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Cusp bifurcation

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David A Roberts edited this page Dec 2, 2015 · 1 revision 




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.






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 two parameters <math>\alpha \in {\mathbb R}^2\ ,</math> where <math>f</math> is 
smooth. 





    

  • Suppose that at <math>\alpha=0</math> the system has an equilibrium <math>x^0=0\ .</math> 
    • 

  • Further assume that its Jacobian matrix <math>A_0=f_x(0,0)</math>  has a simple eigenvalue <math>\lambda_{1}=0 </math> and no other eigenvalues with zero real part.
    • 

  • Finally assume that the quadratic normal form coefficient for the saddle-node bifurcation <math> a(0)=0 \ .</math>
    • 


This bifurcation is 
characterized by two bifurcation conditions <math> \lambda_{1}=0</math> and
<math>a(0) = 0</math> (has codimension two) 
and appears generically in two-parameter families of smooth ODEs. 
Generically, the critical equilibrium <math> x^0 </math> is a triple root of the equation 
<math> f(x,0)=0 </math> and <math>\alpha=0</math> 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 <math> x^0 \ .</math>






One-dimensional Case






To describe the bifurcation analytically, consider the system above with <math>n=1\ ,</math>

<math>

\dot{x} = f(x,\alpha), \ \ \ x \in {\mathbb R}
\ .</math>
If the following nondegeneracy conditions hold:




    

  • 
(CP.1) <math>c(0)=\frac{1}{6}f_{xxx}(0,0) \neq 0\ ,</math>
    • 

  • 
(CP.2) the map <math> (x,\alpha) \mapsto (f(x,\alpha),f_x(x,\alpha),f_{xx}(x,\alpha)))</math> is regular at <math> (x,\alpha)=(0,0) \ ,</math>
    • 






then this system is locally topologically equivalent 
near the origin to the normal form

<math>

\dot{y} = \beta_1 + \beta_2 y + \sigma y^3
\ ,</math>
where <math>y \in {\mathbb R},\ \beta \in {\mathbb R}^2\ ,</math> and 
<math>\sigma= {\rm sign}\ c(0) = \pm 1\ .</math> 


The local bifurcation diagram of the normal form with <math>\sigma=-1</math> is presented in <figref>1Dcusp.gif</figref>. The point <math> \beta=0 </math> is the origin of two branches of the 
saddle-node bifurcation curve: 

<math>

LP_{1,2}=\{(\beta_1,\beta_2): \beta_1=\mp \frac{2}{3\sqrt{3}} \beta_2^{3/2},\ \beta_2 > 0 \},
</math>
which divide the parameter plane into two regions. Inside the wedge between <math> LP_1 </math>
and <math> LP_2 \ ,</math> 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 

<math>

{\mathcal M}=\{(y,\beta) \in {\mathbb R}^3: \beta_1 + \beta_2 y - y^3 =0 \}
</math>
is shown in  <figref>1DcuspManifold.gif</figref>. The projection of this manifold onto the parameter
plane has fold singularities along <math> LP_{1,2} \ ,</math> while the cusp singularity
shows up at the origin. Here we have hysteresis:
A jump to a different stable equilibrium happens at either <math> LP_{1} </math> or
<math> LP_{2} \ ,</math> depending on whether the traced under variation of <math> \beta_1 </math>
equilibrium belongs initially to the upper or lower sheet of <math> {\mathcal M} \ .</math>



The case <math> \sigma=1 </math> can be reduced to the one above by the substitution
<math> t \to -t,\ \beta \to -\beta </math>






Multidimensional Case






In the <math>n</math>-dimensional case with <math>n \geq 2\ ,</math> the Jacobian 
matrix <math>A_0</math> at the cusp bifurcation has 




    

  • a simple zero eigenvalue <math>\lambda_{1}=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+1=n\ .</math> 
According to the Center Manifold Theorem, there is a family of smooth 
one-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 
one-dimensional, hence has the normal form above. 


Moreover, under the non-degeneracy conditions (CP.1) and (CP.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} = \beta_1 + \beta_2 y + \sigma y^3
\ ,</math>

<math>

\dot{y}^s = -y^s
\ ,</math>

<math>

\dot{y}^u = +y^u
\ ,</math>
where <math>y \in {\mathbb R}\ ,</math> <math>y^s \in {\mathbb R}^{n_s}, \ y^u 
\in {\mathbb R}^{n_u}\ .</math>  <figref>2Dcusp.gif</figref> 
shows the phase portraits of the normal form suspension 
when <math>n=2\ ,</math> <math>n_s=1\ ,</math> <math>n_u=0\ ,</math> and <math>\sigma=+1\ .</math>




Cubic Coefficient






The cubic coefficient <math>c(0)\ ,</math> which is involved in the nondegeneracy
condition (CP.1), can be computed for <math>n \geq 1</math> as follows. 
Write the Taylor expansion of <math>f(x,0)</math> at <math>x=0</math> as

<math>

f(x,\alpha)=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 R}^n</math> be a null-vector 
of <math>A_0\ :</math> <math>A_0q=0, \ \langle q, q \rangle =1\ ,</math> 
where <math>\langle p, q \rangle = p^Tq</math> is the standard inner product 
in <math>{\mathbb R}^n\ .</math>
Introduce also the adjoint null-vector <math>p \in {\mathbb R}^n\ :</math> <math>A_0^T p = 0, \ \langle p, q \rangle =1\ .</math> Then
(see, Kuznetsov (2004)) 

<math>

c(0)= \frac{1}{6} \langle p, C(q,q,q) + 3B(q,h_2)\rangle 
\ ,</math>
where  <math> h_{2} \in {\mathbb R}^n</math>
is the solution of the nonsingular <math> (n+1)</math>-dimensional linear system

<math>

\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)
\ .</math>



Standard bifurcation software (e.g. MATCONT) computes <math>c(0)</math> 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. doi:10.1007/978-1-4615-7904-5.
    • 

  • 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.  doi:10.1007/978-1-4757-3978-7.
    • 

  • 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















External Links








See Also






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



Category: Dynamical Systems
Category:Bifurcations
Category:Multiple_Curators

              


              

                
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