Phase Plane Analysis

6 PHASE PLANE

6.0 Introduction

• This chapter introduces two-dimensional nonlinear systems.
• General properties are considered.
• Classification of fixed points is built on knowledge of linear systems (Chapter 5).
• Examples include biology (competition) and physics (conservative systems, reversible systems, pendulum).
• The chapter discusses index theory, a topological method for global phase portrait information.
• Focus is mainly on fixed points; closed orbits and bifurcations are covered in subsequent chapters.

6.1 Phase Portraits

• General form of a vector field on the phase plane:

  $\dot{x} = f(x)$

  where $x = (x<em>1, x</em>2)$, $\dot{x} = (f<em>1(x), f</em>2(x))$.

• $x$ represents a point in the phase plane, $\dot{x}$ is the velocity vector at that point.

• A phase point traces out a solution $x(t)$, corresponding to a trajectory winding through the phase plane.

• The phase plane is filled with trajectories since each point can be an initial condition.

• For nonlinear systems, finding trajectories analytically is generally not possible.

• The goal is to determine qualitative behavior directly from properties of $f(x)$.

• Salient features of a phase portrait:

  1. Fixed points (where $f(x^*) = 0$), corresponding to steady states or equilibria.
  2. Closed orbits, corresponding to periodic solutions where $x(t + T) = x(t)$ for all $t$, for some T > 0.
  3. Arrangement of trajectories near fixed points and closed orbits.
  4. Stability or instability of fixed points and closed orbits.

• Numerical computation:

  • Numerical integration of $\dot{x} = f(x)$ is used.

  • Runge-Kutta method in vector form:

    $x<em>{n+1} = x</em>n + \frac{1}{6}(k<em>1 + 2k</em>2 + 2k<em>3 + k</em>4)$

    where

    $k<em>1 = f(x</em>n)$,

    $k<em>2 = f(x</em>n + \frac{\Delta t}{2}k_1)$,

    $k<em>3 = f(x</em>n + \frac{\Delta t}{2}k_2)$,

    $k<em>4 = f(x</em>n + \Delta t k_3)$.

  • Stepsize $\Delta t = 0.1$ usually provides sufficient accuracy.

• Direction field plots provide a clearer picture of flow direction.

EXAMPLE 6.1.1:

• System: $\dot{x} = 1 + e^{-y} - x$, $\dot{y} = -y$.

• Fixed point: $(x^<em>, y^</em>) = (1, 0)$.

• Stability: Unstable, based on analysis of the equations.

• Nullclines: Curves where $\dot{x} = 0$ or $\dot{y} = 0$ indicate purely horizontal or vertical flow.

  • $\dot{y} = 0$ occurs on the line $y = 0$.
  • $\dot{x} = 0$ occurs on the curve $x = 1 + e^{-y}$.

• The nullclines partition the plane into regions where $\dot{x}$ and $\dot{y}$ have various signs, giving a sense of the overall flow pattern.

• The fixed point is a nonlinear version of a saddle point.

6.2 Existence, Uniqueness, and Topological Consequences

• Existence and Uniqueness Theorem:

  • For the initial value problem $\dot{x} = f(x)$, $x(0) = x<em>0$, if $f$ is continuous and its partial derivatives $\frac{\partial f</em>i}{\partial x<em>j}$, $i, j = 1, …, n$, are continuous for $x$ in some open connected set $D \subset R^n$, then for $x</em>0 \in D$, the initial value problem has a unique solution $x(t)$ on some time interval $(-\tau, \tau)$ about $t = 0$.
  • Existence and uniqueness are guaranteed if $f$ is continuously differentiable.

• Corollary: Different trajectories never intersect.

• In two-dimensional phase spaces, these results have strong topological consequences.

• Poincaré-Bendixson theorem: If a trajectory is confined to a closed, bounded region with no fixed points, it approaches a closed orbit.

6.3 Fixed Points and Linearization

• Extends linearization technique from one-dimensional systems.

• Approximates phase portrait near a fixed point using a linear system.

• Consider the system $\dot{x} = f(x, y)$, $\dot{y} = g(x, y)$ with fixed point $(x^<em>, y^</em>)$.

• Let $u = x - x^<em>, v = y - y^</em>$ denote small disturbances from the fixed point.

