Hopf bifurcation

Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis.

In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the set of solutions (trajectories) to change from being attracted to (or repelled by) a fixed point, and instead become attracted to (or repelled by) an oscillatory, periodic solution. The Hopf bifurcation is a two-dimensional analog of the pitchfork bifurcation.

Many different kinds of systems exhibit Hopf bifurcations, from radio oscillators to railroad bogies. Trailers towed behind automobiles become infamously unstable if loaded incorrectly, or if designed with the wrong geometry. This offers a gut-sense intuitive example of a Hopf bifurcation in the ordinary world, where stable motion becomes unstable and oscillatory as a parameter is varied.

The general theory of how the solution sets of dynamical systems change in response to changes of parameters is called bifurcation theory; the term bifurcation arises, as the set of solutions typically split into several classes. Stability theory pursues the general theory of stability in mechanical, electronic and biological systems.

The conventional approach to locating Hopf bifurcations is to work with the Jacobian matrix associated with the system of differential equations. When this matrix has a pair of complex-conjugate eigenvalues that cross the imaginary axis as a parameter is varied, that point is the bifurcation. That crossing is associated with a stable fixed point "bifurcating" into a limit cycle.

A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf.

Overview

Hopf bifurcations occur in a large variety of dynamical systems described by differential equations. Near such a bifurcation, the dynamical system is approximated by a normal form, canonically expressed as the following time-dependent differential equation:

${\frac {dz}{dt}}=z\left((\lambda +i)+b|z|^{2}\right)$

Here $z$ is the dynamical variable; it is a complex number. The parameter $\lambda$ is real, and $b=\alpha +i\beta$ is a complex parameter. The number $\alpha$ is called the first Lyapunov coefficient. The above has a simple exact solution, given below. This solution exhibits two distinct behaviors, depending on whether $\lambda >0$ or $\lambda <0$. This change of behavior, as a function of $\lambda ,$ is termed the "Hopf bifurcation".

The study of Hopf bifurcations is not so much the study of the above and its solution, as it is the study of the generic phenomenon in systems that are approximated by the above. The generic setting is studied by the examination of eigenvalues of the Jacobian matrix of the differential equations as a parameter is varied near the bifurcation point.

Exact solution

Visualization of the normal form of the supercritical Hopf bifurcation.

The normal form is effectively the Stuart–Landau equation, written with a different parameterization. It has a simple exact solution in polar coordinates. Writing $z=re^{i\theta }$ and considering the real and imaginary parts as distinct, one obtains a pair of ordinary differential equations:

${\frac {d\theta }{dt}}=1+\beta r^{2}$

and

${\frac {dr}{dt}}=r\left(\lambda +\alpha r^{2}\right)$

The first equation has the trivial solution

$\theta (t)=\theta _{0}+t(1+\beta r^{2}(t))$

The second equation can be solved by observing that it is linear in $1/r^{2}$. That is,

${\frac {d}{dt}}\left({\frac {1}{r^{2}}}\right)=-2\left({\frac {\lambda }{r^{2}}}+\alpha \right)$

which is just the shifted exponential equation. Re-arranging gives the generic solution

$r(t)=r_{0}{\sqrt {\frac {\lambda }{\left(\lambda +\alpha r_{0}^{2}\right)e^{-2\lambda t}-\alpha r_{0}^{2}}}}$

Depending on the sign of $\lambda$ and $\alpha$, the trajectory of a point can be seen to spiral in to the origin, spiral out to infinity, or to approach a limit cycle.

Supercritical and subcritical Hopf bifurcations

Dynamics of the Hopf bifurcation near $\lambda =0$. 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. $\omega$ determines the angular dynamics and therefore the direction of winding for the trajectories.

The limit cycle is orbitally stable if the first Lyapunov coefficient $\alpha$ is negative, and if $\lambda >0.$ Then the bifurcation is said to be supercritical. Otherwise it is unstable and the bifurcation is subcritical.

If $\alpha$ is negative then there is a stable limit cycle for $\lambda >0:$

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

where

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

This is the supercritical regime.

If $\alpha$ is positive then there is an unstable limit cycle for $\lambda <0.$ The bifurcation is said to be subcritical. This classification into sub and super-critical bifurcations is analogous to that of the pitchfork bifurcation.

