# Bogdanov-Takens bifurcation

Post-publication activity

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) multiplicity 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.

## 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 $$y \in {\mathbb R}\ ,$$ $$y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}\ .$$

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: $\ \ A_0q=0, \ A_0q_1=q_0, \ A_0^Tp_1=0, \ A_0^Tp_0=p_1$ 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.

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