Hopf bifurcation

File:Hopfbifurcation.png

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:

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

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

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

::$z(t)\; =\; r\; e^\{i\; \backslash omega\; t\}\; \backslash ,$

: where

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

: 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.

Image:Supercritical Hopf bifurcation.gifThe normal form of the supercritical Hopf bifurcation can be expressed intuitively in polar coordinates,

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

where $r(t)$ is the instantaneous amplitude of the oscillation and $\backslash theta(t)$ is its instantaneous angular position.{{cite book |last=Strogatz |first=Steven H. |year=1994 |title=Nonlinear Dynamics and Chaos |publisher=Addison Wesley |isbn=978-0-7382-0453-6 |url-access=registration |url=https://archive.org/details/nonlineardynamic00stro }} The angular velocity $(\backslash omega)$ is fixed. When $\backslash mu>0$, the differential equation for $r(t)$ has an unstable fixed point at $r=0$ and a stable fixed point at $r=\backslash sqrt\backslash mu$. The system thus describes a stable circular limit cycle with radius $\backslash sqrt\; \backslash mu$ and angular velocity $\backslash omega$. When 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.

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

: $$

\begin{align}

\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{align}

and

: $$

\begin{align}

\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{align}

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

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

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

Image:hopf-bif.gif (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,{{citation

| last1 = Guckenheimer | first1 = J.

| last2 = Labouriau | first2 = J.S.

| doi = 10.1007/BF02460693

| issue = 5

| journal = Bulletin of Mathematical Biology

| pages = 937–952

| title = Bifurcation of the Hodgkin and Huxley equations: A new twist

| volume = 55

| year = 1993| s2cid = 189888352

}}. the Selkov model of glycolysis,{{cite web| title=Selkov Model Wolfram Demo| work=[demonstrations.wolfram.com ]|url=http://demonstrations.wolfram.com/HopfBifurcationInTheSelkovModel/|access-date=30 September 2012}} the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator, and in classical electromagnetism.{{Cite journal|last=López|first=Álvaro G|date=2020-12-01|title=Stability analysis of the uniform motion of electrodynamic bodies|url=https://doi.org/10.1088/1402-4896/abcad2|journal=Physica Scripta|language=en|volume=96|issue=1|pages=015506|doi=10.1088/1402-4896/abcad2|s2cid=228919333 |issn=1402-4896}} Hopf bifurcations have also been shown to occur in fission waves.{{Cite journal |last1=Osborne |first1=Andrew G. |last2=Deinert |first2=Mark R. |date=October 2021 |title=Stability instability and Hopf bifurcation in fission waves |journal=Cell Reports Physical Science |language=en |volume=2 |issue=10 |pages=100588 |doi=10.1016/j.xcrp.2021.100588|bibcode=2021CRPS....200588O |s2cid=240589650 |doi-access=free }}

The Selkov model is

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

The figure shows a phase portrait illustrating the Hopf bifurcation in the Selkov model.For detailed derivation, see {{cite book |title= Nonlinear Dynamics and Chaos |last= Strogatz |first= Steven H. |year= 1994 |publisher= Addison Wesley |isbn= 978-0-7382-0453-6 |page= [https://archive.org/details/nonlineardynamic00stro/page/205 205] |url-access= registration |url= https://archive.org/details/nonlineardynamic00stro/page/205 }}

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.{{cite journal|last1=Serajian |first1=Reza |year=2011 |title=Effects of the bogie and body inertia on the nonlinear wheel-set hunting recognized by the hopf bifurcation theory |url=http://www.iust.ac.ir/ijae/article-1-25-en.pdf |journal=International Journal of Automotive Engineering |volume=3 |issue=4 |pages=186–196}}

[https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/c6f658c96c4ab007838a3dba51899a73_MIT18_385JF14_Hopf-Bif.pdf 18.385J / 2.036J Nonlinear Dynamics and Chaos Fall 2014: Hopf Bifurcations]. MIT OpenCourseWare

Consider a system defined by $\backslash ddot\; x\; +\; h(\backslash dot\; x,\; x,\; \backslash mu)\; =\; 0$, where $h$ is smooth and $\backslash mu$ is a parameter. After a linear transform of parameters, we can assume that as $\backslash mu$ increases from below zero to above zero, the origin turns from a spiral sink to a spiral source.

Now, for $\backslash mu\; >\; 0$, we perform a perturbative expansion using two-timing:$$x(t)\; =\; \backslash epsilon\; x\_1(t,\; T)\; +\; \backslash epsilon^2\; x\_2(t,\; T)\; +\; \backslash epsilon^3\; x\_3(t,\; T)\; +\; \backslash cdots$$

where $T\; =\; \backslash nu\; t$ is "slow-time" (thus "two-timing"), and $\backslash epsilon,\; \backslash nu$ are functions of $\backslash mu$. By an argument with harmonic balance (see for details), we can use $\backslash epsilon\; =\; \backslash mu^\{1/2\},\; \backslash nu\; =\; \backslash mu$. Then, plugging in $x(t)$ to $\backslash ddot\; x\; +\; h(\backslash dot\; x,\; x,\; \backslash mu)\; =\; 0$, and expanding up to the $\backslash epsilon^3$ order, we would obtain three ordinary differential equations in $x\_1,\; x\_2,\; x\_3$.

The first equation would be of form $\backslash partial\_\{tt\}\; x\_1\; +\; \backslash omega\_0^2\; x\_1\; =\; 0$, which gives the solution $x\_1(t,\; T)\; =\; A(T)\; \backslash cos(\backslash omega\_0\; t\; +\; \backslash phi(T))$, where $A(T),\; \backslash phi(T)$ are "slowly varying terms" of $x\_1$. Plugging it into the second equation, we can solve for $x\_2(t,\; T)$.

Then plugging $x\_1,\; x\_2$ into the third equation, we would have an equation of form $\backslash partial\_\{tt\}\; x\_3\; +\; \backslash omega\_0^2\; x\_3\; =\; ...$, with the right-hand-side a sum of trigonometric terms. Of these terms, we must set the "resonance term" -- that is, $\backslash cos(\backslash omega\_0\; t),\; \backslash sin(\backslash omega\_0\; t)$ -- to zero. This is the same idea as Poincaré–Lindstedt method. This then provides two ordinary differential equations for $A,\; \backslash phi$, allowing one to solve for the equilibrium value of $A$, as well as its stability.

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

First, we eliminate $y$ from the equations:$$\backslash frac\{d\; x\}\{d\; t\}=\backslash mu\; x+y-x^2\; \backslash implies\; \backslash ddot\; x\; =\; \backslash mu\; \backslash dot\; x\; +\; (-x\; +\; \backslash mu\; y\; +\; 2x^2)-\; 2x\backslash dot\; x\; \backslash implies\; \backslash ddot\; x\; -\; 2\backslash mu\; \backslash dot\; x\; +\; (1+\backslash mu^2)x\; +\; 2x\backslash dot\; x\; -\; (2\; +\; \backslash mu\; )x^2\; =\; 0$$Now, perform the perturbative expansion as described above:$$x(t)\; =\; \backslash epsilon\; x\_1(t,\; T)\; +\; \backslash epsilon^2\; x\_2(t,\; T)\; +\; \backslash cdots$$with $\backslash epsilon\; =\; \backslash mu^\{1/2\},\; T\; =\; \backslash mu\; t$. Expanding up to order $\backslash epsilon^3$, we obtain:$$\backslash begin\{cases\}$$

\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_1x_2 + 2\partial_t(x_1-x_1x_2 - \partial_T x_1)

\end{cases}First equation has solution $x\_1(t,\; T)\; =\; A(T)\; \backslash cos(t\; +\; \backslash phi(T))$. Here $A(T),\; \backslash phi(T)$ are respectively the "slow-varying amplitude" and "slow-varying phase" of the simple oscillation.

Second equation has solution $x\_1(t,\; T)\; =\; B\; \backslash cos(t\; +\; \backslash theta)\; +\; A^2\; -\; \backslash frac\; 13\; A^2(\backslash sin(2t+2\backslash phi)\; +\; \backslash cos(2t+2\backslash phi))$, where $B,\; \backslash theta$ are also slow-varying amplitude and phase. Now, since $x\; =\; \backslash epsilon\; x\_1\; +\; \backslash epsilon^2\; x\_2\; +\; \backslash cdots\; =\; \backslash epsilon(A\; \backslash cos(t\; +\; \backslash phi)\; +\; \backslash epsilon\; B\backslash cos(t+\backslash theta))\; +\; \backslash cdots$, we can merge the two terms $A\; \backslash cos(t\; +\; \backslash phi)\; +\; \backslash epsilon\; B\backslash cos(t+\backslash theta)$ as some $C\; \backslash cos(t\; +\; \backslash xi)$.

Thus, without loss of generality, we can assume $B\; =\; 0$. Thus$$x\_2(t,\; T)\; =\; A^2\; -\; \backslash frac\; 13\; A^2(\backslash sin(2t+2\backslash phi)\; +\; \backslash cos(2t+2\backslash phi))$$Plug into the third equation, we obtain$$\backslash partial\_t^2\; x\_3\; +\; x\_3\; +\; (2A-A^3-2A\text{'})\backslash sin(t+\backslash phi)-\; (2A\backslash phi\text{'}\; +11A^3/3)\backslash cos(t+\backslash phi)+\backslash frac\; 13\; A^3(5\backslash sin(3t+3\backslash phi)-\backslash cos(3t+3\backslash phi))$$Eliminating the resonance terms, we obtain $$A\text{'}\; =\; A-A^3/2,\; \backslash quad\; \backslash phi\text{'}\; =\; -\backslash frac\{11\}\{6\}A^2$$The first equation shows that $A=\; \backslash sqrt\; 2$ is a stable equilibrium. Thus we find that the Hopf bifurcation creates an attracting (rather than repelling) limit cycle.

Plugging in $A=\; \backslash sqrt\; 2$, we have $\backslash phi\; =\; -\backslash frac\{11\}\{3\}T\; +\; \backslash phi\_0$. We can repick the origin of time to make $\backslash phi\_0\; =\; 0$. Now solve for $$\backslash partial\_t^2\; x\_3\; +\; x\_3\; +\; \backslash frac\; 13\; A^3(5\backslash sin(3t+3\backslash phi)-\backslash cos(3t+3\backslash phi))$$yielding$$x\_3\; =\; \backslash frac\{\backslash sqrt\; 2\}\{12\}(5\backslash sin(3t\; +\; 3\backslash phi)-\backslash cos(3t\; +\; 3\backslash phi))$$Plugging in $A=\; \backslash sqrt\; 2$ back to the expressions for $x\_1,\; x\_2$, we have$$x\_1\; =\; \backslash sqrt\; 2\; \backslash cos(t+\backslash phi),\; \backslash quad\; x\_2\; =\; 2-\backslash frac\; 23\; (\backslash sin(2t+2\backslash phi)\; +\; \backslash cos(2t+2\backslash phi))$$Plugging them back to $y\; =\; x^2\; +\; \backslash dot\; x\; -\; \backslash mu\; x$ yields the serial expansion of $y$ as well, up to order $\backslash mu^\{3/2\}$.

Letting $\backslash theta\; :=\; t\; +\; \backslash phi$ for notational neatness, we have$$\backslash begin\{aligned\}$$

x &= +\mu^{1/2} \sqrt{2} \cos\theta + \mu \left(-\frac 23 \sin(2\theta)-\frac 23 \cos(2\theta)+2\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(+\frac 43 \sin(2\theta)-\frac 13 \cos(2\theta)+1\right) + \mu^{3/2}\frac{1}{\sqrt{72}} ( - 5\sin(3\theta) + 7\cos(3\theta)&+36\sin\theta + 28\cos\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.

File:Hopf_and_homoclinic_bifurcation.gif|A Hopf bifurcation occurs in the system $\backslash frac\{d\; x\}\{d\; t\}=\backslash mu\; x+y-x^2$ and $\backslash frac\{d\; y\}\{d\; t\}=-x+\backslash mu\; y+2\; x^2$, when $\backslash mu\; =\; 0$, around the origin. A homoclinic bifurcation occurs around $\backslash mu\; =\; 0.06605695$.

File:Hopf_and_homoclinic_bifurcation_2.gif|A detailed view of the homoclinic bifurcation.

File:Hopf_bifurcation,_with_limit_cycle_up_to_order_3-2..gif|As $\backslash mu$ 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 $\backslash mu^\{3/2\}$.