• Taylor series expansion:

  $\dot{u} = \frac{\partial f}{\partial x}u + \frac{\partial f}{\partial y}v + O(u^2, v^2, uv)$

  $\dot{v} = \frac{\partial g}{\partial x}u + \frac{\partial g}{\partial y}v + O(u^2, v^2, uv)$

• Jacobian matrix at the fixed point $(x^<em>, y^</em>)$: $\begin{pmatrix} \frac{\partial f}{\partial x} &amp; \frac{\partial f}{\partial y} \ \frac{\partial g}{\partial x} &amp; \frac{\partial g}{\partial y} \end{pmatrix}$

• Linearized system:

  $\begin{pmatrix} \dot{u} \ \dot{v} \end{pmatrix} = \begin{pmatrix} \frac{\partial f}{\partial x} &amp; \frac{\partial f}{\partial y} \ \frac{\partial g}{\partial x} &amp; \frac{\partial g}{\partial y} \end{pmatrix} \begin{pmatrix} u \ v \end{pmatrix}$

• The linearized system gives a qualitatively correct picture near $(x^<em>, y^</em>)$ if the fixed point is not a borderline case (center, degenerate node, star, non-isolated fixed points).

EXAMPLE 6.3.1:

• System: $\dot{x} = x - x^3$, $\dot{y} = -2y$.

• Fixed points: $(0, 0), (1, 0), (-1, 0)$.

• Jacobian matrix: $\begin{pmatrix} 1 - 3x^2 &amp; 0 \ 0 &amp; -2 \end{pmatrix}$.

• Classification:

  • $(0, 0)$: Stable node.
  • $(1, 0), (-1, 0)$: Saddle points.

• The x and y equations are uncoupled.

• Phase portrait is symmetric in both x and y axes.

EXAMPLE 6.3.2:

• System: $\dot{x} = -y + ax(x^2 + y^2)$, $\dot{y} = x + ay(x^2 + y^2)$.

• Linearized system predicts a center at the origin for all values of $a$.

• In polar coordinates: $\dot{r} = ar^3$, $\dot{\theta} = 1$.

  • If a < 0: Stable spiral.
  • If $a = 0$: Center.
  • If a > 0: Unstable spiral.

• Centers are sensitive to small nonlinear terms.

• Stars and degenerate nodes can be altered by small nonlinearities, but their stability remains unchanged.

• Robust cases:

  • Repellers (sources): both eigenvalues have a positive real part.
  • Attractors (sinks): both eigenvalues have a negative real part.
  • Saddles: one eigenvalue is positive and one is negative.

• Marginal cases:

  • Centers: both eigenvalues are pure imaginary.
  • Higher-order and non-isolated fixed points: at least one eigenvalue is zero.

• Hyperbolic fixed points: all eigenvalues satisfy $Re(\lambda) \neq 0$. Their stability type is unaffected by small nonlinear terms.

• Nonhyperbolic fixed points are fragile.

• Hartman-Grobman theorem: the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization.

• Structural stability: a phase portrait is structurally stable if its topology is unchanged by small perturbations.

6.4 Rabbits versus Sheep

• Lotka-Volterra model of competition between two species (rabbits and sheep).

• Assumptions:

  1. Each species grows to its carrying capacity in the absence of the other (logistic growth).
  2. Conflicts reduce growth rate, more severely for rabbits.

• Model: $\dot{x} = x(3 - x - 2y)$, $\dot{y} = y(2 - y - x)$ where $x(t)$ is the population of rabbits and $y(t)$ is the population of sheep.

• Fixed points: $(0, 0), (0, 2), (3, 0), (1, 1)$.

• Jacobian matrix: $\begin{pmatrix} 3 - 2x - 2y &amp; -2x \ -y &amp; 2 - 2y - x \end{pmatrix}$.

• Classification:

  • $(0, 0)$: Unstable node.
  • $(0, 2), (3, 0)$: Stable nodes.
  • $(1, 1)$: Saddle point.

• One species generally drives the other to extinction.

• Principle of competitive exclusion: two species competing for the same limited resource typically cannot coexist.

• Basin of attraction: the set of initial conditions $x_0$ such that $x(t) \rightarrow x^*$ as $t \rightarrow \infty$.

• Separatrices: trajectories that comprise the stable manifold of the saddle; they partition phase space into regions of different long-term behavior.

