Hopf bifurcation

Dynamics of the Hopf bifurcation near ${\displaystyle \lambda =0}$. Possible trajectories in red, stable structures in dark blue and unstable structures in dashed light blue. Supercritical Hopf bifurcation: 1a) stable fixed point 1b) unstable fixed point, stable limit cycle 1c) phase space dynamics. Subcritical Hopf bifurcation: 2a) stable fixed point, unstable limit cycle 2b) unstable fixed point 2c) phase space dynamics. ${\displaystyle \omega }$ determines the angular dynamics and therefore the direction of winding for the trajectories.

The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical.

The normal form of a Hopf bifurcation is the following time-dependent differential equation:

${\displaystyle {\frac {dz}{dt}}=z((\lambda +i)+b|z|^{2}),}$ where z, b are both complex and λ is a real parameter.

Write: ${\displaystyle b=\alpha +i\beta .\,}$ The number α is called the first Lyapunov coefficient.

• If α is negative then there is a stable limit cycle for λ > 0:

${\displaystyle z(t)=re^{i\omega t}\,}$

where

${\displaystyle r={\sqrt {-\lambda /\alpha }}{\text{ and }}\omega =1+\beta r^{2}.\,}$

The bifurcation is then called supercritical.

• If α is positive then there is an unstable limit cycle for λ < 0. The bifurcation is called subcritical.

Normal form of the supercritical Hopf bifurcation in Cartesian coordinates.[2]

The normal form of the supercritical Hopf bifurcation can be expressed intuitively in polar coordinates,

${\displaystyle {\frac {dr}{dt}}=(\mu -r^{2})r,~~{\frac {d\theta }{dt}}=\omega }$

where ${\displaystyle r(t)}$ is the instantaneous amplitude of the oscillation and ${\displaystyle \theta (t)}$ is its instantaneous angular position.[3] The angular velocity ${\displaystyle (\omega )}$ is fixed. When ${\displaystyle \mu >0}$, the differential equation for ${\displaystyle r(t)}$ has an unstable fixed point at ${\displaystyle r=0}$ and a stable fixed point at ${\displaystyle r={\sqrt {\mu }}}$. The system thus describes a stable circular limit cycle with radius ${\displaystyle {\sqrt {\mu }}}$ and angular velocity ${\displaystyle \omega }$. When ${\displaystyle \mu <0}$ then ${\displaystyle r=0}$ is the only fixed point and it is stable. In that case, the system describes a spiral that converges to the origin.

The polar coordinates can be transformed into Cartesian coordinates by writing ${\displaystyle x=r\cos(\theta )}$ and ${\displaystyle y=r\sin(\theta )}$.[3] Differentiating ${\displaystyle x}$ and ${\displaystyle y}$ with respect to time yields the differential equations,

${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&={\frac {dr}{dt}}\cos(\theta )-{\frac {d\theta }{dt}}r\sin(\theta )\\&=(\mu -r^{2})r\cos(\theta )-\omega r\sin(\theta )\\&=(\mu -x^{2}-y^{2})x-\omega y\end{aligned}}}$

and

${\displaystyle {\begin{aligned}{\frac {dy}{dt}}&={\frac {dr}{dt}}\sin(\theta )+{\frac {d\theta }{dt}}r\cos(\theta )\\&=(\mu -r^{2})r\sin(\theta )+\omega r\cos(\theta )\\&=(\mu -x^{2}-y^{2})y+\omega x.\end{aligned}}}$

The normal form of the subcritical Hopf is obtained by negating the sign of ${\displaystyle dr/dt}$,

${\displaystyle {\frac {dr}{dt}}=-(\mu -r^{2})r,~~{\frac {d\theta }{dt}}=\omega }$

which reverses the stability of the fixed points in ${\displaystyle r(t)}$. For ${\displaystyle \mu >0}$ the limit cycle is now unstable and the origin is stable.

The Hopf bifurcation in the Selkov system (see article). As the parameters change, a limit cycle (in blue) appears out of a stable equilibrium.

Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment), the Hodgkin–Huxley model for nerve membrane potential,[4] the Selkov model of glycolysis,[5] the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator, and in classical electromagnetism.[6] Hopf bifurcations have also been shown to occur in fission waves.[7]

The Selkov model is

${\displaystyle {\frac {dx}{dt}}=-x+ay+x^{2}y,~~{\frac {dy}{dt}}=b-ay-x^{2}y.}$

The figure shows a phase portrait illustrating the Hopf bifurcation in the Selkov model.[8]

In railway vehicle systems, Hopf bifurcation analysis is notably important. Conventionally a railway vehicle's stable motion at low speeds crosses over to unstable at high speeds. One aim of the nonlinear analysis of these systems is to perform an analytical investigation of bifurcation, nonlinear lateral stability and hunting behavior of rail vehicles on a tangent track, which uses the Bogoliubov method.[9]

Consider the system defined by ${\displaystyle {\frac {dx}{dt}}=\mu x+y-x^{2}}$ and ${\displaystyle {\frac {dy}{dt}}=-x+\mu y+2x^{2}}$. The system has an equilibrium point at origin. When ${\displaystyle \mu }$ increases from negative to positive, the origin turns from a stable spiral point to an unstable spiral point.

First, we eliminate ${\displaystyle y}$ from the equations:

$${\displaystyle {\frac {dx}{dt}}=\mu x+y-x^{2}\implies {\ddot {x}}=\mu {\dot {x}}+(-x+\mu y+2x^{2})-2x{\dot {x}}\implies {\ddot {x}}-2\mu {\dot {x}}+(1+\mu ^{2})x+2x{\dot {x}}-(2+\mu )x^{2}=0}$$

Now, perform the perturbative expansion as described above:

$${\displaystyle x(t)=\epsilon x_{1}(t,T)+\epsilon ^{2}x_{2}(t,T)+\cdots }$$

with ${\displaystyle \epsilon =\mu ^{1/2},T=\mu t}$. Expanding up to order ${\displaystyle \epsilon ^{3}}$, we obtain:

$${\displaystyle {\begin{cases}\partial _{tt}x_{1}+x_{1}=0\\\partial _{tt}x_{2}+x_{2}=2x_{1}^{2}-2x_{1}\partial _{t}x_{1}\\\partial _{tt}x_{3}+x_{3}=4x_{1}x_{2}+2\partial _{t}(x_{1}-x_{1}x_{2}-\partial _{T}x_{1})\end{cases}}}$$

First equation has solution ${\displaystyle x_{1}(t,T)=A(T)\cos(t+\phi (T))}$. Here ${\displaystyle A(T),\phi (T)}$ are respectively the "slow-varying amplitude" and "slow-varying phase" of the simple oscillation.

Second equation has solution ${\displaystyle x_{1}(t,T)=B\cos(t+\theta )+A^{2}-{\frac {1}{3}}A^{2}(\sin(2t+2\phi )+\cos(2t+2\phi ))}$, where ${\displaystyle B,\theta }$ are also slow-varying amplitude and phase. Now, since ${\displaystyle x=\epsilon x_{1}+\epsilon ^{2}x_{2}+\cdots =\epsilon (A\cos(t+\phi )+\epsilon B\cos(t+\theta ))+\cdots }$, we can merge the two terms ${\displaystyle A\cos(t+\phi )+\epsilon B\cos(t+\theta )}$ as some ${\displaystyle C\cos(t+\xi )}$.

Thus, WLOG, we can assume ${\displaystyle B=0}$. Thus

$${\displaystyle x_{2}(t,T)=A^{2}-{\frac {1}{3}}A^{2}(\sin(2t+2\phi )+\cos(2t+2\phi ))}$$

Plug into the third equation, we obtain

$${\displaystyle \partial _{t}^{2}x_{3}+x_{3}+(2A-A^{3}-2A')\sin(t+\phi )-(2A\phi '+11A^{3}/3)\cos(t+\phi )+{\frac {1}{3}}A^{3}(5\sin(3t+3\phi )-\cos(3t+3\phi ))}$$

