Bogdanov-Takens bifurcation

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John Guckenheimer and Yuri A. Kuznetsov (2007), Scholarpedia, 2(1):1854. doi:10.4249/scholarpedia.1854 revision #91072 [link to/cite this article]
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Curator: Yuri A. Kuznetsov

Figure 1: Bogdanov-Takens bifurcation in planar system\[\dot{y}_1=y_2\] and \(\dot{y}_2=\beta_1 + \beta_2y_1+y_1^2-y_1y_2\ .\)

The Bogdanov-Takens (BT) bifurcation is a bifurcation of an equilibrium point in a two-parameter family of autonomous ODEs at which the critical equilibrium has a zero eigenvalue of (algebraic) mulitplicity two. For nearby parameter values, the system has two equilibria (a saddle and a nonsaddle) which collide and disappear via a saddle-node bifurcation. The nonsaddle equilibrium undergoes an Andronov-Hopf bifurcation generating a limit cycle. This cycle degenerates into an orbit homoclinic to the saddle and disappears via a saddle homoclinic bifurcation.

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Definition

Consider an autonomous system of ordinary differential equations (ODEs) \[\tag{1} \dot{x}=f(x,\alpha),\ \ \ x \in {\mathbb R}^n \]

depending on two parameters \(\alpha \in {\mathbb R}^2\ ,\) where \(f\) is smooth.

  • Suppose that at \(\alpha=0\) the system has an equilibrium \(x^0=0\ .\)
  • Assume that its Jacobian matrix \(A_0=f_x(0,0)\) has zero eigenvalue of (algebraic) multiplicity two \(\lambda_{1,2}=0 \ .\)

This bifurcation is characterized by two bifurcation condition \(\lambda_1=\lambda_2=0\) (has codimension two) and appears generically in two-parameter families of smooth ODEs.

Generically, the critical equilibrium \( x^0 \) is a double root of the equation \( f(x,0)=0 \) and \(\alpha=0\) is the origin in the parameter plane of

Moreover, these bifurcations are nondegenerate and no other bifurcation occur in a small fixed neighbourhood of \( x^0 \) for parameter values sufficiently close to \(\alpha=0\ .\) In this neighbourhood, the system has at most two equilibria and one limit cycle.

Two-dimensional Case

To describe the BT-bifurcation analytically, consider the system (1) with \(n=2\ ,\) \[ \dot{x} = f(x,\alpha), \ \ \ x \in {\mathbb R}^2 \ .\] If the following nondegeneracy conditions hold:

  • (BT.1) \(a(0)b(0) \neq 0\ ,\) where \( a(0) \) and \( b(0)\) are certain quadratic coefficients (see below),
  • (BT.2) the map \( (x,\alpha) \mapsto (f(x,\alpha),{\rm Tr}(f_x(x,\alpha)),\det(f_x(x,\alpha)))\) is regular at \( (x,\alpha)=(0,0) \ ,\)

then this system is locally topologically equivalent near the origin to the normal form \[ \dot{y}_1 = y_2 \ ,\] \[ \dot{y}_2 = \beta_1 + \beta_2 y_1 + y_1^2 + \sigma y_1y_2 \ ,\] where \(y=(y_1,y_2)^T \in {\mathbb R}^2,\ \beta=(\beta_1,\beta_2)^T \in {\mathbb R}^2\ ,\) and \(\sigma= {\rm sign}\ a(0)b(0) = \pm 1\ .\)

