Bifurcation
Atlas

The most common way equilibria are created or destroyed. It is what you see when the system has no special structure.

• μ < 0: no equilibria; every trajectory drifts to −∞.
• μ > 0: two equilibria x* = ±√μ. The upper one is stable, the lower one unstable.
• μ = 0: they collide in one half-stable point and annihilate.

Just past the bifurcation (μ slightly negative) a ghost remains: trajectories crawl through the bottleneck near x = 0 for a time of about π/√|μ|.

Onset of repetitive firing in class-I neurons; the outbreak threshold in the spruce budworm model; tangent bifurcations of periodic windows.

Appears when one equilibrium is forced to exist for every μ, usually the "nothing there" state x = 0.

• The equilibria x* = 0 and x* = μ exist on both sides of the bifurcation.
• At μ = 0 they pass through each other and exchange stability: 0 is stable for μ < 0, μ is stable for μ > 0.

Nothing appears or disappears; the two equilibria simply trade roles.

The laser threshold; the epidemic threshold R₀ = 1; the logistic map at r = 1.

When the system looks the same after x → −x, the right-hand side is odd in x: the quadratic term is ruled out and the cubic term decides. x = 0 is an equilibrium for every μ.

• Supercritical (−x³): for μ > 0 the origin loses stability and two stable states ±√μ branch off. The transition is soft and continuous.
• Subcritical (+x³): two unstable states ±√−μ exist for μ < 0 and shrink onto the origin. Past μ = 0 the state jumps away; with a stabilising −x⁵ term you get hysteresis.

Breaking the symmetry with a small constant term unfolds the pitchfork into a saddle-node plus a disconnected branch (the imperfect bifurcation, a slice of the cusp).

Buckling of an Euler strut; ferromagnet below the Curie point; convection-roll direction in Rayleigh–Bénard.

In the plane, a spiral equilibrium can switch from spiralling in to spiralling out. In polar coordinates the radius follows the normal form above while the angle turns at a steady speed ω; the figure uses ω = 1 and ℓ₁ = ∓1. The sign of the cubic coefficient ℓ₁ decides the type.

• Supercritical (ℓ₁ < 0): a stable limit cycle of radius √(−μ/ℓ₁) is born for μ > 0. Its size grows like √μ and its period starts near 2π/ω.
• Subcritical (ℓ₁ > 0): an unstable cycle surrounds the stable focus for μ < 0 and shrinks onto it. Past μ = 0 the state jumps to a distant attractor.

Unlike the steady-state cases, the new object is a periodic orbit, not an equilibrium, so the diagram plots the cycle's radius.

Onset of oscillation in the Brusselator and other chemical clocks; aeroelastic flutter; the van der Pol oscillator as its damping changes sign.

• μ > 0: fixed points x* = ±√μ. The upper is stable, the lower unstable.
• μ = 0: the graph of f touches the diagonal (tangency) and the pair merges.
• μ < 0: no fixed points; orbits squeeze through the narrow channel between the graph and the diagonal.

The channel is the mechanism of type-I intermittency: long laminar phases whose mean length scales like 1/√|μ|.

The period-3 window of the logistic map opens by a saddle-node of f³ at r = 1 + √8.

• Fixed points x* = 0 and x* = μ exist for every μ.
• At μ = 0 they pass through each other and swap stability.

The logistic map x ↦ rx(1 − x) at r = 1: extinction x = 0 hands stability to x = 1 − 1/r. Writing r = 1 + μ gives this normal form after rescaling.

• For an odd map f(−x) = −f(x), x* = 0 is a fixed point for every μ, and it loses stability at μ = 0.
• Supercritical (−x³): stable fixed points ±√μ appear for μ > 0.
• Subcritical (+x³): unstable fixed points ±√−μ exist for μ < 0 and merge into the origin.

Poincaré maps of symmetric flows, such as a symmetric periodic orbit splitting into a pair of mirror-image orbits.

