Andronov Hopf Bifurcation

Andronov-Hopf bifurcation is the birth of a Limit Cycle from an equilibrium in dynamical systems generated by ODEs, when the equilibrium changes stability via a pair of purely imaginary Eigenvalues. The bifurcation can be supercritical or subcritical, resulting in stable or unstable (within an invariant two-dimensional manifold) Limit Cycle, respectively.

Supercritical Andronov-Hopf bifurcation in the plane.

Subcritical Andronov-Hopf bifurcation in the plane

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Definition

Consider an autonomous system of Ordinary Differential Equations:

$$\dot{x}=f(x,\alpha ),x\in {\mathbb{R}}^n$$

depending on a parameter $\alpha \in R,$ where $f$ is smooth.

• Suppose that for all sufficiently small $|\alpha |$ the system has a family of equilibria $x0(\alpha ).$
• Further assume that its Jacobian Matrix $A(\alpha )=f_x(x^0(\alpha ),\alpha )$ has one pair of complex eigenvalues

$${\lambda }_{1,2}(\alpha )=\mu (\alpha )\pm i\omega (\alpha )$$

that becomes purely imaginary when $\alpha =0,$ i.e., $\mu (0)=0$ and $\omega (0)=\omega 0>0.$ Then, generically, as $\alpha$ passes through $\alpha =0,$ the equilibrium changes stability and a unique limit cycle bifurcates from it. This bifurcation is characterized by a single bifurcation condition $Re\lambda 1,2=0$ (has codimension one) and appears generically in one-parameter families of smooth ODEs.

Two-dimensional Case

To describe the bifurcation analytically, consider the system above with $n=2,$

$$x\dot{}1=f1(x1,x2,\alpha ),$$ $$x\dot{}2=f2(x1,x2,\alpha ).$$

If the following nondegeneracy conditions hold:

• (AH.1) $l_1(0)\ne 0,$ where $l1(\alpha )$ is the first Lyapunov coefficient (see below);
• (AH.2) ${\mu }^{\prime }(0)\ne 0,$

then this system is locally topologically equivalent near the equilibrium to the normal form

$${\dot{y}}_1=\beta y_1-y_2+\sigma y_1(y_1^2+y_2^2),$$ $$y\dot{}2=y1+\beta y2+\sigma y2(y21+y22),$$

where $y=(y1,y2)T\in R2,\beta \in R,$ and $\sigma =signl1(0)=\pm 1.$

• If $\sigma =-1,$ the normal form has an equilibrium at the origin, which is asymptotically stable for $\beta \le 0$ (weakly at $\beta =0$ ) and unstable for $\beta >0.$ Moreover, there is a unique and stable circular limit cycle that exists for $\beta >0$ and has radius $\beta \surd .$ This is a supercritical Andronov-Hopf bifurcation (see Figure 1).
• If $\sigma =+1,$ the origin in the normal form is asymptotically stable for $\beta <0$ and unstable for $\beta \ge 0$ (weakly at $\beta =0$ ), while a unique and unstable limit cycle exists for $\beta <0.$ This is a subcritical Andronov-Hopf bifurcation (see Figure 2).

$"\\usepackage{tikz}\n\\usetikzlibrary{arrows.meta}\n\\begin{document}\n\\begin{tikzpicture}[scale=1.2, >=Stealth, thick]\n\n Subcritical (sigma = +1)\n\\begin{scope}[xshift=6.5cm]\n \\node[font=\\bfseries\\small] at (0, 3.1) {Subcritical $(\\sigma=+1)$};\n \\draw[->] (-2.3,0) -- (2.6,0) node[right, font=\\small] {$\\beta$};\n \\draw[->] (0,-0.3) -- (0,2.9) node[above, font=\\small] {amplitude};\n \\node[font=\\footnotesize] at (-0.2,-0.22) {$0$};\n$

source code

Multi-dimensional Case

In the $n$ -dimensional case with $n\ge 3,$ the Jacobian matrix $A0=A(0)$ has

• a simple pair of purely imaginary eigenvalues ${\lambda }_{1,2}=\pm i{\omega }_0,{\omega }_0>0,$ as well as
• $n_s$ eigenvalues with $Re\lambda j<0,$ and
• $n_u$ eigenvalues with $Re\lambda j>0,$

with $n_s+n_u+2=n.$ According to the Center Manifold Theorem, there is a family of smooth two-dimensional invariant manifolds $Wc\alpha$ near the origin. The $n$ -dimensional system restricted on $Wc\alpha$ is two-dimensional, hence has the normal form above.

Moreover, under the non-degeneracy conditions (AH.1) and (AH.2), the $n$ -dimensional system is locally topologically equivalent near the origin to the suspension of the normal form by the standard saddle, i.e.

$$y\dot{}1=\beta y1-y2+\sigma y1(y21+y22),$$ $$y\dot{}2=y1+\beta y2+\sigma y2(y21+y22),$$ $$y\dot{}s=-ys,$$ $$y\dot{}u=+yu,$$

where $y=(y1,y2)T\in R2,$ $ys\in Rns,yu\in Rnu.$ Figure 3 shows the phase portraits of the normal form suspension when $n=3,$ $ns=1,$ $nu=0,$ and $\sigma =-1.$

First Lyapunov Exponent|Lyapunov Coefficient

Whether Andronov-Hopf bifurcation is subcritical or supercritical is determined by $\sigma ,$ which is the sign of the first lyapunov coefficient $l1(0)$ of the dynamical system near the equilibrium. This coefficient can be computed at $\alpha =0$ as follows. Write the Taylor expansion of $f(x,0)$ at $x=0$ as

$$f(x,0)=A0x+12B(x,x)+16C(x,x,x)+O(\Vert x\Vert 4),$$

where $B(x,y)$ and $C(x,y,z)$ are the multilinear functions with components

$$Bj(x,y)=\sum k,l=1n\partial 2fj(\xi ,0)\partial \xi k\partial \xi l\mid \mid \mid \mid \xi =0xkyl,$$ $$Cj(x,y,z)=\sum k,l,m=1n\partial 3fj(\xi ,0)\partial \xi k\partial \xi l\partial \xi m\mid \mid \mid \mid \xi =0xkylzm,$$

where $j=1,2,\dots ,n.$ Let $q\in Cn$ be a complex eigenvector of $A0$ corresponding to the eigenvalue $i\omega 0:$ $A0q=i\omega 0q.$ Introduce also the adjoint eigenvector $p\in Cn:$ $AT0p=-i\omega 0p,$ $⟨p,q⟩=1.$ Here $⟨p,q⟩=p¯Tq$ is the inner product in $Cn.$ Then (see, for example, Kuznetsov (2004))

$$l1(0)=12\omega 0Re[⟨p,C(q,q,q¯)⟩-2⟨p,B(q,A-10B(q,q¯))⟩+⟨p,B(q¯,(2i\omega 0In-A0)-1B(q,q))⟩],$$

where $In$ is the unit $n\times n$ matrix. Note that the value (but not the sign) of $l1(0)$ depends on the scaling of the eigenvector $q.$ The normalization $⟨q,q⟩=1$ is one of the options to remove this ambiguity. Standard bifurcation software (e.g. MATCONT) computes $l1(0)$ automatically.

For planar smooth ODEs with

$$x=\left(\begin{array}{c}u\\ v\end{array}\right),f(x,0)=\left(\begin{array}{cc}0& -{\omega }_0\\ {\omega }_0& 0\end{array}\right)\left(\begin{array}{c}u\\ v\end{array}\right)+\left(\begin{array}{c}P(u,v)\\ Q(u,v)\end{array}\right),$$

the setting $q=p=12\surd (1-i)$ leads to the formula

$$l1(0)=18\omega 0(Puuu+Puvv+Quuv+Qvvv)$$ $$+18\omega 20[Puv(Puu+Pvv)-Quv(Quu+Qvv)-PuuQuu+PvvQvv],$$

where the lower indices mean partial derivatives evaluated at $x=0$ (cf. Guckenheimer and Holmes, 1983).

Some Important Examples

The first Lyapunov coefficient can be found easily in some simple but important examples (Izhikevich 2007). Here $a,b>0$ are positive parameters and all derivatives should be evaluated at the critical equilibrium.

System	Condition	$\mathrm{sign}{l}_1(0)$

$${\dot{x}}_1=F(x_1)-x_2$$ $$x\dot{}2=a(x1-b)$$

|

$$F^{\prime }=0$$

|

$$\mathrm{sign}{F}^{\prime \prime \prime }$$

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|

$${\dot{x}}_1=F(x_1)-x_2$$ $$x\dot{}2=a(bx1-x2)$$

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$$F^{\prime }=a$$

and $b>a$ |

$$\mathrm{sign}\left[F^{\prime \prime \prime }+(F^{\prime \prime }{)}^2/(b-a)\right]$$

|

|

$${\dot{x}}_1=F(x_1)-x_2$$ $$x\dot{}2=a(G(x1)-x2)$$

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$$F^{\prime }=a$$

and $G\prime >a$ |

{\rm sign}\left[F'''+F''(F''-G'')/(G'-a)\right] $$ | ## Other Cases Andronov-Hopf bifurcation occurs also in infinitely-dimensional ODEs generated by [PDEs](http://www.scholarpedia.org/article/Partial_Differential_Equations "Partial Differential Equations") and [DDEs](http://www.scholarpedia.org/article/Delay_Differential_Equations "Delay Differential Equations"), to which the [Center Manifold Theorem](http://www.scholarpedia.org/article/Center_Manifold_Theorem "Center Manifold Theorem") applies. An analogue of the Andronov-Hopf bifurcation - called **[Neimark-Sacker bifurcation](http://www.scholarpedia.org/article/Neimark-Sacker_bifurcation "Neimark-Sacker bifurcation")** - occurs in generic dynamical systems generated by iterated maps when the critical [fixed point](http://www.scholarpedia.org/article/Fixed_point "Fixed point") has a pair of simple eigenvalues $\mu_{1,2}=e^{\pm i \theta} \ .$