Cusp bifurcation
John Guckenheimer and Yuri A. Kuznetsov (2007), Scholarpedia, 2(4):1852. | doi:10.4249/scholarpedia.1852 | revision #91186 [link to/cite this article] |
The cusp bifurcation is a bifurcation of equilibria in a two-parameter family of autonomous ODEs at which the critical equilibrium has one zero eigenvalue and the quadratic coefficient for the saddle-node bifurcation vanishes.
At the cusp bifurcation point two branches of saddle-node bifurcation curve meet tangentially, forming a semicubic parabola. For nearby parameter values, the system can have three equilibria which collide and disappear pairwise via the saddle-node bifurcations. The cusp bifurcation implies the presence of a hysteresis phenomenon.
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Definition
Consider an autonomous system of ordinary differential equations (ODEs) \[ \dot{x}=f(x,\alpha),\ \ \ x \in {\mathbb R}^n \] depending on two parameters \(\alpha \in {\mathbb R}^2\ ,\) where \(f\) is smooth.
- Suppose that at \(\alpha=0\) the system has an equilibrium \(x^0=0\ .\)
- Further assume that its Jacobian matrix \(A_0=f_x(0,0)\) has a simple eigenvalue \(\lambda_{1}=0 \) and no other eigenvalues with zero real part.
- Finally assume that the quadratic normal form coefficient for the saddle-node bifurcation \( a(0)=0 \ .\)
This bifurcation is characterized by two bifurcation conditions \( \lambda_{1}=0\) and \(a(0) = 0\) (has codimension two) and appears generically in two-parameter families of smooth ODEs. Generically, the critical equilibrium \( x^0 \) is a triple root of the equation \( f(x,0)=0 \) and \(\alpha=0\) is the origin in the parameter plane of two branches of saddle-node bifurcation curve. Crossing each branch results in a pairwise collision and disappearance of equilibria. These bifurcations are nondegenerate and no more than three equilibria exist in a neighbourhood of \( x^0 \ .\)
One-dimensional Case
To describe the bifurcation analytically, consider the system above with \(n=1\ ,\) \[ \dot{x} = f(x,\alpha), \ \ \ x \in {\mathbb R} \ .\] If the following nondegeneracy conditions hold:
- (CP.1) \(c(0)=\frac{1}{6}f_{xxx}(0,0) \neq 0\ ,\)
- (CP.2) the map \( (x,\alpha) \mapsto (f(x,\alpha),f_x(x,\alpha),f_{xx}(x,\alpha)))\) is regular at \( (x,\alpha)=(0,0) \ ,\)
then this system is locally topologically equivalent near the origin to the normal form \[ \dot{y} = \beta_1 + \beta_2 y + \sigma y^3 \ ,\] where \(y \in {\mathbb R},\ \beta \in {\mathbb R}^2\ ,\) and \(\sigma= {\rm sign}\ c(0) = \pm 1\ .\)
The local bifurcation diagram of the normal form with \(\sigma=-1\) is presented in Figure 2. The point \( \beta=0 \) is the origin of two branches of the saddle-node bifurcation curve: \[ LP_{1,2}=\{(\beta_1,\beta_2): \beta_1=\mp \frac{2}{3\sqrt{3}} \beta_2^{3/2},\ \beta_2 > 0 \}, \] which divide the parameter plane into two regions. Inside the wedge between \( LP_1 \) and \( LP_2 \ ,\) there are three equilibria, two stable and one unstable. Outside the wedge, there is a single equilibrium, which is stable. If we approach the cusp point from inside the wedge, all three equilibria merge together.
