scholarly journals Periodic Points and Chaotic-Like Dynamics of Planar Maps Associated to Nonlinear Hill's Equations with Indefinite Weight

2002 ◽  
Vol 9 (2) ◽  
pp. 339-366
Author(s):  
Duccio Papini ◽  
Fabio Zanolin
Keyword(s):  
Fixed Points ◽  
Periodic Points ◽  
Indefinite Weight ◽  
Suitable Property ◽  
Planar Maps ◽  
Hill’S Equations

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.

scholarly journals Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight

2021 ◽  
Vol 379 (2191) ◽  
pp. 20190385
Author(s):  
Alessandro Margheri ◽  
Carlota Rebelo ◽  
Fabio Zanolin
Keyword(s):  
Fixed Points ◽  
Complex Dynamics ◽  
Indefinite Weight ◽  
Multiplicity Results ◽  
Birkhoff Theorem ◽  
Topological Horseshoes ◽  
Indefinite Weights ◽  
Planar Systems ◽  
Planar Maps ◽  
Area Preserving

In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive–contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré–Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré–Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

scholarly journals Fixed points and periodic points of semiflows of holomorphic maps

2003 ◽  
Vol 2003 (4) ◽  
pp. 217-260
Author(s):  
Edoardo Vesentini
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Complex Banach Space ◽  
Periodic Points ◽  
General Question ◽  
Open Unit ◽  
Holomorphic Maps ◽  
Open Unit Ball ◽  
Spin Factor ◽  
Cartan Factor

Letϕbe a semiflow of holomorphic maps of a bounded domainDin a complex Banach space. The general question arises under which conditions the existence of a periodic orbit ofϕimplies thatϕitself is periodic. An answer is provided, in the first part of this paper, in the case in whichDis the open unit ball of aJ∗-algebra andϕacts isometrically. More precise results are provided when theJ∗-algebra is a Cartan factor of type one or a spin factor. The second part of this paper deals essentially with the discrete semiflowϕgenerated by the iterates of a holomorphic map. It investigates how the existence of fixed points determines the asymptotic behaviour of the semiflow. Some of these results are extended to continuous semiflows.

scholarly journals Dynamical properties of some adic systems with arbitrary orderings

2016 ◽  
Vol 37 (7) ◽  
pp. 2131-2162 ◽  
Author(s):  
SARAH FRICK ◽  
KARL PETERSEN ◽  
SANDI SHIELDS
Keyword(s):  
Probability Measure ◽  
Fixed Points ◽  
Invariant Probability Measure ◽  
Random Order ◽  
Periodic Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Necessary And Sufficient

We consider arbitrary orderings of the edges entering each vertex of the (downward directed) Pascal graph. Each ordering determines an adic (Bratteli–Vershik) system, with a transformation that is defined on most of the space of infinite paths that begin at the root. We prove that for every ordering the coding of orbits according to the partition of the path space determined by the first three edges is essentially faithful, meaning that it is one-to-one on a set of paths that has full measure for every fully supported invariant probability measure. We also show that for every$k$the subshift that arises from coding orbits according to the first$k$edges is topologically weakly mixing. We give a necessary and sufficient condition for any adic system to be topologically conjugate to an odometer and use this condition to determine the probability that a random order on a fixed diagram, or a diagram constructed at random in some way, is topologically conjugate to an odometer. We also show that the closure of the union over all orderings of the subshifts arising from codings of the Pascal adic by the first edge has superpolynomial complexity, is not topologically transitive, and has no periodic points besides the two fixed points, while the intersection over all orderings consists of just four orbits.

scholarly journals Periodic Points and Fixed Points for the Weaker(ϕ,φ)-Contractive Mappings in Complete Generalized Metric Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Chi-Ming Chen ◽  
W. Y. Sun
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Contractive Mapping ◽  
Periodic Points ◽  
Generalized Metric Spaces ◽  
Complete Metric Spaces ◽  
Contractive Mappings ◽  
Generalized Metric

We introduce the notion of weaker(ϕ,φ)-contractive mapping in complete metric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature.

scholarly journals Fixed points of elliptic reversible transformations with integrals

1996 ◽  
Vol 16 (4) ◽  
pp. 683-702
Author(s):  
Xianghong Gong
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Periodic Points ◽  
General Transformation ◽  
Reversible Transformation ◽  
Level Curves ◽  
Hyperbolic Complex ◽  
Holomorphic Transformation ◽  
Real Analytic ◽  
Real Hyperplane

AbstractWe show that for a certain family of integrable reversible transformations, the curves of periodic points of a general transformation cross the level curves of its integrals. This leads to the divergence of the normal form for a general reversible transformation with integrals. We also study the integrable holomorphic reversible transformations coming from real analytic surfaces in ℂ2 with non-degenerate complex tangents. We show the existence of real analytic surfaces with hyperbolic complex tangents, which are contained in a real hyperplane, but cannot be transformed into the Moser—Webster normal form through any holomorphic transformation.

On the Periodic Boundary Value Problem and Chaotic-like Dynamics for Nonlinear Hill’s Equations

2004 ◽  
Vol 4 (1) ◽  
Author(s):  
Duccio Papini ◽  
Fabio Zanolin
Keyword(s):  
Periodic Boundary ◽  
Order Differential Equation ◽  
Linear Equations ◽  
Two Dimensional ◽  
Indefinite Weight ◽  
Asymptotically Linear ◽  
Complicated Structure ◽  
The Rich ◽  
Hill’S Equations ◽  
New Applications

AbstractWe present some results which show the rich and complicated structure of the solutions of the second order differential equation ẍ + w(t)g(x) = 0 when the weight w(t) changes sign and g is sufficiently far from the linear case. New applications, motivated by recent studies on the superlinear Hill’s equation in [57, 58, 59], are then proposed for some asymptotically linear equations and for some sublinear equations with a sign-indefinite weight. Our results are based on a fixed point theorem for maps which satisfy a stretching condition along the paths on two-dimensional cells.

scholarly journals FIXED POINTS OF POLYNOMIALS OVER DIVISION RINGS

2021 ◽  
pp. 1-7
Author(s):  
ADAM CHAPMAN ◽  
SOLOMON VISHKAUTSAN
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Periodic Points ◽  
Conjugacy Classes ◽  
Sufficient Condition ◽  
Discrete Dynamics ◽  
Polynomial Equations ◽  
Commutative Case ◽  
Division Rings

Abstract We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$ , where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

scholarly journals Jungck theorem for triangular maps and related results

2000 ◽  
Vol 1 (1) ◽  
pp. 83 ◽  
Author(s):  
M. Grinc ◽  
L. Snoha
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Periodic Points ◽  
Fixed Point Property ◽  
Compact Interval ◽  
Invariance Property ◽  
Point Property ◽  
Dimensional Cube ◽  
Triangular Maps

<p>We prove that a continuous triangular map G of the n-dimensional cube I<sup>n</sup> has only fixed points and no other periodic points if and only if G has a common fixed point with every continuous triangular map F that is nontrivially compatible with G. This is an analog of Jungck theorem for maps of a real compact interval. We also discuss possible extensions of Jungck theorem, Jachymski theorem and some related results to more general spaces. In particular, the spaces with the fixed point property and the complete invariance property are considered.</p>

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