Jacobian

The Hopf bifurcation can be understood by examining the eigenvalues of the Jacobian matrix for the normal form. This is most readily done by re-writing the normal form in Cartesian coordinates $z=x+iy$. It then has the form

${\begin{aligned}{\dot {x}}&=\lambda x-y+\left(\alpha x-\beta y\right)\left(x^{2}+y^{2}\right)\\{\dot {y}}&=x+\lambda y+\left(\beta x+\alpha y\right)\left(x^{2}+y^{2}\right)\\\end{aligned}}$

where the shorthand ${\dot {x}}=dx/dt$ and ${\dot {y}}=dy/dt$ is used. The Jacobian is

$J=\left[{\begin{array}{cc}{\frac {\partial {\dot {x}}}{\partial x}}&{\frac {\partial {\dot {x}}}{\partial y}}\\{\frac {\partial {\dot {y}}}{\partial x}}&{\frac {\partial {\dot {y}}}{\partial y}}\end{array}}\right]$

This is a bit odious to compute:

$J=\left[{\begin{array}{cc}\lambda +3\alpha x^{2}-2\beta xy+\alpha y^{2}\quad &-1-3\beta y^{2}+2\alpha xy-\beta x^{2}\\1+3\beta x^{2}+2\alpha xy+\beta y^{2}\quad &\lambda +3\alpha y^{2}+2\beta xy+\alpha x^{2}\end{array}}\right]$

The stable point was previously identified to be located at $x=y=0$, at which location the Jacobian takes the particularly simple form:

$J=\left[{\begin{array}{cc}\lambda &-1\\1&\lambda \end{array}}\right]$

The corresponding characteristic polynomial is

$0=\det[J-wI]=\left(\lambda -w\right)^{2}+1$

which has solutions

$w=\lambda \pm i$

Here, $w$ is a pair of complex conjugate eigenvalues of the Jacobian. When the parameter $\lambda$ is negative, the real part of the eigenvalues is (obviously) negative. As the parameter crosses zero, the real part vanishes: this is the Hopf bifurcation. As previously seen, the limit cycle arises as $\lambda$ goes positive.

All Hopf bifurcations have this general form: the Jacobian matrix has a pair of complex-conjugate eigenvalues that cross the imaginary axis as the pertinent parameter is varied.

Linearization

The above computation of the Jacobian can be significantly simplified by working in the tangent plane, tangent to the stable point. The stable point is located at $(x,y)=(0,0)$ and so one can "linearize" the differential equation by dropping all terms that are higher than linear order. This gives

${\begin{aligned}{\dot {x}}&=\lambda x-y\\{\dot {y}}&=x+\lambda y\\\end{aligned}}$

The Jacobian is computed exactly as before; nothing has changed, except to make the calculations simpler. The linearized differential equation can be recognized as being given by a Lie derivative defined on the tangent bundle. Because all tangent bundles are always symplectic manifolds, it is common to formulate bifurcation theory in terms of symplectic geometry.

Examples

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, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator, and in classical electromagnetism. Hopf bifurcations have also been shown to occur in fission waves.

The Selkov model is

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

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

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.

Formal definition of a Hopf bifurcation

The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a fixed point is known as the Hopf bifurcation. The following theorem works for fixed points with one pair of conjugate nonzero purely imaginary eigenvalues. It tells the conditions under which this bifurcation phenomenon occurs.

Theorem (see section 11.2 of ). Let $J$ be the Jacobian of a continuous parametric dynamical system evaluated at a stable point. Suppose that all eigenvalues of $J$ have negative real part except for one conjugate pair, varying as $\rho \pm i\epsilon$ for some function $\rho$ of the parameters. A Hopf bifurcation arises when this eigenvalue pair cross the imaginary axis. This occurs as $\rho$ changes from negative to positive as the system parameters are varied.

Routh–Hurwitz criterion

The Routh–Hurwitz criterion (section I.13 of ) gives necessary conditions for a Hopf bifurcation to occur.

Sturm series

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

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

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

Propositions

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

Proposition 2. If all Hurwitz determinants $c_{i,0}$ (for all $i$ in $\{0,~\dots ~,~k-2\}$ are positive, $c_{k-1,0}=0$ and $c_{k-2,1}<0$ 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.

