# Andronov-Hopf Bifurcation

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Curator: Yuri A. Kuznetsov

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.

## Definition

Consider an autonomous system of ordinary differential equations (ODEs) $\dot{x}=f(x,\alpha),\ \ \ x \in {\mathbb R}^n$ depending on a parameter $$\alpha \in {\mathbb R}\ ,$$ where $$f$$ is smooth.

• Suppose that for all sufficiently small $$|\alpha|$$ the system has a family of equilibria $$x^0(\alpha)\ .$$
• 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 $${\rm Re}\ \lambda_{1,2}=0$$ (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 $$n=2\ ,$$ $\dot{x}_1 = f_1(x_1,x_2,\alpha) \ ,$ $\dot{x}_2 = f_2(x_1,x_2,\alpha) \ .$ If the following nondegeneracy conditions hold:

• (AH.1) $$l_1(0) \neq 0\ ,$$ where $$l_1(\alpha)$$ is the first Lyapunov coefficient (see below);
• (AH.2) $$\mu'(0) \neq 0\ ,$$

then this system is locally topologically equivalent near the equilibrium to the normal form $\dot{y}_1 = \beta y_1 - y_2 + \sigma y_1(y_1^2+y_2^2) \ ,$ $\dot{y}_2 = y_1 + \beta y_2 + \sigma y_2(y_1^2+y_2^2) \ ,$ where $$y=(y_1,y_2)^T \in {\mathbb R}^2,\ \beta \in {\mathbb R}\ ,$$ and $$\sigma= {\rm sign}\ l_1(0) = \pm 1\ .$$

• If $$\sigma=-1\ ,$$ the normal form has an equilibrium at the origin, which is asymptotically stable for $$\beta \leq 0$$ (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 $$\sqrt{\beta}\ .$$ 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 $$\beta \geq 0$$ (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).

## Multi-dimensional Case

In the $$n$$-dimensional case with $$n \geq 3\ ,$$ the Jacobian matrix $$A_0=A(0)$$ 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 $${\rm Re}\ \lambda_j < 0\ ,$$ and
• $$n_u$$ eigenvalues with $${\rm Re}\ \lambda_j > 0\ ,$$

with $$n_s+n_u+2=n\ .$$ According to the Center Manifold Theorem, there is a family of smooth two-dimensional invariant manifolds $$W^c_{\alpha}$$ near the origin. The $$n$$-dimensional system restricted on $$W^c_{\alpha}$$ 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. $\dot{y}_1 = \beta y_1 - y_2 + \sigma y_1(y_1^2+y_2^2) \ ,$ $\dot{y}_2 = y_1 + \beta y_2 + \sigma y_2(y_1^2+y_2^2) \ ,$ $\dot{y}^s = -y^s \ ,$ $\dot{y}^u = +y^u \ ,$ where $$y=(y_1,y_2)^T \in {\mathbb R}^2\ ,$$ $$y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}\ .$$ Figure 3 shows the phase portraits of the normal form suspension when $$n=3\ ,$$ $$n_s=1\ ,$$ $$n_u=0\ ,$$ and $$\sigma=-1\ .$$

## First Lyapunov Coefficient

Whether Andronov-Hopf bifurcation is subcritical or supercritical is determined by $$\sigma\ ,$$ which is the sign of the first Lyapunov coefficient $$l_1(0)$$ 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 $f(x,0)=A_0x + \frac{1}{2}B(x,x) + \frac{1}{6}C(x,x,x) + O(\|x\|^4),$ where $$B(x,y)$$ and $$C(x,y,z)$$ are the multilinear functions with components $\ \ B_j(x,y) =\sum_{k,l=1}^n \left. \frac{\partial^2 f_j(\xi,0)}{\partial \xi_k \partial \xi_l}\right|_{\xi=0} x_k y_l \ ,$ $C_j(x,y,z) =\sum_{k,l,m=1}^n \left. \frac{\partial^3 f_j(\xi,0)}{\partial \xi_k \partial \xi_l \partial \xi_m}\right|_{\xi=0} x_k y_l z_m \ ,$ where $$j=1,2,\ldots,n\ .$$ Let $$q\in {\mathbb C}^n$$ be a complex eigenvector of $$A_0$$ corresponding to the eigenvalue $$i\omega_0\ :$$ $$A_0q=i\omega_0 q\ .$$ Introduce also the adjoint eigenvector $$p \in {\mathbb C}^n\ :$$ $$A_0^T p = - i\omega_0 p\ ,$$ $$\langle p, q \rangle =1\ .$$ Here $$\langle p, q \rangle = \bar{p}^Tq$$ is the inner product in $${\mathbb C}^n\ .$$ Then (see, for example, Kuznetsov (2004)) $l_1(0)= \frac{1}{2\omega_0} {\rm Re}\left[\langle p,C(q,q,\bar{q}) \rangle - 2 \langle p, B(q,A_0^{-1}B(q,\bar{q}))\rangle + \langle p, B(\bar{q},(2i\omega_0 I_n-A_0)^{-1}B(q,q))\rangle \right],$ where $$I_n$$ is the unit $$n \times n$$ matrix. Note that the value (but not the sign) of $$l_1(0)$$ depends on the scaling of the eigenvector $$q\ .$$ The normalization $$\langle q, q \rangle =1$$ is one of the options to remove this ambiguity. Standard bifurcation software (e.g. MATCONT) computes $$l_1(0)$$ automatically.

For planar smooth ODEs with $x=\left(\begin{matrix} u \\ v \end{matrix}\right),\ \ f(x,0)=\left(\begin{matrix} 0 & -\omega_0 \\ \omega_0 & 0\end{matrix}\right)\left(\begin{matrix} u \\ v \end{matrix}\right) + \left(\begin{matrix} P(u,v)\\ Q(u,v)\end{matrix}\right),$ the setting $$q=p=\frac{1}{\sqrt{2}}\left(\begin{matrix} 1 \\ -i\end{matrix}\right)$$ leads to the formula $l_1(0)=\frac{1}{8\omega_0}(P_{uuu}+P_{uvv}+Q_{uuv}+Q_{vvv})$ $\ \ \ \ +\frac{1}{8\omega_0^2}\left[P_{uv}(P_{uu}+P_{vv}) -Q_{uv}(Q_{uu}+Q_{vv})-P_{uu}Q_{uu}+P_{vv}Q_{vv}\right],$ 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 $${\rm sign\ }l_1(0)$$ $\dot{x}_1 = F(x_1)-x_2$ $\dot{x}_2 = a(x_1-b)$ $F'=0$ ${\rm sign\ }F'''$ $\dot{x}_1 = F(x_1)-x_2$ $\dot{x}_2 = a(bx_1-x_2)$ $F'=a$ and $$b>a$$ ${\rm sign}\left[F'''+(F'')^2/(b-a)\right]$ $\dot{x}_1 = F(x_1)-x_2$ $\dot{x}_2 = a(G(x_1)-x_2)$ $F'=a$ and $$G'>a$$ ${\rm sign}\left[F'''+F''(F''-G'')/(G'-a)\right]$

## Other Cases

Andronov-Hopf bifurcation occurs also in infinitely-dimensional ODEs generated by PDEs and DDEs, to which the Center Manifold Theorem applies. An analogue of the Andronov-Hopf bifurcation - called Neimark-Sacker bifurcation - occurs in generic dynamical systems generated by iterated maps when the critical fixed point has a pair of simple eigenvalues $$\mu_{1,2}=e^{\pm i \theta} \ .$$