Eliminating the resonance terms, we obtain

$${\displaystyle A'=A-A^{3}/2,\quad \phi '=-{\frac {11}{6}}A^{2}}$$

The first equation shows that ${\displaystyle A={\sqrt {2}}}$ is a stable equilibrium. Thus we find that the Hopf bifurcation creates an attracting (rather than repelling) limit cycle. Plugging in ${\displaystyle A={\sqrt {2}}}$, we have ${\displaystyle \phi =-{\frac {11}{3}}T+\phi _{0}}$. We can repick the origin of time to make ${\displaystyle \phi _{0}=0}$. Now solve for

$${\displaystyle \partial _{t}^{2}x_{3}+x_{3}+{\frac {1}{3}}A^{3}(5\sin(3t+3\phi )-\cos(3t+3\phi ))}$$

yielding

$${\displaystyle x_{3}={\frac {\sqrt {2}}{12}}(5\sin(3t+3\phi )-\cos(3t+3\phi ))}$$

Plugging in ${\displaystyle A={\sqrt {2}}}$ back to the expressions for ${\displaystyle x_{1},x_{2}}$, we have

$${\displaystyle x_{1}={\sqrt {2}}\cos(t+\phi ),\quad x_{2}=2-{\frac {2}{3}}(\sin(2t+2\phi )+\cos(2t+2\phi ))}$$

Plugging them back to ${\displaystyle y=x^{2}+{\dot {x}}-\mu x}$ yields the serial expansion of ${\displaystyle y}$ as well, up to order ${\displaystyle \mu ^{3/2}}$.

Letting ${\displaystyle \theta$ :=t+\phi } for notational neatness, we have

${\displaystyle {\begin{aligned}x&=&\mu ^{1/2}{\sqrt {2}}\cos \theta +&\mu \left(2-{\frac {2}{3}}\sin(2\theta )-{\frac {2}{3}}\cos(2\theta )\right)+&\mu ^{3/2}{\frac {1}{\sqrt {72}}}(5\sin(3\theta )-\cos(3\theta ))+&O(\mu ^{2})\\y&=&-\mu ^{1/2}{\sqrt {2}}\sin \theta +&\mu \left(1+{\frac {4}{3}}\sin(2\theta )-{\frac {1}{3}}\cos(2\theta )\right)+&\mu ^{3/2}{\frac {1}{\sqrt {72}}}(36\sin \theta +28\cos \theta -5\sin(3\theta )+7\cos(3\theta ))+&O(\mu ^{2})\end{aligned}}}$

This provides us with a parametric equation for the limit cycle. This is plotted in the illustration on the right.

• Examples of bifurcations
• 
  A Hopf bifurcation occurs in the system ${\displaystyle {\frac {dx}{dt}}=\mu x+y-x^{2}}$ and ${\displaystyle {\frac {dy}{dt}}=-x+\mu y+2x^{2}}$, when ${\displaystyle \mu =0}$, around the origin. A homoclinic bifurcation occurs around ${\displaystyle \mu =0.06605695}$.
• 
  A detailed view of the homoclinic bifurcation.
• 
  As ${\displaystyle \mu }$ increases from zero, a stable limit cycle emerges out of the origin via Hopf bifurcation. Here we plot the limit cycle parametrically, up to order ${\displaystyle \mu ^{3/2}}$.

Let ${\displaystyle p_{0},~p_{1},~\dots ~,~p_{k}}$ be Sturm series associated to a characteristic polynomial ${\displaystyle P}$. They can be written in the form:

${\displaystyle p_{i}(\mu )=c_{i,0}\mu ^{k-i}+c_{i,1}\mu ^{k-i-2}+c_{i,2}\mu ^{k-i-4}+\cdots }$

The coefficients ${\displaystyle c_{i,0}}$ for ${\displaystyle i}$ in ${\displaystyle \{1,~\dots ~,~k\}}$ correspond to what is called Hurwitz determinants.[13] Their definition is related to the associated Hurwitz matrix.

Proposition 1. If all the Hurwitz determinants ${\displaystyle c_{i,0}}$ are positive, apart perhaps ${\displaystyle c_{k,0}}$ then the associated Jacobian has no pure imaginary eigenvalues.

Proposition 2. If all Hurwitz determinants ${\displaystyle c_{i,0}}$ (for all ${\displaystyle i}$ in ${\displaystyle \{0,~\dots ~,~k-2\}}$ are positive, ${\displaystyle c_{k-1,0}=0}$ and ${\displaystyle c_{k-2,1}<0}$ then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair.

The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition.