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## Summary

In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point. Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis.

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

## Overview

### 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 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:

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

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

• If α is negative then there is a stable limit cycle for λ > 0:
$z(t)=re^{i\omega t}\,$
where
$r={\sqrt {-\lambda /\alpha }}{\text{ and }}\omega =1+\beta r^{2}.\,$
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.

### Intuition

Normal form of the supercritical Hopf bifurcation in Cartesian coordinates.

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

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

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

#### Cartesian coordinates

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

{\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

{\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}}

#### Subcritical case

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

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

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

### Example

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, 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 phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right.

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.

## 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 $J_{0}$  be the Jacobian of a continuous parametric dynamical system evaluated at a steady point $Z_{e}$ . Suppose that all eigenvalues of $J_{0}$  have negative real part except one conjugate nonzero purely imaginary pair $\pm i\beta$ . A Hopf bifurcation arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.

## Routh–Hurwitz criterion

Routh–Hurwitz criterion (section I.13 of ) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea.

### 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 follows:

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

The characteristic polynomial (in $\lambda$ ) of the linearization at (0,0) is equal to:

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

The coefficients are: $a_{0}=1,a_{1}=-\mu ,a_{2}=1$
The associated Sturm series is:

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

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

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

The above proposition 2 tells that one must have:

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

Because 1 > 0 and −1 < 0 are obvious, one can conclude that a Hopf bifurcation may occur for Van der Pol oscillator if $\mu =0$ .