Andronov-Hopf bifurcation

• Prof. Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

Figure 1: Supercritical Andronov-Hopf bifurcation in the plane.

Figure 2: Subcritical Andronov-Hopf bifurcation in the plane.

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.

Figure 3: Supercritical Hopf bifurcation in the 3D-space.

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 $\mu_{1,2}=e^{\pm i \theta} \ .$

References

• A.A. Andronov, E.A. Leontovich, I.I. Gordon, and A.G. Maier (1971) Theory of Bifurcations of Dynamical Systems on a Plane. Israel Program Sci. Transl.
• V.I. Arnold (1983) Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren Math. Wiss., 250, Springer
• J. Guckenheimer and P. Holmes (1983) Nonlinear Oscillations, Dynamical systems and Bifurcations of Vector Fields. Springer
• E.M. Izhikevich (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press.
• Yu.A. Kuznetsov (2004) Elements of Applied Bifurcation Theory, Springer, 3rd edition.
• J. Marsden and M. McCracken (1976) Hopf Bifurcation and its Applications. Springer

Internal references

• Willy Govaerts, Yuri A. Kuznetsov, Bart Sautois (2006) MATCONT. Scholarpedia, 1(9):1375.
• James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

External Links

• Yuri A. Kuznetsov's website

See Also

Bifurcations, Center Manifold Theorem, Dynamical Systems, Equilibria, MATCONT, Ordinary Differential Equations, XPPAUT

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Accepted on: 2006-10-02 12:15:31 GMT