Has no counterpart for one-dimensional flows: near x* the orbit jumps from one side to the other on every step. At μ = 0 a period-2 orbit {a, −a} branches off, and the orbit visits its two points in turn.

• Supercritical (+x³ here): a stable 2-cycle a = ±√μ is born for μ > 0.
• Subcritical (−x³ here): an unstable 2-cycle ±√−μ exists for μ < 0 and collapses onto x*.

The 2-cycle points are drawn as a branch in the diagram. In the cobweb, the path settles onto a square that bounces between them.

The logistic map at r = 3, the first step of the Feigenbaum cascade; as the Poincaré map of a flow it is the period doubling of a limit cycle.

The fixed point switches from pulling orbits in along a spiral to pushing them out. Instead of a periodic orbit, an invariant closed curve of radius about √μ appears around the fixed point. The figure uses θ₀ = 0.5.

• Supercritical: a stable invariant circle for μ > 0.
• Subcritical: an unstable circle for μ < 0 that shrinks onto the fixed point.
• Nondegeneracy: the turn θ₀ per step must not be p/q of a full circle with q ≤ 4. Those cases are the strong resonances.

On the circle the dynamics is like a circle map: rotation can be quasi-periodic (orbits fill the curve) or phase-locked (a periodic orbit on it), giving Arnold tongues. As the Poincaré map of a flow, this is the birth of a torus from a limit cycle.

The delayed logistic map; periodically forced oscillators; the torus step of the Ruelle–Takens route.

A stable cycle and an unstable cycle move towards each other, merge and vanish. It is the saddle-node of A.1.a, but for limit cycles.

• μ < −1/4: only the resting state at the origin.
• −1/4 < μ < 0: rest and a large stable cycle coexist. The unstable cycle between them is the threshold that decides where an orbit goes.
• μ > 0: the inner cycle has shrunk onto the origin (a subcritical Hopf), so only the large cycle is left.

Together with the subcritical Hopf this gives hysteresis: raising μ past 0 starts large oscillations at once; lowering μ again keeps them going until μ = −1/4.

The Hodgkin–Huxley neuron, where resting and repetitive firing coexist; flutter that starts and stops at different flow speeds.

A limit cycle grows until it touches a saddle point. At that moment the cycle becomes a homoclinic orbit: a path that leaves the saddle and comes back to it. A moment later the cycle is gone.

• At μ = −1 a Hopf bifurcation creates a small stable cycle around (1, 0).
• As μ rises the cycle grows and passes closer and closer to the saddle at the origin, where motion is very slow. Its period grows without bound.
• At μ_c ≈ −0.8645 the saddle's outgoing path (red) closes onto its incoming path (teal). Past μ_c orbits escape.

The amplitude stays finite while the period blows up. That is the fingerprint that tells a homoclinic ending from a Hopf ending, where the amplitude shrinks to zero at a finite period.

Strogatz, Example 8.4.3; spiking neurons that stop through a saddle loop; the homoclinic explosion of the Lorenz system at r ≈ 13.93.

A saddle-node that happens on a closed curve. When the two equilibria merge and vanish, nothing blocks the way round any more, and the motion becomes a periodic orbit around the whole circle.

• ω < 1: a stable rest point θ* = arcsin ω and an unstable one at π − arcsin ω. Every orbit comes to rest.
• ω = 1: they merge at θ = π/2.
• ω > 1: endless rotation. The orbit still crawls through the bottleneck near π/2, so the period starts infinitely long.

Compare with Hopf: there the oscillation is born with zero amplitude and a finite period; here it is born with full amplitude and an infinite period.

The theta neuron and class-I excitability; an overdamped pendulum under constant torque; Josephson junctions; Adler's equation for phase locking of fireflies.

Two saddle-node curves meet in a sharp point at the origin of the (r, h) plane.

• Inside the wedge: three equilibria, two stable and one unstable between them. The system is bistable.
• Outside: one stable equilibrium.
• Sweeping h at fixed r > 0 gives hysteresis: the state jumps at one fold and jumps back only at the other.

Along h = 0 the pitchfork of A.1.c reappears. The cusp is what the pitchfork becomes when its symmetry is broken, which is why a real pitchfork is always slightly imperfect.

Zeeman's catastrophe machine; outbreaks in the spruce budworm model; hysteresis loops in magnets and buckling.

The point where a saddle-node curve touches a Hopf curve. Three curves leave it, and the first three kinds of event in this atlas are tied together here.

• Above the saddle-node curve (β₁ > β₂²/4): no equilibria.
• Below it: a saddle and a second equilibrium. Crossing the Hopf line (β₁ = 0, β₂ < 0) turns that equilibrium unstable and a stable cycle appears.
• The cycle grows until it hits the saddle on the homoclinic curve, β₁ ≈ −(6/25)β₂² near the origin. There it disappears, exactly as in C.2.

The shaded strip between the Hopf and homoclinic curves is where the cycle lives. The homoclinic curve is drawn from a numerical computation, because the formula is only exact close to the origin.

Predator–prey models with an Allee effect; neuron models switching between resting and spiking; lasers with injected signal.

Also called the generalised Hopf point: where a Hopf bifurcation switches from supercritical to subcritical.

• On the line β₁ = 0 a Hopf bifurcation happens. It is supercritical for β₂ < 0 and subcritical for β₂ > 0.
• A fold-of-cycles curve β₁ = −β₂²/4 leaves the Bautin point on the subcritical side.
• Between that curve and the Hopf line, rest and a large stable cycle coexist, separated by an unstable cycle.

This is the parameter-plane view of C.1. Its slice at β₂ = 1 is exactly the fold-of-cycles normal form, with its hysteresis loop.

Neurons that switch between rest and repetitive firing with hysteresis; the onset of oscillation in some chemical reactors.

The period doubling of B.2 happens again and again: period 1 becomes 2 at r₁ = 3, then 4 at r₂ ≈ 3.4495, 8 at r₃ ≈ 3.5441, and so on. The doublings pile up at r_∞ ≈ 3.5699.

• Each gap is about δ ≈ 4.6692 times smaller than the one before (Feigenbaum's constant).
• δ is the same for every smooth map with a single quadratic hump. This universality is why the number shows up in experiments.
• Past r_∞ the motion is chaotic, interrupted by periodic windows.

The cobweb on the right draws the transient faintly and the last 64 steps brightly, so you can count the period directly.

Rayleigh–Bénard convection (Libchaber's experiments); a dripping faucet; driven nonlinear circuits.

At r_c = 1 + √8 ≈ 3.8284 a period-3 orbit is born through a saddle-node of the third iterate f³ (B.1.a applied to f³). This opens the famous period-3 window.

• Just below r_c: the orbit spends long laminar phases that look almost like period 3, squeezing through the narrow channel left by the vanished orbit, and then breaks into a chaotic burst.
• The mean laminar length grows like 1/√(r_c − r).
• Above r_c: a clean period 3.

This is type-I intermittency (Pomeau and Manneville). Move r towards r_c and watch the calm spells in the time series get longer.

The onset of turbulence in convection; the Lorenz system near r ≈ 166; Belousov–Zhabotinsky reactions.

A chaotic attractor suddenly changes size, or disappears, when it touches an unstable periodic orbit (Grebogi, Ott and Yorke).

• Interior crisis, r ≈ 3.8568: inside the period-3 window, three small chaotic bands touch the unstable period-3 orbit (the red marks on the right) and abruptly widen to fill the whole interval.
• Boundary crisis, r = 4: the attractor reaches x = 1, which maps onto the unstable fixed point 0. For r > 4 orbits escape, and only transient chaos remains.

The right panel shows how often the attractor visits each x. Before the interior crisis there are three separate bands; after it they have merged into one wide band.

Lasers and driven pendulums whose chaotic signal suddenly grows; crisis-induced intermittency.