Abstract
The main purpose of this paper is to investigate dynamical systems \({F : \mathbb{R}^2 \rightarrow \mathbb{R}^2}\) of the form F(x, y) = (f(x, y), x). We assume that \({f : \mathbb{R}^2 \rightarrow \mathbb{R}}\) is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x0 = (x 0, y 0), such that the orbit
is bounded, O(x0) converges provided that the set of fixed point of F is totally disconnected and F does not admit periodic orbits of prime period two. The obtained result is used to show that all aperiodic orbits can be removed from the dynamics of the map H of Hénon. The goal can be achieved by perturbing H so that the perturbed map H 1 does not have any periodic point of prime period two.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T.M. Apostol, Mathematical Analysis. Second Edition, Addison-Wesley Publishing Co. (1974).
Bashurov V.V., Ogibin V.N.: Conditions for the convergence of iterative processes on the real axis. U.S.S.R. Comp. Math. and Math. Phys. 6(5), 178–184 (1966)
Chu S.C., Moyer R.D.: On continuous functions, commuting functions and fixed points. Fund. Math. 59, 91–95 (1966)
R.L. Devaney, An introduction to Chaotic Dynamical Systems. Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company (1989).
Di Lena G.: Convergenza globale del metodo delle approssimazioni successive in R m per una classe di funzioni. Boll. U.M.I. 5(18-A), 235–241 (1981)
Di Lena G., Franco D., Messano B.: An extension of Chu-Moyer’s theorem to two variable functions. Nonlinear Analysis 69, 3525–3536 (2008)
G. Di Lena, B. Messano, D. Roux, On the Successive Approximation Method for Isotone Functions. Boll. U.M.I. (7) 8-A (1994), 169–180.
Di Lena G., Messano B., Zitarosa A.: On the iterative process x n+1 = f(x n , x n–1). Rend. Sem. Mat. Univ. Padova 80, 139–150 (1988)
Hénon M.: A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50, 69–77 (1976)
E. Kreyszig, Advanced Engineering Mathematics. Seventh Edition (1993), 501.
Marotto F.R.: Chaotic behavior in the Hénon attractor. Comm. Math. Phys. 68, 716–729 (1979)
Messano B.: On the successive approximations method and on the interative process x n+1 = f(x + n, x n–1). Rend. di Matematica 7, 281–297 (1987)
B. Messano, Continuous Functions from an Arcwise Connected Tree into Itself: Periodic Points, Global Convergence, Plus-Global Convergence. Ricerche di Matematica XXXVIII, fasc. 2 (1989), 199–205.
B. Messano, Questioni relative alle equazioni funzionali del tipo \({A(x) - A(\tau(x)) = \varphi(x)}\) e problema di Goursat per l’equazione u xy = 0. Rend. Circ. Matem. Palermo XXXIX Serie II (1990), 281–298.
Messano B.: On functional equations of the form \({A(x) - A(\tau(x)) = \varphi(x)}\) and Goursat problem for the equation \({\frac{\partial^2z(x,y)}{\partial{x}\partial{y}} = f(x, y)}\). Rend. di Matematica XII, 521–543 (1992)
Messano B., Zitarosa A.: Minimal sets of generalized dynamical systems. Rend. Mat. Accad. Lincei LXXXIII, 1–5 (1989)
R.N. Pederson, The Jordan Curve Theorem for Piecewise Smooth Curves. The American Mathematical Monthly 76, No. 6 (1969), 605–610.
H.L. Royden, Real Analysis. MacMillan Publishing (1968), 47.
Sibirsky K.S.: Introduction to Topological Dynamics. Noordoff International Publishing, Leyden (1975)
Ženíšek A.: Green’s Theorem from the viewpoint of applications. Applications of Mathematics 44, 55–80 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
After the paper had been written, Prof. Giovanni Di Lena passed away. Basilio, Davide and Mario are truly devastated by this sudden and unexpected loss. They dedicate the paper to the loving memory of Giovanni.
Rights and permissions
About this article
Cite this article
Di Lena, G., Franco, D., Martelli, M. et al. From Chaos to Global Convergence. Mediterr. J. Math. 8, 473–489 (2011). https://doi.org/10.1007/s00009-010-0091-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-010-0091-7