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 $\backslash pm\; i\backslash beta$. A Hopf bifurcation arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.

Routh–Hurwitz criterion (section I.13 of ) gives necessary conditions so that a Hopf bifurcation occurs.

Let $p\_0,~p\_1,~\backslash dots~,~p\_k$ 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 $\backslash \{1,~\backslash dots~,~k\backslash \}$ correspond to what is called Hurwitz determinants. Their definition is related to the associated Hurwitz matrix.

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 $\backslash \{0,~\backslash dots~,~k-2\backslash \}$ are positive, $c\_\{k-1,0\}=0$ and 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.

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 y x -1 \\

1 & 0

\end{pmatrix}.

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

:$$

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

The coefficients are:

$a\_0=1,\; a\_1=-\backslash mu,\; a\_2=1$

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 $\backslash mu\; =\; 0$.

• Reaction–diffusion systems

{{reflist|30em|refs=

{{cite book |title= Solving Ordinary Differential Equations I: Nonstiff Problems|last1= Hairer |first1= E. | last2= Norsett| first2 =S. P.|last3= Wanner | first3=G. |year= 1993|publisher= Springer-Verlag|location= New York|edition=Second |isbn=978-3-540-56670-0 }}

{{cite book |last1=Hale |first1=J. |last2=Koçak |first2=H. |year=1991 |title=Dynamics and Bifurcations |series=Texts in Applied Mathematics |volume=3 |location=Berlin |publisher=Springer-Verlag |isbn=978-3-540-97141-2 |url-access=registration |url=https://archive.org/details/dynamicsbifurcat0000hale }}

{{cite journal |last1= Kahoui|first1= M. E. |last2= Weber |first2= A. |year=2000 |title= Deciding Hopf bifurcations by quantifier elimination in a software component architecture|journal=Journal of Symbolic Computation |volume=30 |issue= 2 |pages= 161–179 |doi= 10.1006/jsco.1999.0353|doi-access= free }}

}}

• {{cite journal |last1=Guckenheimer |first1=J. |author-link=John Guckenheimer |last2=Myers |first2=M. |last3=Sturmfels |first3=B. |author-link3=Bernd Sturmfels |year=1997 |title=Computing Hopf Bifurcations I |journal=SIAM Journal on Numerical Analysis |volume=34 |issue=1 |pages=1–21 |doi=10.1137/S0036142993253461 |citeseerx=10.1.1.52.1609 }}
• {{cite book |last1=Hale |first1=J. |author-link=Jack K. Hale|last2=Koçak |first2=H. |year=1991 |title=Dynamics and Bifurcations |series=Texts in Applied Mathematics |volume=3 |location=Berlin |publisher=Springer-Verlag |isbn=978-3-540-97141-2 |url-access=registration |url=https://archive.org/details/dynamicsbifurcat0000hale }}
• {{cite book |first1=Brian D. |last1=Hassard |first2=Nicholas D. |last2=Kazarinoff |author-link2=Nicholas D. Kazarinoff|first3=Yieh-Hei |last3=Wan |title=Theory and Applications of Hopf Bifurcation |location=New York |publisher=Cambridge University Press |year=1981 |isbn=0-521-23158-2 |url=https://books.google.com/books?id=3wU4AAAAIAAJ }}
• {{cite book |last=Kuznetsov |first=Yuri A. |author-link=Yuri A. Kuznetsov|year=2004 |title=Elements of Applied Bifurcation Theory |location=New York |publisher=Springer-Verlag |edition=Third |isbn=978-0-387-21906-6 }}
• {{cite book |last=Strogatz |first=Steven H. |author-link=Steven Strogatz|year=1994 |title=Nonlinear Dynamics and Chaos |publisher=Addison Wesley |isbn=978-0-7382-0453-6 |url-access=registration |url=https://archive.org/details/nonlineardynamic00stro }}

{{commons category|Hopf bifurcations}}

• [https://web.archive.org/web/20061103045904/http://www.egwald.com/nonlineardynamics/bifurcations.php#hopfbifurcation The Hopf Bifurcation]
• [http://www.scholarpedia.org/article/Andronov-Hopf_bifurcation Andronov–Hopf bifurcation page] at Scholarpedia

Category:Bifurcation theory

Category:Circuit theorems