13.3 The Hopf bifurcation
Theorem 13.3.1 (Hopf). Let \(x' = f(x, \mu )\) be a smooth planar system with an equilibrium at \(0\) for all \(\mu \) near \(0\), whose Jacobian has eigenvalues \(\alpha (\mu ) \pm i\beta (\mu )\) with \(\alpha (0) = 0\), \(\beta (0) > 0\) and \(\alpha '(0) > 0\) (the pair crosses the imaginary axis with non-zero speed). Then, provided a certain number \(\ell _1\), the first Lyapunov coefficient, computed from the second and third derivatives of \(f\) at \((0, 0)\), is non-zero:
- (i)
- if \(\ell _1 < 0\) (supercritical), for small \(\mu > 0\) there is a unique stable limit cycle near \(0\), of amplitude proportional to \(\sqrt \mu \) and period close to \(2\pi /\beta (0)\);
- (ii)
- if \(\ell _1 > 0\) (subcritical), for small \(\mu < 0\) there is a unique unstable limit cycle near \(0\), of amplitude proportional to \(\sqrt {-\mu }\).
The proof transforms the system, by a near-identity change of variables, to the polar normal form \[ r' = \alpha (\mu )r + \ell _1r^3 + O(r^5), \qquad \theta ' = \beta (\mu ) + O(r^2) , \] in which the cycle is the positive root of \(\alpha (\mu ) + \ell _1r^2 = 0\), \(r \approx \sqrt {-\alpha (\mu )/\ell _1} \approx \sqrt {\alpha '(0)\mu /|\ell _1|}\); the radial equation is a pitchfork in \(r\). We omit the computation of \(\ell _1\), which is long but mechanical. Example 12.1.2 is the supercritical normal form with \(\alpha = 1\), and Problem 12.7.1 shows the subcritical pattern. In the supercritical case the oscillation grows smoothly from zero amplitude as \(\mu \) passes \(0\) (a soft onset); in the subcritical case the system jumps to a large oscillation and shows hysteresis (a hard onset), which is why subcritical Hopf bifurcations are dangerous in engineering, for instance in the flutter of aircraft wings.
Example 13.3.2. The Brusselator, a model of an autocatalytic chemical reaction, is \(x' = a - (b + 1)x + x^2y\), \(y' = bx - x^2y\), with equilibrium \((a, b/a)\) and Jacobian \[ J = \begin {bmatrix} b - 1 & a^2\\ -b & -a^2\end {bmatrix}, \qquad \tr J = b - 1 - a^2, \qquad \det J = a^2 > 0 . \] The equilibrium is a stable focus for \(b < 1 + a^2\) and an unstable one for \(b > 1 + a^2\); at \(b_c = 1 + a^2\) the eigenvalues are \(\pm ia\), crossing with speed \(\alpha '(b_c) = \frac 12\). The bifurcation is supercritical, and for \(b\) slightly above \(b_c\) the concentrations oscillate on a stable limit cycle whose amplitude grows like \(\sqrt {b - b_c}\) (Software 13.5.1): a chemical clock.
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