 by default, enable efficient numerical integration and phase portrait generation for autonomous systems.[16] These tools automate grid evaluation and trajectory plotting, making construction accessible for higher-dimensional or nonlinear cases while preserving the qualitative insights of hand-sketched portraits.[18]
For Non-Autonomous Systems
Non-autonomous systems are described by ordinary differential equations (ODEs) of the form \dot{x} = f(x, t), where the right-hand side explicitly depends on time t, distinguishing them from autonomous systems where the dynamics are time-independent.[19] Unlike autonomous cases, trajectories in non-autonomous systems cannot form closed orbits in the standard phase space because time continuously increases, making the flow irreversible and preventing periodic behavior without accounting for the temporal dimension.[20] To construct phase portraits, the phase space must be extended to include time as an additional coordinate, transforming the system into (x, t) in \mathbb{R}^{n+1}, where the extended equations become \dot{x} = f(x, t) and \dot{t} = 1.[19] In the extended phase space, trajectories are plotted as curves parameterized by time, revealing the evolution of the system in this higher-dimensional space; for periodically forced systems with period T, the space can be compactified into a cylinder by identifying t modulo T, simplifying visualization while preserving the dynamics.[20] An alternative construction method involves stroboscopic maps, which sample the state x at discrete, fixed time intervals (e.g., multiples of T), producing a discrete dynamical system that approximates the continuous flow and allows for phase portrait-like representations in the original n-dimensional space.[19] These maps are particularly useful for analyzing long-term behavior, such as attractors, in systems where full higher-dimensional plots are impractical. A key challenge in non-autonomous phase portraits arises from the non-reversible nature of the flow, which precludes closed orbits even in two-dimensional state spaces extended by time, as trajectories cannot loop back due to the monotonic increase in t.[20] For example, in forced oscillators like the driven van der Pol equation \ddot{x} + \epsilon (x^2 - 1) \dot{x} + x = \epsilon a \sin(\Omega t), the periodic forcing introduces quasi-periodic or chaotic trajectories that spiral or fill regions in the extended space without closing.[19] To address dimensionality and reveal underlying structures, Poincaré sections serve as a reduction technique: trajectories are intersected with a hypersurface transverse to the flow (e.g., at fixed t \mod T), yielding a lower-dimensional map whose points trace the return map, facilitating the identification of periodic orbits, stability, and chaos.[20] This method, originally developed for periodic systems, transforms the continuous non-autonomous flow into a discrete portrait analogous to that of an autonomous system.[19]Types and Features
Nullclines and Trajectories
In phase portraits of two-dimensional autonomous dynamical systems, nullclines are curves in the phase plane where the time derivative of one variable vanishes. For a system \dot{x} = f(x, y), \dot{y} = g(x, y), the x-nullcline consists of points where f(x, y) = 0, so \dot{x} = 0, and the y-nullcline is where g(x, y) = 0, so \dot{y} = 0.[15] Along the x-nullcline, the vector field is vertical, pointing upward if \dot{y} > 0 or downward if \dot{y} < 0, while on the y-nullcline, it is horizontal, pointing right if \dot{x} > 0 or left if \dot{x} < 0.[15] The intersections of these nullclines identify fixed points, where both \dot{x} = 0 and \dot{y} = 0.[15] Nullclines divide the phase plane into regions where the signs of \dot{x} and \dot{y} are constant, aiding in sketching the overall flow without solving the system explicitly.[21] Trajectories, or solution curves, are the integral curves of the vector field that represent the paths traced by solutions (x(t), y(t)) in the phase plane as time t varies.[1] Under standard assumptions, such as the right-hand side being locally Lipschitz continuous, the Picard–Lindelöf theorem guarantees the local existence and uniqueness of solutions for initial value problems, implying that through each point in the phase space, there passes exactly one trajectory.[22] This uniqueness ensures that trajectories cannot intersect or cross each other in the phase plane, as such an intersection would violate the distinct evolution of solutions from different initial conditions.[1] Trajectories are parameterized by time, providing a qualitative picture of the system's dynamics, often plotted without explicit time labels to emphasize the geometric structure.[23] The direction of flow along trajectories is determined by the sign of the vector field components: in regions where f(x, y) > 0, x increases (flow to the right), while f(x, y) < 0 indicates decrease (flow to the left); similarly for g(x, y) and the vertical direction.[1] Arrows are typically drawn tangent to the trajectories to indicate the orientation of increasing time, revealing whether solutions approach or depart from certain regions.[24] This directional information, combined with nullclines, allows for the qualitative construction of the phase portrait by integrating the flow across regions.