6.5 Conservative Systems

• Newton’s law $F = ma$.
• Particle of mass $m$ moving along the x-axis, subject to a nonlinear force $F(x)$.
• Equation of motion: $m\ddot{x} = F(x)$.
• Potential energy: $F(x) = -\frac{dV}{dx}$.
• Energy conservation: $E = \frac{1}{2}m\dot{x}^2 + V(x)$ is constant.
• Conserved quantity: a real-valued continuous function $E(x)$ that is constant on trajectories, i.e., $\frac{dE}{dt} = 0$.
• Conservative systems cannot have any attracting fixed points.

EXAMPLE 6.5.2:

• Particle of mass $m = 1$ moving in a double-well potential $V(x) = -\frac{1}{2}x^2 + \frac{1}{4}x^4$.

• Equation of motion: $\ddot{x} = x - x^3$ or the vector field $\dot{x} = y$, $\dot{y} = -x + x^3$.

• Equilibrium points: $(0, 0), (\pm 1, 0)$.

• Jacobian matrix: $\begin{pmatrix} 0 &amp; 1 \ -1 + 3x^2 &amp; 0 \end{pmatrix}$.

• Classification:

  • $(0, 0)$: Saddle point.
  • $(\pm 1, 0)$: Centers.

• Homoclinic orbits: trajectories that start and end at the same fixed point.

• Nonlinear Centers Theorem: For the system $\dot{x} = f(x)$, where $x = (x, y) \in R^2$, if there exists a conserved quantity $E(x)$ and $x^<em>$ is an isolated fixed point which is a local minimum of $E$, then all trajectories sufficiently close to $x^</em>$ are closed.

6.6 Reversible Systems

• Many mechanical systems have time-reversal symmetry.
• System of the form $m\ddot{x} = F(x)$ is symmetric under time reversal.
• Reversible system: invariant under $t \rightarrow -t$ and $y \rightarrow -y$.
• Examples: systems of the form $\dot{x} = f(x, y)$, $\dot{y} = g(x, y)$, where $f$ is odd in $y$ and $g$ is even in $y$.
• Nonlinear Centers Theorem for Reversible Systems: If the origin $x^* = 0$ is a linear center for the continuously differentiable system $\dot{x} = f(x, y)$, and the system is reversible, then sufficiently close to the origin, all trajectories are closed curves.
• Heteroclinic trajectories or saddle connections: connect different saddle points.
• General definition of reversibility: a mapping $R(x)$ satisfies $R^2(x) = x$.

EXAMPLE 6.6.3:

• System: $\dot{x} = -x - y - cos(y)$, $\dot{y} = y - x - cos(x)$.
• Reversible, but not conservative.

6.7 Pendulum

• Equation of motion: $\ddot{\theta} + \frac{g}{L}sin(\theta) = 0$.
• Nondimensionalized equation: $\ddot{\theta} + sin(\theta) = 0$.
• Corresponding system in the phase plane: $\dot{\theta} = v$, $\dot{v} = -sin(\theta)$.
• Fixed points: $(\theta^<em>, v^</em>) = (k\pi, 0)$, where $k$ is any integer.
• Nonlinear Center Theorem implies the origin is a nonlinear center.
• Energy function: $E = \frac{1}{2}v^2 - cos(\theta)$.
• Cylindrical Phase Space: incorporating the geometric difference between linear velocity and the angle, it is more illuminating to look at the phase portrait on a cylinder.
• Damping: $\ddot{\theta} + b\dot{\theta} + sin(\theta) = 0$, where b > 0 is the damping strength.

6.8 Index Theory

• Provides global information about the phase portrait.

• Index of a Closed Curve: an integer that measures the winding of the vector field on C.

  $I<em>C = \frac{1}{2\pi} [\phi]</em>C$

• Where $\phi$ is a smooth vector field in phase plane and C is closed curve.

• Index of a Point: $\begin{aligned} I = I_C\end{aligned}$

• Properties: A closed curve C surrounds n isolated fixed points $x^<em><em>1 , … , x^</em></em>n$:

  $\begin{aligned} I<em>C = I</em>1 + I<em>2 + … + I</em>n \end{aligned}$

  Where $I<em>k$ is the index of $x^*</em>k$, for $k = 1, …, n$.

• Theorem 6.8.2: Any closed orbit in the phase plane must enclose fixed points whose indices sum to +1. Has many practical consequences in determining systems dynamics.
  Example 6.8.1-6, discuss Index theory and its applications. Integral formula for the index of a curve.