Qualitative methods

Qualitative methods, also known as the qualitative theory of differential equations, study differential equations and systems of differential equations without solving them. Instead of formulas for the solutions, they derive the behaviour directly from the equation: the equilibria, their stability, the periodic orbits, and the changes of behaviour (bifurcations) that occur as parameters vary. The approach complements the analytical and numerical methods discussed in the article Differential equation.

The field was founded by Henri Poincaré in the 1880s, whose geometric methods grew out of the three-body problem of celestial mechanics; Aleksandr Lyapunov placed the notion of stability on a rigorous footing in 1892.

Equilibria and stability

For an autonomous equation or system dy/dt = f(y), an equilibrium (fixed point) is a point y* at which f(y*) = 0: a solution placed there never changes. An equilibrium is stable when solutions starting close enough stay close for all time, and asymptotically stable when they are drawn back to it as t → ∞; otherwise it is unstable.

For a single first-order equation y′ = f(y), stability is read from the derivative at the equilibrium: f′(y*) < 0 means a small displacement is pushed back, so the equilibrium is stable; f′(y*) > 0 means it grows, so the equilibrium is unstable. For instance, y′ = −y has the stable equilibrium 0, while y′ = y has the unstable equilibrium 0.

For systems the same question is answered by linearising about the equilibrium. Writing u = y − y* and expanding f to first order gives the linear system

\mathbf{u}'=A\mathbf{u},\qquad A=D\mathbf{f}(\mathbf{x}^{*});\qquad \operatorname{Re}(\lambda)<0\ \text{for all eigenvalues } \lambda\ \Rightarrow\ \mathbf{x}^{*}\ \text{asymptotically stable}

where A = Df(y*) is the Jacobian matrix of f at y*. In the plane the eigenvalues of A determine the local phase portrait: two negative real eigenvalues give a stable node; complex eigenvalues with negative real part give a stable spiral; real eigenvalues of opposite signs give a saddle, which is unstable; and purely imaginary eigenvalues give a centre, around which the linearised solutions oscillate. For hyperbolic equilibria, at which no eigenvalue has zero real part, the Hartman–Grobman theorem guarantees that the nonlinear system behaves near the equilibrium exactly like its linearisation.

The pendulum in the phase plane

The undamped pendulum

\frac{d^{2}\theta}{dt^{2}}+\frac{g}{L}\sin\theta=0

has equilibria wherever sin θ = 0, namely the downward position θ = 0 and the upright position θ = π. Written as a first-order system in the phase plane (θ, v),

\frac{d\theta}{dt}=v,\qquad \frac{dv}{dt}=-\frac{g}{L}\sin\theta,\qquad \text{equilibria at }(\theta,v)=(0,0)\ \text{and}\ (\pi,0)

and linearised about each equilibrium,

\theta\approx 0:\ \ \frac{d^{2}\theta}{dt^{2}}+\frac{g}{L}\theta=0;\qquad \phi=\theta-\pi\approx 0:\ \ \frac{d^{2}\phi}{dt^{2}}-\frac{g}{L}\phi=0

Near the downward position the linearised equation is the harmonic oscillator, so the phase plane is filled with the closed loops of periodic swinging: a centre. Near the upright position the linearised equation has exponential solutions, so the equilibrium is a saddle: only the two special trajectories that approach along its stable directions ever reach it, and every other nearby state falls away.

Bifurcations

As a parameter changes, the qualitative picture can change abruptly at a bifurcation value: equilibria may appear, merge or vanish. In a saddle-node bifurcation a stable and an unstable equilibrium collide and annihilate each other.

The harvested logistic equation models a population N that grows logistically and is harvested at the constant rate H:

\frac{dN}{dt}=rN\Bigl(1-\frac{N}{K}\Bigr)-H,\qquad H_{c}=\frac{rK}{4}

Its equilibria are found by setting dN/dt = 0; the discriminant of the resulting quadratic counts them:

rN\Bigl(1-\frac{N}{K}\Bigr)-H=0\ \iff\ \frac{r}{K}N^{2}-rN+H=0;\qquad \Delta=r^{2}-\frac{4rH}{K}\ \Rightarrow\ \text{two equilibria if }H<H_{c}=\frac{rK}{4},\ \text{one if }H=H_{c},\ \text{none if }H>H_{c}

For H < H_c two equilibria coexist, one attracting and one repelling. As H increases towards H_c = rK/4 they move together and merge at the saddle-node bifurcation; beyond H_c no equilibrium remains and every population declines to extinction. This is the collapse of an overharvested fishery.

Periodic orbits and limit cycles

In two dimensions a bounded solution eventually approaches an equilibrium or a periodic orbit (Poincaré–Bendixson theorem). Isolated periodic orbits are called limit cycles: nearby trajectories spiral onto them, so the system settles into self-sustained oscillation whatever the initial state. Limit cycles are intrinsically nonlinear, since a linear system oscillates with an amplitude fixed by its initial conditions rather than by the equation itself. The van der Pol oscillator

\ddot{x}-\mu\,(1-x^{2})\,\dot{x}+x=0,\qquad \mu>0

has, for every μ > 0, an unstable equilibrium at the origin and one stable limit cycle: at small amplitudes the term −μ(1 − x²) feeds energy in, and at large amplitudes it dissipates energy, so all trajectories settle onto the same periodic motion.

Chaos

For systems of three or more equations, bounded solutions need not be periodic: they can be aperiodic yet deterministic, with such sensitivity to initial conditions that nearby states separate exponentially fast. Such motion is called chaotic. Edward Lorenz found the first example in 1963 in a simple model of atmospheric convection, the system that now bears his name:

\frac{dx}{dt}=\sigma\,(y-x),\qquad \frac{dy}{dt}=x\,(\rho-z)-y,\qquad \frac{dz}{dt}=x\,y-\beta\,z

For σ = 10, ρ = 28 and β = 8/3 its solutions trace the famous butterfly-shaped chaotic attractor. Because nearby states separate exponentially, the long-term behaviour is unpredictable in practice although the equations are deterministic: this is why weather prediction beyond a limited horizon is impossible. Chaos has since been found throughout physics, biology, chemistry and economics.

Further reading

• Qualitative theory of differential equations, Wikipedia
• Phase plane, Wikipedia
• Henri Poincaré, MacTutor History of Mathematics