Autonomous planar system

An autonomous planar system is an ordinary differential equationwhose vector field does not depend explicitly on time.

The nullclines meet at the origin and $P=(2-k,k(2-k))$. At the origin, the Jacobian matrix has trace three, determinant $2-k>0$ and discriminant $1+4k>0$, giving an unstable node. At $P$, its determinant is $k-2<0$, giving a saddle equilibrium. The horizontal component is positive below the parabolic nullcline and negative above it; the vertical component is positive above the straight nullcline and negative below it. These signs determine the directional regions of the phase portrait.

The quadrant is not positively invariant: nonzero points on either axis have an outward-pointing component. Restricting a drawing to that quadrant does not keep its trajectories there. For $k>2$, only the saddle at the origin remains in the quadrant, and an exponential Dulac multiplier for a quadratic planar system excludes nonconstant periodic orbits there.

The phase plane is the two-dimensional state space of an autonomous planar system. Its oriented trajectories, equilibria, nullclines, and invariant sets display the system's qualitative dynamics.

An equilibrium point, or fixed point, of $\dot{z}=f(z)$ is a state $z_{\ast }$ with $f(z_{\ast })=0$.

A homogeneous equilibrium in a spatial evolution equation is a steady state independent of position. Its stability against homogeneous perturbations need not imply stability against spatially varying modes.

A hyperbolic equilibrium point is a source when all its eigenvalues have positive real parts. It is a sink equilibrium for the reversed flow. Every sufficiently close nonstationary trajectory leaves its neighborhood in forward time. This is distinct from an external forcing source.

A hyperbolic equilibrium point is a sink when all its eigenvalues have negative real parts. Nearby trajectories converge exponentially to it. In a planar system this includes both stable nodes and stable foci. The condition is stronger than neutral stability: a centre equilibrium is not a sink.

A three-dimensional hyperbolic equilibrium point is a saddle-focus when its eigenvalues include a nonreal pair and a real eigenvalue with the opposite sign of real part. In the displayed orientation it has a one-dimensional unstable manifold and a two-dimensional spiralling stable manifold; reversing time interchanges the stable and unstable dimensions. The ratio $-{\lambda }_s/{\lambda }_u$ controls local transverse contraction in a Shilnikov return map.

For a hyperbolic equilibrium of a smooth planar system, the eigenvalues of the Jacobian matrix determine local stability. If its determinant and trace are positive, both eigenvalues have positive real part and the equilibrium is a repeller.

A planar equilibrium point whose Jacobian matrix has nonreal conjugate eigenvalues with positive real part. Its nearby linearized trajectories rotate while their amplitude grows exponentially; the smooth nonlinear system has the same local qualitative behavior when the equilibrium is hyperbolic.

A node equilibrium with a repeated nonzero real eigenvalue. If the planar Jacobian matrix has a one-dimensional eigenspace, its linearized solutions contain a term $t{e}^{\lambda t}$ and the node is also called improper. A negative repeated eigenvalue still gives asymptotic stability; a repeated value does not by itself indicate a bifurcation.

A planar equilibrium with real nonzero eigenvalues of the same sign. Negative eigenvalues give a stable node; positive eigenvalues give an unstable node. Repeated real eigenvalues may give a degenerate or improper node, depending on the eigenspace.

A planar node whose Jacobian matrix has a repeated nonzero real eigenvalue and only one independent eigenvector is an improper node. In suitable real coordinates its linearization of a dynamical system has the displayed form, with solutionsIt is a stable node for $\lambda <0$ and an unstable node for $\lambda >0$. The factor $t$ makes generic trajectories tangent to the unique eigendirection as they approach the equilibrium in the attracting time direction. For a nonlinear autonomous system with this hyperbolic equilibrium, the sign of $\lambda$ still determines local stability; the local phase portrait has the same topological type as its linearization.

A planar equilibrium whose linearization has a complex-conjugate pair of nonreal eigenvalues with nonzero real part. Negative real part gives a stable focus; positive real part gives an unstable focus.

A planar hyperbolic equilibrium is a saddle when the Jacobian determinant is negative. It has one stable and one unstable eigendirection and corresponding one-dimensional invariant manifolds.

A center equilibrium of a planar dynamical system is surrounded by closed trajectories. It is stable but not asymptotically stable.

Let $F$ be a continuously differentiable planar vector field with an isolated equilibrium point $p$, and let the twice continuously differentiable first integral $H$ satisfy $F\cdot \nabla H=0$ and $\nabla H(p)=0$. If the Hessian matrix $D^{2}H(p)$ is positive or negative definite, then $p$ is a centre equilibrium.

For a positive-definite matrix, sufficiently short rays from $p$ have strictly increasing $H$ with radial derivative bounded below by a positive constant times distance. Every sufficiently small positive level difference therefore meets each ray exactly once, and the implicit function theorem makes these intersections a simple closed curve. These regular level sets shrink to $p$; use $-H$ in the negative-definite case. The vector field is tangent and nonzero on each such compact curve. Its speed is bounded away from zero, so a traversal takes finite time and each curve is a periodic orbit. An arbitrary strict local extremum without these regularity and nondegeneracy conditions does not by itself supply this argument.

A planar equilibrium is a stable node when its two Jacobian eigenvalues are real and negative. Every nearby trajectory approaches it, tangent asymptotically to an eigendirection.

A planar equilibrium is an unstable node when its two Jacobian eigenvalues are real and positive. Every nearby nonstationary trajectory moves away from it in forward time.

A planar equilibrium is a stable spiral when its Jacobian has a complex-conjugate pair of eigenvalues with negative real part. Nearby nonstationary trajectories spiral toward it.