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Antimonotonicity: Concurrent creation and annihilation of periodic orbits. (English) Zbl 0765.58020

It is well-known that periodic orbits of the logistic family of one- dimensional maps \(f_ \lambda=\lambda-x^ 2\) are created monotonically as the parameter \(\lambda\) is increased. An antimonotonicity in the creation and annihilation of periodic orbits is indeed reflected in numerical simulations of the Hénon family \(H_ \lambda(x,y)=(\lambda- x^ 2+0.3y, x)\), which is a one-parameter family of dissipative diffeomorphisms of the plane. The main purpose of this paper is to show a general conclusion as the following Antimonotonicity Theorem: In any neighborhood of a nondegenerate, homoclinic-tangency parameter value of a one-parameter family of dissipative \(C^ 3\) diffeomorphisms of the plane, there must be both infinitely many orbit-creation and infinitely many orbit-annihilation parameter values.
This theorem is an immediate consequence of a so-called Bubble Lemma, which is involved in a tedious discussion for its proof.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
34C23 Bifurcation theory for ordinary differential equations
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