Phase portrait

A phase portrait displays representative orbits in a dynamical system's state space, together with the direction of evolution and features such as equilibrium points, nullclines and invariant sets. For an autonomous differential equation, one vector field determines the motion at each state. Curves represent possible histories from different initial states, rather than successive snapshots of different physical bodies.

For $\dot{x}=v$, $\dot{v}=-{\omega }_0^{2}x$, the curves $v^2+{\omega }_0^2{x}^2=\text{constant}$ are closed ellipses. Adding damping makes mechanical energy decrease and turns the nearby oscillatory trajectories inward. The picture immediately distinguishes repeated motion from convergence even before the equations are explicitly solved. With uniqueness, distinct trajectories cannot cross at the same complete state and then continue along different futures; intersections visible in a projection can occur because hidden state coordinates have been omitted.

For $\dot{x}=\alpha x-y+y^3$, $\dot{y}=-x$, the energy $H=x^2-y^2+y^4/2$ satisfies $\dot{H}=2\alpha x^2$. At $\alpha =0$, its compact level sets give two families of closed periodic orbits around $(0,\pm 1)$ and an outer family around both; $H=0$ is a figure-eight homoclinic orbit pair through the saddle point at the origin. The two minima of $H$ are center equilibria for zero damping, stable foci for small negative $\alpha$, and unstable foci for small positive $\alpha$. For $\alpha \ne 0$, strict monotonicity along nonconstant solutions excludes periodic orbits and homoclinic orbits.

For unit-mass motion in $V(x)=3x^2-2x^3$, the mechanical energy is $E=v^2/2+V(x)$. The origin is a center equilibrium, and $(1,0)$ is a saddle equilibrium. Energies $0<E<1$ have trapped periodic orbits around the origin as well as a separate escaping component to the right. The barrier energy has a homoclinic orbit returning to the saddle after a turn at $x=-1/2$, plus two saddle branches on $x>1$. Above the barrier, particles traverse the well and eventually escape to the right.

For a planar autonomous differential equation $\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$, an $x$-nullcline is a set where $f=0$ and a $y$-nullcline is a set where $g=0$. Their intersections are equilibrium points. Away from intersections, the vector field is vertical on an $x$-nullcline and horizontal on a $y$-nullcline.

For $\dot{x}=x-y$, $\dot{y}=x+y$, the nullclines are $y=x$ and $y=-x$. On $y=x$ the vertical velocity is $2x$, and on $y=-x$ the horizontal velocity is $2x$. Signs in the four intervening sectors complete a qualitative phase portrait.

A nullcline need not be a trajectory. At $(1,1)$ in this example, the vertical vector $(0,2)$ crosses the diagonal nullcline rather than following it. Invariance of a curve requires tangency of the full vector field, not merely vanishing of one coordinate velocity.

This reflection-symmetric planar system has equilibria $(0,\pm 1)$ and, for $|a|\le 1$, $(\pm \sqrt{1-a^2},a)$. Its divergence is the constant $-2a$, excluding periodic orbits when $a\ne 0$ by the Bendixson-Dulac criterion. At $a=0$ it is Hamiltonian withand the nonzero equilibria are nonlinear centers. At $a=\pm 1$ the symmetric equilibria undergo pitchfork bifurcations.