The equilibrium manifold of the normal form \[ {\mathcal M}=\{(y,\beta) \in {\mathbb R}^3: \beta_1 + \beta_2 y - y^3 =0 \} \] is shown in Figure 1. The projection of this manifold onto the parameter plane has fold singularities along \( LP_{1,2} \ ,\) while the cusp singularity shows up at the origin. Here we have hysteresis: A jump to a different stable equilibrium happens at either \( LP_{1} \) or \( LP_{2} \ ,\) depending on whether the traced under variation of \( \beta_1 \) equilibrium belongs initially to the upper or lower sheet of \( {\mathcal M} \ .\)
The case \( \sigma=1 \) can be reduced to the one above by the substitution
\( t \to -t,\ \beta \to -\beta \)
Multidimensional Case
In the \(n\)-dimensional case with \(n \geq 2\ ,\) the Jacobian matrix \(A_0\) at the cusp bifurcation has
- a simple zero eigenvalue \(\lambda_{1}=0\ ,\) as well as
- \(n_s\) eigenvalues with \({\rm Re}\ \lambda_j < 0\ ,\) and
- \(n_u\) eigenvalues with \({\rm Re}\ \lambda_j > 0\ ,\)
with \(n_s+n_u+1=n\ .\) According to the Center Manifold Theorem, there is a family of smooth one-dimensional invariant manifolds \(W^c_{\alpha}\) near the origin. The \(n\)-dimensional system restricted on \(W^c_{\alpha}\) is one-dimensional, hence has the normal form above.
Moreover, under the non-degeneracy conditions (CP.1) and (CP.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. \[ \dot{y} = \beta_1 + \beta_2 y + \sigma y^3 \ ,\] \[ \dot{y}^s = -y^s \ ,\] \[ \dot{y}^u = +y^u \ ,\] where \(y \in {\mathbb R}\ ,\) \(y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}\ .\) Figure 3 shows the phase portraits of the normal form suspension when \(n=2\ ,\) \(n_s=1\ ,\) \(n_u=0\ ,\) and \(\sigma=+1\ .\)
Cubic Coefficient
The cubic coefficient \(c(0)\ ,\) which is involved in the nondegeneracy condition (CP.1), can be computed for \(n \geq 1\) as follows. Write the Taylor expansion of \(f(x,0)\) at \(x=0\) as \[ f(x,\alpha)=A_0x + \frac{1}{2}B(x,x) + \frac{1}{6}C(x,x,x) + O(\|x\|^4), \] 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 \(q\in {\mathbb R}^n\) be a null-vector of \(A_0\ :\) \(A_0q=0, \ \langle q, q \rangle =1\ ,\) where \(\langle p, q \rangle = p^Tq\) is the standard inner product in \({\mathbb R}^n\ .\) Introduce also the adjoint null-vector \(p \in {\mathbb R}^n\ :\) \(A_0^T p = 0, \ \langle p, q \rangle =1\ .\) Then (see, Kuznetsov (2004)) \[ c(0)= \frac{1}{6} \langle p, C(q,q,q) + 3B(q,h_2)\rangle \ ,\] where \( h_{2} \in {\mathbb R}^n\) is the solution of the nonsingular \( (n+1)\)-dimensional linear system \[ \left(\begin{array}{cc} A_0 & q\\ p^{T} & 0 \end{array} \right) \left(\begin{array}{c} h_{2}\\s\end{array}\right)= \left(\begin{array}{c} -B(q,q)\\0\end{array}\right) \ .\]
Standard bifurcation software (e.g. MATCONT) computes \(c(0)\) automatically.
Other Cases
Cusp bifurcation occurs also in infinitely-dimensional ODEs generated by PDEs and DDEs, to which the Center Manifold Theorem applies. The nomenclature and analysis of cusp bifurcations is based upon cusps in singularity theory where they appear as one of Thom's seven elementary catastrophes.
References
- V.I. Arnold (1983) Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren Math. Wiss., 250, Springer.
- M. Golubitsky and V. Guillemin (1973), Stable Mappings and their Singularities, 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.
- R. Thom (1975), Structural Stability and Morphogenesis, Benjamin.
- R. Thom (1977), Stabilité structurelle et morphogénèse : essai d’une théorie générale des modèles, Intereditions.
Internal references
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Willy Govaerts, Yuri A. Kuznetsov, Bart Sautois (2006) MATCONT. Scholarpedia, 1(9):1375.
- James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
- Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- Emmanuil E. Shnol (2007) Stability of equilibria. Scholarpedia, 2(3):2770.
- Bard Ermentrout (2007) XPPAUT. Scholarpedia, 2(1):1399.
External Links
See Also
Saddle-node Bifurcation, Bifurcations, Center Manifold Theorem, Dynamical Systems, Equilibria, MATCONT, Ordinary Differential Equations, XPPAUT