Andronov-Hopf bifurcation

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Yuri A. Kuznetsov (2006), Scholarpedia, 1(10):1858.

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Curator: Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

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

Figure 2: Fig.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.

1 Definition
2 Two-dimensional Case
3 Multi-dimensional Case
4 First Lyapunov Coefficient
- 4.1 Some Important Examples
5 Other Cases
6 References
7 External Links
8 See Also

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 $n=2$ ,

$\dot{x}_1 = f(x_1,x_2,\alpha)$ ,

$\dot{x}_2 = f(x_1,x_2,\alpha)$ .

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

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

$\dot{y}_2 = y_1 + \beta y_2 + \sigma y_1(y_1^2+y_2^2)$ ,

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 Fig.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 Fig.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.

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

$\dot{y}_2 = y_1 + \beta y_2 + \sigma y_1(y_1^2+y_2^2)$ ,

$\dot{y}^s = -y^s$ ,

$\dot{y}^u = +y^u$ ,

where $y=(y_1,y_2)^T \in {\mathbb R}^2$ , $y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}$ . The figure 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 $i\omega_0$ : $A_0q=i\omega_0 q$ , $\langle q, q \rangle =1$ . 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. 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 $x=0$ (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

Andronov-Hopf bifurcation

From Scholarpedia

Contents

Definition

Two-dimensional Case

Multi-dimensional Case

First Lyapunov Coefficient

Some Important Examples

Other Cases

References

External Links

See Also

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