Andronov Hopf bifurcation - davidar/scholarpedia GitHub Wiki

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.

Table of Contents

Definition

Consider an autonomous system of ordinary differential equations (ODEs)

<math>
\dot{x}=f(x,\alpha),\ \ \ x \in {\mathbb R}^n </math> depending on a parameter <math>\alpha \in {\mathbb R}\ ,</math> where <math>f</math> is smooth.
  • Suppose that for all sufficiently small <math>|\alpha|</math> the system has a family of equilibria <math>x^0(\alpha)\ .</math>
  • Further assume that its Jacobian matrix <math>A(\alpha)=f_x(x^0(\alpha),\alpha)</math> has one pair of complex eigenvalues
<math>
\lambda_{1,2}(\alpha)=\mu(\alpha) \pm i\omega(\alpha) </math> that becomes purely imaginary when <math>\alpha=0\ ,</math> i.e., <math>\mu(0)=0</math> and <math>\omega(0)=\omega_0>0\ .</math> Then, generically, as <math>\alpha</math> passes through <math>\alpha=0\ ,</math> the equilibrium changes stability and a unique limit cycle bifurcates from it. This bifurcation is characterized by a single bifurcation condition <math>{\rm Re}\ \lambda_{1,2}=0</math> (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 <math>n=2\ ,</math>

<math>
\dot{x}_1 = f_1(x_1,x_2,\alpha) \ ,</math>
<math>
\dot{x}_2 = f_2(x_1,x_2,\alpha) \ .</math> If the following nondegeneracy conditions hold:
  • (AH.1) <math>l_1(0) \neq 0\ ,</math> where <math>l_1(\alpha)</math> is the first Lyapunov coefficient (see below);
  • (AH.2) <math>\mu'(0) \neq 0\ ,</math>
then this system is locally topologically equivalent near the equilibrium to the normal form
<math>
\dot{y}_1 = \beta y_1 - y_2 + \sigma y_1(y_1^2+y_2^2) \ ,</math>
<math>
\dot{y}_2 = y_1 + \beta y_2 + \sigma y_2(y_1^2+y_2^2) \ ,</math> where <math>y=(y_1,y_2)^T \in {\mathbb R}^2,\ \beta \in {\mathbb R}\ ,</math> and <math>\sigma= {\rm sign}\ l_1(0) = \pm 1\ .</math>
  • If <math>\sigma=-1\ ,</math> the normal form has an equilibrium at the origin, which is asymptotically stable for <math>\beta \leq 0</math> (weakly at <math>\beta=0</math>) and unstable for <math>\beta>0\ .</math> Moreover, there is a unique and stable circular limit cycle that exists for <math>\beta>0</math> and has radius <math>\sqrt{\beta}\ .</math> This is a supercritical Andronov-Hopf bifurcation (see <figref>SuperHopf.gif</figref>).
  • If <math>\sigma=+1\ ,</math> the origin in the normal form is asymptotically stable for <math>\beta<0</math> and unstable for <math>\beta \geq 0</math> (weakly at <math>\beta=0</math>), while a unique and unstable limit cycle exists for <math>\beta <0\ .</math> This is a subcritical Andronov-Hopf bifurcation (see <figref>SubHopf.gif</figref>).

Multi-dimensional Case

In the <math>n</math>-dimensional case with <math>n \geq 3\ ,</math> the Jacobian matrix <math>A_0=A(0)</math> has

  • a simple pair of purely imaginary eigenvalues <math>\lambda_{1,2}=\pm i \omega_0, \ \omega_0>0\ ,</math> as well as
  • <math>n_s</math> eigenvalues with <math>{\rm Re}\ \lambda_j < 0\ ,</math> and
  • <math>n_u</math> eigenvalues with <math>{\rm Re}\ \lambda_j > 0\ ,</math>
with <math>n_s+n_u+2=n\ .</math> According to the Center Manifold Theorem, there is a family of smooth two-dimensional invariant manifolds <math>W^c_{\alpha}</math> near the origin. The <math>n</math>-dimensional system restricted on <math>W^c_{\alpha}</math> is two-dimensional, hence has the normal form above.