[21] Among trajectories, special cases include heteroclinic and homoclinic orbits, which connect fixed points and play key roles in global dynamics. A heteroclinic orbit is a trajectory that asymptotically approaches one fixed point as t \to -\infty and a different fixed point as t \to +\infty.[25] In contrast, a homoclinic orbit connects a single fixed point to itself, approaching it as t \to \pm \infty.[25] Limit cycles are another important type: isolated closed trajectories that correspond to periodic solutions, where nearby trajectories spiral toward (stable limit cycle) or away from (unstable limit cycle) the cycle, indicating sustained oscillations in the system.[2] These orbits often form separatrices in the phase portrait, delineating basins of attraction or enabling complex behaviors like chaos in perturbed systems.[23]Fixed Points and Stability
In dynamical systems, fixed points, also known as equilibrium points, are solutions where the state vector remains constant over time, satisfying \dot{\mathbf{x}} = 0 or equivalently f(\mathbf{x}^*) = 0 for the vector field f.[26] These points represent the locations in phase space where trajectories may converge, diverge, or remain stationary, providing critical insights into the system's long-term behavior without requiring full trajectory integration.[26] To assess local stability near a fixed point \mathbf{x}^*, the system is linearized using the Jacobian matrix Df(\mathbf{x}^*), which approximates the nonlinear dynamics via the first-order Taylor expansion.[27] The eigenvalues \lambda_i of this matrix determine the stability type: all real parts negative indicate a stable sink (attracting trajectories), all positive a source (repelling), mixed signs a saddle (unstable with stable and unstable manifolds), and purely imaginary a center (periodic orbits, neutrally stable in linear case).[27] This linearization theorem, under hyperbolicity (no zero real parts), ensures the nonlinear phase portrait qualitatively matches the linear one locally via topological conjugacy, as established by the Hartman-Grobman theorem.[27] For two-dimensional systems, stability is classified using the trace \tau = \lambda_1 + \lambda_2 and determinant \Delta = \lambda_1 \lambda_2 of the Jacobian, where the fixed point is asymptotically stable if \tau < 0 and \Delta > 0 (both eigenvalues have negative real parts), unstable if \tau > 0 and \Delta > 0 or \Delta < 0, and requires further analysis if \tau = 0 or \Delta = 0 (degenerate cases like nodes, foci, or lines of equilibria).[14] \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc = \Delta, \quad a + d = \tau This trace-determinant plane divides into regions: for \Delta > 0, spirals if \tau^2 < 4\Delta (complex eigenvalues), nodes if \tau^2 > 4\Delta (real eigenvalues of the same sign), and degenerate cases on the parabola \tau^2 = 4\Delta; for \Delta < 0, saddles (real distinct eigenvalues of opposite signs).[14] Fixed points often occur at nullcline intersections, aiding their identification in phase portraits. Global stability extends local analysis by confirming attraction from the entire phase space or a basin, often using Lyapunov functions V(\mathbf{x})—positive definite functions with negative definite time derivatives \dot{V} < 0 along trajectories—proving asymptotic stability if V is radially unbounded.[28] Alternatively, index theory, via the Poincaré index (winding number of the vector field around a closed curve enclosing the fixed point), provides topological constraints: the sum of indices over all fixed points equals the Euler characteristic (1 for the plane), enabling global portrait reconstruction and ruling out certain configurations without limit cycles if indices mismatch.[29] In phase portraits, global stability manifests as all trajectories converging to the fixed point, contrasting local behavior near isolated equilibria.Examples
Linear Systems
Linear systems of ordinary differential equations (ODEs) are typically expressed in the form \dot{\mathbf{x}} = A \mathbf{x}, where \mathbf{x} is the state vector, A is a constant n \times n matrix, and the dot denotes time derivative.[30] For two-dimensional systems (n=2), phase portraits provide a visual representation of the trajectories in the phase plane, determined by the eigenvalues of A. These eigenvalues dictate the qualitative behavior near the origin, which is the sole fixed point at \mathbf{x} = \mathbf{0}.[30] The classification of phase portraits for 2D linear systems relies on the eigenvalues \lambda_1, \lambda_2 of A:- Nodes: Real eigenvalues of the same sign. If both negative (\lambda_1 < \lambda_2 < 0), trajectories approach the origin along eigenlines, forming a stable node; if both positive ($0 < \lambda_1 < \lambda_2), they diverge, yielding an unstable node.[30]
- Saddles: Real eigenvalues of opposite signs (\lambda_1 < 0 < \lambda_2). Trajectories approach along the stable eigenline and diverge along the unstable one, creating hyperbolic paths.[30]
- Spirals (or foci): Complex conjugate eigenvalues \lambda = \alpha \pm i\beta with \beta \neq 0. If \alpha < 0, spirals inward to the origin (stable spiral); if \alpha > 0, outward (unstable spiral).[30]
- Centers: Purely imaginary eigenvalues \lambda = \pm i\beta (\alpha = 0). Trajectories form closed elliptical orbits around the origin, indicating neutral stability with periodic motion.[30]