The local bifurcation diagram of the normal form with \(\sigma=-1\) is presented in Figure 1. The point \( \beta=0 \) separates two branches of the saddle-node bifurcation curve: \[ T_{+}=\{(\beta_1,\beta_2): \beta_1=\frac{1}{4}\beta_2^2,\ \beta_2>0 \} \] and \[ T_{-}=\{(\beta_1,\beta_2): \beta_1=\frac{1}{4}\beta_2^2,\ \beta_2<0 \} \ .\] The half-line \[ H=\{(\beta_1,\beta_2): \beta_1=0,\ \beta_2<0 \} \] corresponds to the Andronov-Hopf bifurcation that generates a stable limit cycle. This cycle exists and remains hyperbolic between the line \( H \) and a smooth curve \[ P=\{(\beta_1,\beta_2): \beta_1=-\frac{6}{25}\beta_2^2 + O(|\beta_2|^3),\ \beta_2<0 \} \ ,\] at which a saddle homoclinic bifurcation occurs. When the cycle approaches the homoclinic orbit, its period tends to infinity.

The case \( \sigma=1 \) can be reduced to the one above by the substitution \( t \to -t, \ y_2 \to -y_2 \ .\) This does not affect the bifurcation curves but the limit cycle becomes unstable.

Multidimensional Case

In the \(n\)-dimensional case with \(n \geq 2\ ,\) the Jacobian matrix \(A_0\) at the Bogdanov-Takens bifurcation has

  • a zero eigenvalue \(\lambda_{1,2}=0\) with (algebraic) multiplicity two, 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 (BT.1) and (BT.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. \[ \dot{y}_1 = y_2 \ ,\] \[ \dot{y}_2 = \beta_1 + \beta_2 y_1 + y_1^2 + \sigma y_1y_2 \ ,\] \[ \dot{y}^s = -y^s \ ,\] \[ \dot{y}^u = +y^u \ ,\] where \(y \in {\mathbb R}\ ,\) \(y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}\ .\)

Quadratic Coefficients

The quadratic coefficients \(a(0)\) and \(b(0)\ ,\) which are involved in the nondegeneracy condition (BT.1), can be computed for \(n \geq 2\) as follows. Write the Taylor expansion of \(f(x,0)\) at \(x=0\) as \[ f(x,0)=A_0x + \frac{1}{2}B(x,x) + O(\|x\|^3) \ ,\] where \(B(x,y)\) is the bilinear function with components \[ \ \ B_j(x,y) =\sum_{k,l=1}^n \left. \frac{\partial^2 f_j(\xi,0)}{\partial \xi_k \partial \xi_l}\right|_{\xi=0} x_k y_l \ ,\] where \(j=1,2,\ldots,n\ .\) Let \(q_0, q_1, p_0, p_1\in {\mathbb R}^n\) be nonzero vectors that satisfy: \[ \ \ A_0q=0, \ A_0q_1=q_0, \ A_0p_1=0, \ A_0p_0=p_1 \] and are normalized so that \[ \langle p_0, q_0 \rangle = \langle p_1, q_1 \rangle = 1,\ \langle p_0, q_1 \rangle = \langle p_1, q_0 \rangle = 0 \ ,\] where \(\langle p, q \rangle = p^Tq\) is the standard inner product in \({\mathbb R}^n\ .\) Then (see, for example, Kuznetsov (2004)) \[ a(0)= \frac{1}{2} \langle p_1, B(q_0,q_0))\rangle,\ \ b(0)= \langle p_0, B(q_0,q_0))\rangle + \langle p_1, B(q_0,q_1))\rangle \ .\] Standard bifurcation software (e.g. MATCONT) computes \(a(0)\) and \(b(0)\)automatically.

Other Cases

Bogdanov-Takens bifurcation occurs also in infinitely-dimensional ODEs generated by PDEs and DDEs, to which the Center Manifold Theorem applies.

References

  • 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
  • Yu.A. Kuznetsov (2004) Elements of Applied Bifurcation Theory, Springer, 3rd edition.

Internal references

External Links

See Also

Andronov-Hopf Bifurcation, Saddle-node Bifurcation, Bifurcations, Center Manifold Theorem, Dynamical Systems, Equilibria, MATCONT, Ordinary Differential Equations, Homoclinic Bifurcation, XPPAUT

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