Example

Consider the classical Van der Pol oscillator written with ordinary differential equations:

$\left\{{\begin{array}{l}{\dfrac {dx}{dt}}=\mu (1-y^{2})x-y,\\{\dfrac {dy}{dt}}=x.\end{array}}\right.$

The Jacobian matrix associated to this system is

$J={\begin{bmatrix}-\mu (-1+y^{2})&-2\mu yx-1\\1&0\end{bmatrix}}.$

The characteristic polynomial (in $\lambda$) of the Jacobian at the stable point $(x,y)=(0,0)$ is

$P(\lambda )=\lambda ^{2}-\mu \lambda +1.$

The associated Sturm series is

${\begin{array}{l}p_{0}(\lambda )=a_{0}\lambda ^{2}-a_{2}\\p_{1}(\lambda )=a_{1}\lambda \end{array}}$

with coefficients $a_{0}=1,a_{1}=-\mu ,a_{2}=1.$

The Sturm polynomials can be written as (here $i=0,1$):

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

For the Van der Pol oscillator, the coefficients are

$c_{0,0}=1,\qquad c_{1,0}=-\mu ,\qquad c_{0,1}=-1.$

A Hopf bifurcation can occur when proposition 2 is satisfied; in the present case, proposition 2 requires that

$c_{0,0}>0,\qquad c_{1,0}=0,\qquad c_{0,1}<0.$

Clearly, the first and third conditions are satisfied; the second condition states that a Hopf bifurcation occurs for the Van der Pol oscillator when $\mu =0$.

Serial expansion method

The serial expansion method provides a way for obtaining explicit solutions containing a Hopf bifurcation by means of a perturbative expansion in the order parameter.

Consider a system defined by ${\ddot {x}}+h({\dot {x}},x,\mu )=0$, where $h$ is smooth and $\mu$ is a parameter. The parameter should be written so that as $\mu$ increases from below zero to above zero, the origin turns from a spiral sink to a spiral source. A linear transform of parameters may be needed to place the equation into this form. For $\mu >0$, a perturbative expansion is performed using two-timing:

$x(t)=\epsilon x_{1}(t,T)+\epsilon ^{2}x_{2}(t,T)+\epsilon ^{3}x_{3}(t,T)+\cdots$

where $T=\nu t$ is "slow-time" (thus "two-timing"), and $\epsilon ,\nu$ are functions of $\mu$. By an argument of harmonic balance (see for details), one may use $\epsilon =\mu ^{1/2},\nu =\mu$. Placing the perturbative expansion for $x(t)$ into ${\ddot {x}}+h({\dot {x}},x,\mu )=0$, and keeping terms up to the $\epsilon ^{3}$ produces three ordinary differential equations in $x_{1},x_{2},x_{3}$.

The first equation is of form $\partial _{tt}x_{1}+\omega _{0}^{2}x_{1}=0$, which is solved by $x_{1}(t,T)=A(T)\cos(\omega _{0}t+\phi (T)).$ The $A(T),\phi (T)$ are "slowly varying" functions of $T$. Inserting this into the second equation allows it to be solved for $x_{2}(t,T)$.

Then plugging the solutions for $x_{1},x_{2}$ into the third equation, an equation of form $\partial _{tt}x_{3}+\omega _{0}^{2}x_{3}=\cdots$ is obtained, with the right-hand-side a sum of trigonometric terms. Of these terms, the "resonance term", the one containing $\cos(\omega _{0}t),\sin(\omega _{0}t)$ must be set to zero. This is the same idea as in the Poincaré–Lindstedt method. This provides two ordinary differential equations for $A,\phi$, allowing one to solve for the equilibrium value of $A$, as well as its stability.

