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.
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 \( \mu_{1,2}=e^{\pm i \theta} \ .\)
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
See Also
Bifurcations, Center Manifold Theorem, Dynamical Systems, Equilibria, MATCONT, Ordinary Differential Equations, XPPAUT