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

<math>
\dot{y}_1 = \beta y_1 - y_2 + \sigma y_1(y_1^2+y_2^2) \ ,</math>
<math>
\dot{y}_2 = y_1 + \beta y_2 + \sigma y_2(y_1^2+y_2^2) \ ,</math>
<math>
\dot{y}^s = -y^s \ ,</math>
<math>
\dot{y}^u = +y^u \ ,</math> where <math>y=(y_1,y_2)^T \in {\mathbb R}^2\ ,</math> <math>y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}\ .</math> <figref>3DHopf.gif</figref> shows the phase portraits of the normal form suspension when <math>n=3\ ,</math> <math>n_s=1\ ,</math> <math>n_u=0\ ,</math> and <math>\sigma=-1\ .</math>

First Lyapunov Coefficient

Whether Andronov-Hopf bifurcation is subcritical or supercritical is determined by <math>\sigma\ ,</math> which is the sign of the first Lyapunov coefficient <math>l_1(0)</math> of the dynamical system near the equilibrium. This coefficient can be computed at <math>\alpha=0</math> as follows. Write the Taylor expansion of <math>f(x,0)</math> at <math>x=0</math> as

<math>
f(x,0)=A_0x + \frac{1}{2}B(x,x) + \frac{1}{6}C(x,x,x) + O(\|x\|^4), </math> where <math>B(x,y)</math> and <math>C(x,y,z)</math> are the multilinear functions with components
<math>
\ \ 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 \ ,</math>
<math>
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 \ ,</math> where <math>j=1,2,\ldots,n\ .</math> Let <math>q\in {\mathbb C}^n</math> be a complex eigenvector of <math>A_0</math> corresponding to the eigenvalue <math>i\omega_0\ :</math> <math>A_0q=i\omega_0 q\ .</math> Introduce also the adjoint eigenvector <math>p \in {\mathbb C}^n\ :</math> <math>A_0^T p = - i\omega_0 p\ ,</math> <math> \langle p, q \rangle =1\ .</math> Here <math>\langle p, q \rangle = \bar{p}^Tq</math> is the inner product in <math>{\mathbb C}^n\ .</math> Then (see, for example, Kuznetsov (2004))
<math>
l_1(0)= \frac{1}{2\omega_0} {\rm Re}\left[\langle], </math> where <math>I_n</math> is the unit <math>n \times n</math> matrix. Note that the value (but not the sign) of <math>l_1(0)</math> depends on the scaling of the eigenvector <math>q\ .</math> The normalization <math> \langle q, q \rangle =1</math> is one of the options to remove this ambiguity. Standard bifurcation software (e.g. MATCONT) computes <math>l_1(0)</math> automatically.

For planar smooth ODEs with

<math>
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), </math> the setting <math> q=p=\frac{1}{\sqrt{2}}\left(\begin{matrix} 1 \\ -i\end{matrix}\right) </math> leads to the formula
<math>
l_1(0)=\frac{1}{8\omega_0}(P_{uuu}+P_{uvv}+Q_{uuv}+Q_{vvv}) </math>
<math>\ \ \ \ +\frac{1}{8\omega_0^2}\left[P_{uv}(P_{uu}+P_{vv})],
</math> where the lower indices mean partial derivatives evaluated at <math>x=0</math> (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 <math>a,b>0</math> are positive parameters and all derivatives should be evaluated at the critical equilibrium.

System Condition <math>{\rm sign\ }l_1(0)</math>
<math>
\dot{x}_1 = F(x_1)-x_2 </math>
<math>
\dot{x}_2 = a(x_1-b) </math>
<math>F'=0</math>
<math>{\rm sign\ }F</math>
<math>
\dot{x}_1 = F(x_1)-x_2 </math>
<math>
\dot{x}_2 = a(bx_1-x_2) </math>
<math>F'=a</math> and <math>b>a</math>
<math>{\rm sign}\left[F+(F)^2/(b-a)\right]</math>
<math>
\dot{x}_1 = F(x_1)-x_2 </math>
<math>
\dot{x}_2 = a(G(x_1)-x_2) </math>
<math>F'=a</math> and <math>G'>a</math>
<math>{\rm sign}\left[F+F(F-G)/(G'-a)\right]</math>


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 <math> \mu_{1,2}=e^{\pm i \theta} \ .</math>

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

Category:Bifurcations

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