Example of serial expansion

Consider the system defined by

$\left\{{\begin{array}{l}{\dot {x}}&=\mu x+y-x^{2}\\{\dot {y}}&=-x+\mu y+2x^{2}\\\end{array}}\right.$

This system has an equilibrium point at origin. When $\mu$ increases from negative to positive, the origin turns from a stable spiral point to an unstable spiral point. Eliminating $y$ from the equations gives a singe second-order differential equation

${\ddot {x}}-2\mu {\dot {x}}+(1+\mu ^{2})x+2x{\dot {x}}-(2+\mu )x^{2}=0.$

The perturbative expansion to be performed is

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

with

$\epsilon =\mu ^{1/2},\qquad T=\mu t.$

Expanding up to order $\epsilon ^{3}$ results in

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

The first equation has the solution

$x_{1}(t,T)=A(T)\cos(t+\phi (T)).$

Here $A(T),\phi (T)$ are respectively the "slow-varying amplitude" and "slow-varying phase" of the simple oscillation. The second equation has solution

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

where $B,\theta$ are also slow-varying amplitude and phase. The $B$ and $\theta$ terms can be absorbed into $A$ and $\phi ;$ equivalently, $B=0$ can be set without loss of generality. To demonstrate this, the perturbative expansion is written as

${\begin{aligned}x&=\epsilon x_{1}+\epsilon ^{2}x_{2}+\cdots \\&=\epsilon (A\cos(t+\phi )+\epsilon B\cos(t+\theta ))+\cdots \\\end{aligned}}$

Basic trigonometry allows the two cosines to be merged into one:

$A\cos(t+\phi )+\epsilon B\cos(t+\theta )=C\cos(t+\xi )$

for some $C$ and $\xi .$ But this has exactly the same form as $A.$ Thus, the $B$ term can be eliminated by redefining $A$ to be $C$ and $\xi$ to be $\phi .$ The solution to the second equation is thus

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

Plugging through the third equation gives

${\begin{aligned}\partial _{tt}x_{3}+x_{3}&=(2A-A^{3}-2A')\sin(t+\phi )\\&\quad -(2A\phi '+11A^{3}/3)\cos(t+\phi )\\&\quad +{\frac {1}{3}}A^{3}(5\sin(3t+3\phi )-\cos(3t+3\phi ))\end{aligned}}$

Eliminating the resonance term gives

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

where the prime denotes differentiation by the slow time $T.$ The first equation shows that $A={\sqrt {2}}$ is a stable equilibrium. The Hopf bifurcation creates an attracting (rather than repelling) limit cycle.

Plugging in $A={\sqrt {2}}$ gives $\phi =-{\frac {11}{3}}T+\phi _{0}$. The time coordinate can be shifted so that $\phi _{0}=0$. The third equation becomes

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

giving a solution

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

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

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

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

After writing $\theta :=t+\phi$ the solution is

${\begin{aligned}x&=\mu ^{1/2}{\sqrt {2}}\cos \theta +\mu \left(-{\frac {2}{3}}\sin(2\theta )-{\frac {2}{3}}\cos(2\theta )+2\right)\\&\qquad +\mu ^{3/2}{\frac {1}{\sqrt {72}}}\left(5\sin(3\theta )-\cos(3\theta )\right)+{\mathcal {O}}(\mu ^{2})\\\end{aligned}}$

and

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

This provides 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 ${\frac {dx}{dt}}=\mu x+y-x^{2}$ and ${\frac {dy}{dt}}=-x+\mu y+2x^{2}$, when $\mu =0$, around the origin. A homoclinic bifurcation occurs around $\mu =0.06605695$.
• 
  A detailed view of the homoclinic bifurcation.
• 
  As $\mu$ increases from zero, a stable limit cycle emerges out of the origin via Hopf bifurcation. The limit cycle is plotted parametrically, up to order $\mu ^{3/2}$.

See also

• Reaction–diffusion systems

Further reading

• Guckenheimer, J.; Myers, M.; Sturmfels, B. (1997). "Computing Hopf Bifurcations I". SIAM Journal on Numerical Analysis. 34 (1): 1–21. CiteSeerX 10.1.1.52.1609. doi:10.1137/S0036142993253461.
• Hale, J.; Koçak, H. (1991). Dynamics and Bifurcations. Texts in Applied Mathematics. Vol. 3. Berlin: Springer-Verlag. ISBN 978-3-540-97141-2.
• Hassard, Brian D.; Kazarinoff, Nicholas D.; Wan, Yieh-Hei (1981). Theory and Applications of Hopf Bifurcation. New York: Cambridge University Press. ISBN 0-521-23158-2.
• Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory (Third ed.). New York: Springer-Verlag. ISBN 978-0-387-21906-6.
• Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Addison Wesley. ISBN 978-0-7382-0453-6.

External links

• The Hopf Bifurcation
• Andronov–Hopf bifurcation page at Scholarpedia