Bifurcations in Dynamical Systems

Bifurcations: Qualitative Changes in Dynamical Systems

What is a Bifurcation

The word "bifurcation" (from Latin bifurcus — forked) in mathematics means a qualitative change in the behavior of a dynamical system with a small change in a parameter. Before the bifurcation — one picture of behavior, after — fundamentally different.

Bifurcations are a mathematical language for describing catastrophic changes in nature and technology: loss of stability of a structure when the critical load is exceeded, transition from laminar flow to turbulent, sudden collapse of a biological population when the catch quota is exceeded, transition of the market from a stable state to chaotic.

Saddle-Node Bifurcation (fold bifurcation)

System: $x' = \mu - x^2$.

For $\mu > 0$: two equilibria $x^* = \pm\sqrt{\mu}$. At $x^* = +\sqrt{\mu}$: $f'(x) = -2x|_{x = \sqrt{\mu}} = -2\sqrt{\mu} < 0$ → stable. At $x^* = -\sqrt{\mu}$: $f' = +2\sqrt{\mu} > 0$ → unstable.

For $\mu = 0$: one equilibrium $x^* = 0$, semi-stable (neutral). Bifurcation point.

For $\mu < 0$: no real equilibria.

Physically: at $\mu = 0$ the stable and unstable equilibria "collide" and "annihilate." If the parameter of the system (for example, the load on a structure) exceeds the critical value, the equilibrium disappears and the system "falls" into a distant attractive state — catastrophe.

Transcritical Bifurcation

System: $x' = \mu x - x^2$. Equilibria: $x^* = 0$ and $x^* = \mu$.

For $\mu < 0$: $x^* = 0$ is stable ($f' = \mu < 0$), $x^* = \mu < 0$ is unstable.

For $\mu = 0$: both equilibria merge at $x^* = 0$.

For $\mu > 0$: $x^* = 0$ is unstable, $x^* = \mu > 0$ is stable.

At the bifurcation point, equilibria exchange stability. In ecology, this bifurcation models the invasion of a new species: for $\mu > 0$ (favorable environment) the new species stably coexists with the original one.

Pitchfork Bifurcation

System: $x' = \mu x - x^3$ (supercritical pitchfork).

For $\mu \leq 0$: the only stable equilibrium $x^* = 0$.

For $\mu > 0$: $x^* = 0$ is unstable, two new stable equilibria $x^* = \pm\sqrt{\mu}$ appear. The system "chooses" one of two symmetric states — spontaneous symmetry breaking.

Physical example — longitudinal buckling: For load $P < P_{cr}$ the rod remains straight ($x^* = 0$ is stable). For $P > P_{cr}$ the straight state loses stability, the rod bends to one side ($x^* = \pm\sqrt{P - P_{cr}}$).

This behavior describes Euler's problem of the critical load of a column, solved back in the XVIII century. Pitchfork bifurcation is its mathematical generalization.

Subcritical pitchfork ($x' = \mu x + x^3$): equilibria $\pm\sqrt{-\mu}$ for $\mu < 0$ are unstable. As $\mu \to 0$ they disappear and $x^* = 0$ loses stability. This is a more dangerous bifurcation: loss of stability occurs "suddenly" and with finite amplitude.

Hopf Bifurcation

Hopf bifurcation — birth of a limit cycle (stable periodic oscillations) from an equilibrium state.

Normal form: $\dot{x} = \alpha x - \beta y + (\mu - x^2 - y^2)x$, $\dot{y} = \beta x + \alpha y + (\mu - x^2 - y^2)y$.

In polar coordinates: $\dot{r} = (\mu - r^2)r$, $\dot{\theta} = \beta$.

For $\mu < 0$: only stable point $r = 0$ (all trajectories converge to zero).

For $\mu > 0$: $r = 0$ is unstable, a stable limit cycle $r = \sqrt{\mu}$ appears.

Physical examples:

• Oscillator on a tube/transistor: at zero supply there are no oscillations ($r = 0$); when the threshold is exceeded, stable harmonic oscillations arise (limit cycle).
• Heart oscillations: with changes in ion concentration, the transition from static state to cyclic (beating) — Hopf bifurcation.
• Flutter of a wing when exceeding flow speed.

René Thom's Catastrophe Theory

Thom (1972) classified stable bifurcations (those not disappearing with small perturbations of the equation). If the number of control parameters $\leq 4$, only 7 types of elementary catastrophes exist: fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, parabolic umbilic.

Catastrophe theory found applications in economics (crises), biology (morphogenesis — the formation of organs), geology (earthquakes as dynamical catastrophes).

Bifurcation Diagrams in Population Dynamics

Logistic growth equation with delay: $N_{n+1} = r N_n (1 - N_n/K)$. For $r < 1$: extinction. For $1 < r < 3$: stable equilibrium. For $3 < r < 1+\sqrt{6} \approx 3.449$: oscillations with period 2 (pitchfork bifurcation). For $r > 3.57$: chaos. Bifurcation diagram is plotted: parameter $r$ on the horizontal axis, attractors $N_n$ for large $n$ on the vertical. Characteristic structure: cascade of period doubling ($1 \to 2 \to 4 \to 8 \to \ldots$) with Feigenbaum constant $\delta \approx 4.669$: the ratio of the lengths of consecutive parameter intervals up to the next doubling. This constant is universal — identical for a wide class of mappings, indicating a deep unity of chaotic systems.

Question for reflection: Subcritical pitchfork bifurcation is more dangerous than supercritical because the system suddenly "jumps" to a distant state. Can the system be returned to its original state simply by reversing the parameter? What is "hysteresis" in the context of bifurcations?