# Some Codimension-Two Bifurcations for Maps, Leading to Chaos

• G. Iooss
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 24)

## Abstract

Consider a family of mappings F μ in a Banach space, depending on a real parameter μ, and assume that 0 is a fixed point of F°. The derivative at this point being noted T°, an elementary bifurcation occurs in each of these cases:
1. i)

1 is a simple eigenvalue of T°, the remaining part of its spectrum being of modulus less than 1. This is the “saddle-node” bifurcation where, while μ crosses 0, two fixed points (a saddle point and a node) meet together and disappear. By some extra nonlocal phenomenon this can produce “intermittency” of a simple kind [1].

2. ii)

−1 is a simple eigenvalue of T°, the remaining part of its spectrum being of modulus less than 1. This is the “flip” bifurcation where, while μ crosses 0, the fixed point is changed from a node to a saddle, and two periodic points appear of period 2 for μ on one side of 0, they are node (resp.saddle) if the fixed point is a saddle (resp.node). A succession of such bifurcations seems to be frequent in physics (for instance in hydrodynamics [2]).

3. iii)

λ° and $${\bar \lambda _ \circ }$$ are simple eigenvalues of T° on the unit circle, $$\lambda _ \circ ^n\, \ne \,1$$ for n=l,2,3,4, the remaining part of the spectrum being of modulus less than 1. This is the “Hopf bifurcation” for maps which lead, while μ. crosses 0, to the creation of an invariant circle under F μ , growing from 0, attracting (resp. repelling) if it appears on the side where the fixed point is repelling (resp. attracting). The dynamic on the invariant circle depends on μ and is related to the rotation number of the diffeomorphism of the circle (the restriction of F μ to the invariant circle) [3], [4].

## Keywords

Hopf Bifurcation Phase Portrait Periodic Point Chaotic Behavior Homoclinic Orbit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Pomeau, Y., P. Mannevilie, Intermittency and the Lorenz model, Phys.Lett. 75 A, 1, 1979
2. [2]
Giglio, M., S. Musazzi, U. Perini, Transition to chaos via a well ordered sequence of period doubling bifurcations, Phys.Rev.Lett. 47, 243, 1981
3. [3]
looss, G., Bifurcation of maps and applications, North Holland Math. Studies, 36, 1979
4. [4]
Marsden, J.E., M.Mc Cracken, The Hopf bifurcation and its applications, Applied Math. Sci. 19, Springer Verlag, 1976Google Scholar
5. [5]
Chenciner, A., Bifurcations de difféomorphismes de M2, au voisinage d’un point fixe elliptique. Chaotic behavior of deterministic system, Les Houches Summer School 1981, G.IOOSS, R.HELLEMAN, R.STORA ed., North Holland, (to appear in 1983 )Google Scholar
6. [6]
Arnold, V., Chapitres supplémentaires de la Théorie des équations différentielles ordinaires, Mir, Moscou, 1980Google Scholar
7. [7]
Bogdanov, R., Trudy Sem. Petrov, 2, 1976, 23
8. [8]
Iooss, G., Persistance d’un cercle invariant par une application voisine de “l’application temps T” d’un champ de vecteurs intégrable. Partie I: En dehors de la bifurcation de Hopf, Partie II: Voisinage d’une bifurcation de Hopf. Exemples d’applications, C.R. Acad.Sci.Paris, t.296,I, 27–30 et 113–116, 1983Google Scholar
9. [9]
Gambaudo, J -M., Perturbation de “l’application temps T” d’un champ de vecteurs intégrable de 2, C.R. Acad. Sci. Paris, I, (to appear)Google Scholar
10. [10]
Newhouse, S.E., The creation of non-trivial recurrence in the dynamics of diffeomorphisms. Chaotic behavior of deterministic systems, Les Houches Summer School 1981, G.IOOSS, R.HELLEMAN, R.STORA ed., North Holland, (to appear in 1983 )Google Scholar
11. [11]
Chencîner, A., G.Ioossr Bifurcations de tores invariants, Arch. Rat. Mech. Anal. 69, 2, 109–198, and 71, 301–306, 1979
12. [12]
Ruelle, D., F. Takens, On the nature of turbulence, Com. Math. Phys. 20, 167–192 and 23, 343–344, 1971.