Codimension-two bifurcations

In a generic autonomous system of ordinary differential equations depending upon parameters,

$$ \tag{a1} {\dot{x} } = f ( x, \alpha ) , $$

where $ f : {\mathbf R ^ {n} \times \mathbf R ^ {p}} \rightarrow {\mathbf R ^ {n}} $ is smooth, two codimension-one singularities of equilibria are possible:

1) fold or saddle-node, where the equilibrium has a simple eigenvalue $ \lambda _ {1} = 0 $ and no other eigenvalues with zero real part, and the restriction of (a1) to the centre manifold near the equilibrium at the critical parameter values has the form

$$ \tag{a2} {\dot{u} } = au ^ {2} + O ( \left | u \right | ^ {3} ) , \quad u \in \mathbf R ^ {1} , $$

with $ a \neq 0 $.

2) Hopf, where the equilibrium has a simple pair of purely imaginary eigenvalues $ \lambda _ {1,2} = \pm i \omega _ {0} $, $ \omega _ {0} > 0 $, and no other eigenvalues with zero real part, and the restriction of (a1) to the centre manifold at the critical parameter values can be transformed near the equilibrium to the form

$$ \tag{a3} \left \{ \begin{array}{l} { {\dot \rho } = l _ {1} \rho ^ {3} + O ( \rho ^ {4} ) , \ \ } \\ { {\dot \varphi } = 1, \ \ } \end{array} \right . $$

where $ ( \rho, \varphi ) $ are polar coordinates and the first Lyapunov coefficient $ l _ {1} \neq 0 $.

The transformation can be performed by smooth invertible changes of variables and a reparametrization of time. These singularities take place at certain codimension-one manifolds in the parameter space of (a1). Crossing these manifolds results in either fold or Hopf bifurcation (cf. Bifurcation). Along these manifolds, the coefficient of the corresponding normal form may vanish or extra eigenvalues may approach the imaginary axis, thus giving rise to the following five codimension-two bifurcations of equilibria in ordinary differential systems [a1], [a5]:

<tbody> </tbody> Name Bifurcation conditions cusp $ \lambda _ {1} = 0 $, $ a = 0 $ generalized Hopf (Bautin) $ \lambda _ {1,2} = \pm i \omega _ {0} $, $ l _ {1} = 0, \omega _ {0} > 0 $, Bogdanov–Takens $ \lambda _ {1,2} = 0 $ fold-Hopf (zero-Hopf) $ \lambda _ {1} = 0 $, $ \lambda _ {2,3} = \pm i \omega _ {0,} \omega _ {0} > 0 $ Hopf–Hopf (double Hopf) $ \lambda _ {1,2} = \pm i \omega _ {1} $, $ \lambda _ {3,4} = \pm i \omega _ {2,} \omega _ {1,2} > 0 $

Each of these cases is characterized by two bifurcation conditions and, generically, appears on codimension-two manifolds in the parameter space of (a1). For example, in generic two-parameter systems, codimension-two bifurcations happen at isolated points in the parameter plane.

Assume that at $ \alpha = ( \alpha _ {1} , \alpha _ {2} ) = ( 0,0 ) $, the two-parameter system

$$ \tag{a4} {\dot{x} } = f ( x, \alpha ) , \quad f : {\mathbf R ^ {n} \times \mathbf R ^ {2}} \rightarrow {\mathbf R ^ {n}} , $$

has the equilibrium $ x = 0 $ exhibiting a codimension-two bifurcation. A normal form of the restriction of a generic system (a4) to its $ n _ {c} $-

dimensional invariant centre manifold can be found using the following table:

