## Periodic points and chaotic-like dynamics of planar maps associated to nonlinear Hill’s equations with indefinite weight

Venue: | Georgian Mathematical J |

Citations: | 4 - 3 self |

### BibTeX

@ARTICLE{Papini_periodicpoints,

author = {Duccio Papini and Fabio Zanolin},

title = {Periodic points and chaotic-like dynamics of planar maps associated to nonlinear Hill’s equations with indefinite weight},

journal = {Georgian Mathematical J},

year = {},

pages = {339--366}

}

### OpenURL

### Abstract

Abstract. We prove some results about the existence of fixed points, periodic points and chaotic-like dynamics for a class of planar maps which satisfy a suitable property of “arc expansion ” type. We also outline some applications to the nonlinear Hill’s equations with indefinite weight.

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Citation Context ...ros at all (according to any sequence of 1’s and 0’s fixed in advance) in the intervals where q(t) < 0. Further results in this direction and for the superlinear indefinite case have been obtained in =-=[16]-=-, [50], [51]. In particular, a detailed investigation concerning the existence of chaotic-like oscillatory solutions of (1.1) is contained in the recent article [16] by Capietto, Dambrosio and Papini.... |

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Citation Context ...uperlinear growth at infinity, thus including the case of equation (1.2). Recent developments along Butler’s work have been obtained in [49], [50], [51]. In the past ten years, starting with Lassoued =-=[40]-=-, the study of boundary value problems for superlinear equations with an indefinite weight has received much attention in the literature. Besides the case of partial differential equations that we don... |

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Citation Context ...ained for the periodic solutions of Hamiltonian systems and, in particular, second order vector differential equations of the form ü + ∇uV (t, u) = 0, using critical point theory (see [5], [6], [22], =-=[25]-=-, [26], [43]). Other applications of variational methods to the periodic problem for the equation ¨x + q(t)|x| p−1 x = 0 (1.3) are contained in [41] and [57]. In [68], Terracini and Verzini showed tha... |

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Citation Context ..., b) ≤ 2rot(z; a, b) + 1 where rot(z; a, b) = 1 2π ∫b a y(t) 2 + q(t)g(x(t))x(t) y(t) 2 + x(t) 2 Second, thanks to the assumption of superlinear growth at infinity for g(x), we know (see, e.g., [14], =-=[29]-=-, [33], [44], [65] ) that in the interval of positivity of q(t) solutions having at most j zeros are uniformly bounded (by a constant depending on j) in the C 1 -norm on [a, b]. By virtue of these fac... |

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Citation Context ...and further results in this direction will be given in [53]. At the end of this introductory section, we should also mention two recent articles by Kennedy and Yorke [35] and Kennedy, Koçak and Yorke =-=[34]-=- in which the authors develop a general topological framework in order to deduce the presence of chaotic dynamics for a broad variety of different situations. In particular, in [34], the authors defin... |

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Citation Context ...matica, la Probabilità e le loro Applicazioni and supported by MIUR-Cofin2001, Project “Sistemi dinamici non lineari ed applicazioni fisiche”. A summary of the main results of this paper and of [51], =-=[52]-=- was presented during some lectures at the “Third Turin Fortnight on Nonlinear Analysis”, September 24-28, 2001. The authors thank Prof. Anna Capietto and the organizers for the pleasant hospitality. ... |

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Citation Context ... is the case of equation (1.3). However, we point out that in our results the oddness or the homogeneity of g(x) are not required. More general assumptions for g(x) in terms of time-mappings, like in =-=[23]-=-, are given in [52]. Following the notation in [52] we denote by Ai, for i = 1, 2, 3, 4, the open quadrants of the plane, counted in the counterclockwise sense starting from A1 = {(x, y) : x > 0, y > ... |

6 |
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6 |
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Citation Context ...ve been obtained for the periodic solutions of Hamiltonian systems and, in particular, second order vector differential equations of the form ü + ∇uV (t, u) = 0, using critical point theory (see [5], =-=[6]-=-, [22], [25], [26], [43]). Other applications of variational methods to the periodic problem for the equation ¨x + q(t)|x| p−1 x = 0 (1.3) are contained in [41] and [57]. In [68], Terracini and Verzin... |

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Infinitely many solutions for a Floquet-type BVP with superlinearity indefinite in sign
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Citation Context ...iodic solutions to equation (1.1) for a function g(x) having a superlinear growth at infinity, thus including the case of equation (1.2). Recent developments along Butler’s work have been obtained in =-=[49]-=-, [50], [51]. In the past ten years, starting with Lassoued [40], the study of boundary value problems for superlinear equations with an indefinite weight has received much attention in the literature... |

6 |
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Citation Context ...solutions to equation (1.1) for a function g(x) having a superlinear growth at infinity, thus including the case of equation (1.2). Recent developments along Butler’s work have been obtained in [49], =-=[50]-=-, [51]. In the past ten years, starting with Lassoued [40], the study of boundary value problems for superlinear equations with an indefinite weight has received much attention in the literature. Besi... |

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Citation Context ...ons to equation (1.1) for a function g(x) having a superlinear growth at infinity, thus including the case of equation (1.2). Recent developments along Butler’s work have been obtained in [49], [50], =-=[51]-=-. In the past ten years, starting with Lassoued [40], the study of boundary value problems for superlinear equations with an indefinite weight has received much attention in the literature. Besides th... |

6 |
Chaotic-like oscillatory solutions for planar processes, with application to nonlinear Hill’s equations with indefinite weight, in preparation
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Citation Context ...uations, including some cases of asymptotically linear systems (provided that a suitable gap between zero and infinity is assumed). More details and further results in this direction will be given in =-=[53]-=-. At the end of this introductory section, we should also mention two recent articles by Kennedy and Yorke [35] and Kennedy, Koçak and Yorke [34] in which the authors develop a general topological fra... |