<tbody> </tbody> Bifurcation $ n _ {c} $ Normal form cusp 1 $ {\dot{u} } = \beta _ {1} + \beta _ {2} u + \sigma u ^ {3} + O ( u ^ {4} ) $ generalized Hopf 2 $ \begin{array}{c} { {\dot \rho } = \beta _ {1} \rho + \beta _ {2} \rho ^ {3} + \sigma \rho ^ {5} + O ( \rho ^ {6} ),} \\ { {\dot \varphi } = 1} \end{array} $ Bogdanov–Takens 2 $ \begin{array}{c} { {\dot{y} } _ {1} = y _ 2,} \\ { {\dot{y} } _ {2} = \beta _ {1} + \beta _ {2} y _ {1} + y _ {1} ^ {2} + \sigma y _ {1} y _ {2} + O ( \| y \| ^ {3} )} \end{array} $ fold-Hopf 3 $ \begin{array}{c} { {\dot{u} } = \beta _ {1} + u ^ {2} + \sigma \rho ^ {2} + O ( \| {( u, \rho )} \| ^ {4} ),} \\ { {\dot \rho } = \rho ( \beta _ {2} + \theta u + u ^ {2} ) + O ( \| {( u, \rho )} \| ^ {4} ),} \\ { {\dot \psi } = \omega _ {1} + O ( \| {( u, \rho )} \| )} \end{array} $ double Hopf 4 $ \begin{array}{c} { {\dot{r} } _ {1} = r _ {1} ( \beta _ {1} + p _ {11} r _ {1} ^ {2} + p _ {12} r _ {2} ^ {2} + s _ {1} r _ {2} ^ {4} ) + O ( \| {( r _ {1} ,r _ {2} )} \| ^ {6} ),} \\ { {\dot{r} } _ {2} = r _ {2} ( \beta _ {2} + p _ {21} r _ {1} ^ {2} + p _ {22} r _ {2} ^ {2} + s _ {2} r _ {1} ^ {4} ) + O ( \| {( r _ {1} ,r _ {2} )} \| ^ {6} ),} \\ { {\dot \varphi } _ {1} = \omega _ {1} + O ( \| {( r _ {1} ,r _ {2} )} \| ),} \\ { {\dot \varphi } _ {2} = \omega _ {2} + O ( \| {( r _ {1} ,r _ {2} )} \| )} \end{array} $

Here, $ u $ and $ y _ {i} $ are real variables, $ ( \rho, \psi ) $ and $ ( r _ {i} , \varphi _ {i} ) $ are polar coordinates, and $ \sigma = \pm 1 $. Moreover, the functions $ \theta = \theta ( \beta ) $, $ \omega _ {i} = \omega _ {i} ( \beta ) $, $ p _ {ij} = p _ {ij} ( \beta ) $, and $ s _ {i} = s _ {i} ( \beta ) $ are smooth for $ i,j = 1,2 $. The $ O $- terms can depend smoothly on the parameters $ ( \beta _ {1} , \beta _ {2} ) $, and are smooth $ 2 \pi $- periodic functions of $ \psi, \varphi _ {i} $ if polar coordinates are used.

To transform a generic system (a4) restricted to its centre manifold to the corresponding normal form near the origin, one should apply invertible, smooth and smoothly dependent on the parameters changes of variables, time reparametrizations (possibly reversing time), and invertible smooth parameter changes (see [a6]).

For the cusp, generalized Hopf (also called Bautin) and Bogdanov–Takens bifurcations the given normal forms are locally topologically equivalent (cf. Equivalence of dynamical systems) near the origin to the truncated normal forms obtained by dropping the $ O $- terms in the corresponding equations [a1], [a2]. On the contrary, for the fold-Hopf (also called zero-Hopf) and Hopf–Hopf (or double Hopf) bifurcations, the $ O $- terms do determine certain topological features of the parameter and phase portraits and, in general, cannot be eliminated. The bifurcation diagrams of the truncated normal forms can be found in [a2], [a4], [a6], together with all necessary non-degeneracy conditions.

Detection of codimension-two equilibrium bifurcations in a system of ordinary differential equations allows one to predict such global phenomena as hysteresis, invariant tori, limit cycle and homoclinic bifurcations, and chaotic attractors (cf. also Chaos) by means of algebraic computations at the critical equilibrium.

In discrete-time dynamical systems, various codimension-two bifurcations of fixed points are possible, including strong resonances (see [a1], [a3], [a2], [a6]).

References

[a1]	V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Grundlehren math. Wiss. , 250 , Springer (1983) (In Russian)
[a2]	V.I. Arnol'd, V.S. Afraimovich, Yu.S. Il'yashenko, L.P. Shil'nikov, "Bifurcation theory" V.I. Arnol'd (ed.) , Dynamical Systems V , Encycl. Math. Sci. , Springer (1994) (In Russian) Zbl 0791.00009
[a3]	D.K. Arrowsmith, C.M. Place, "An introduction to dynamical systems" , Cambridge Univ. Press (1990) MR1069752 Zbl 0702.58002
[a4]	S.-N. Chow, C. Li, D. Wang, "Normal forms and bifurcations of planar vector fields" , Cambridge Univ. Press (1994) MR1290117
[a5]	J. Guckenheimer, Ph. Holmes, "Nonlinear oscillations, dynamical systems and bifurcations of vector fields" , Springer (1983) MR0709768 Zbl 0515.34001
[a6]	Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995) MR1344214 Zbl 0829.58029