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 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 Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 101
Author(s):  
Alicia Cordero ◽  
Marlon Moscoso-Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Convergence Properties ◽  
Parameter Spaces ◽  
Numerical Tests ◽  
The Family ◽  
Dynamics Stability

In this paper, we present a new parametric family of three-step iterative for solving nonlinear equations. First, we design a fourth-order triparametric family that, by holding only one of its parameters, we get to accelerate its convergence and finally obtain a sixth-order uniparametric family. With this last family, we study its convergence, its complex dynamics (stability), and its numerical behavior. The parameter spaces and dynamical planes are presented showing the complexity of the family. From the parameter spaces, we have been able to determine different members of the family that have bad convergence properties, as attracting periodic orbits and attracting strange fixed points appear in their dynamical planes. Moreover, this same study has allowed us to detect family members with especially stable behavior and suitable for solving practical problems. Several numerical tests are performed to illustrate the efficiency and stability of the presented family.

scholarly journals Invariant curves around a parabolic fixed point at infinity

1990 ◽  
Vol 10 (2) ◽  
pp. 209-229 ◽  
Author(s):  
Dov Aharonov ◽  
Uri Elias
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Invariant Curves ◽  
The Stability ◽  
Area Preserving ◽  
Parabolic Fixed Point

AbstractThe stability of a fixed point of an area-preserving transformation in the plane is characterized by the invariant curves which surround it. The existence of invariant curves had been extensively studied for elliptic fixed points. Here we study the similar problem for parabolic fixed points. In particular we are interested in the case where the fixed point is at infinity.

scholarly journals On a positive solution for $(p,q)$-Laplace equation with Nonlinear

2019 ◽  
Vol 38 (4) ◽  
pp. 219-133
Author(s):  
Abdellah Zerouali ◽  
Belhadj Karim ◽  
Omar Chakrone ◽  
Abdelmajid Boukhsas
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solution ◽  
Laplace Equation ◽  
Existence Results ◽  
Indefinite Weight ◽  
Principal Eigenvalues ◽  
Steklov Eigenvalue ◽  
Indefinite Weights ◽  
Steklov Eigenvalue Problem

In the presentp aper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator $(p,q)$-Laplacian with indefinite weights. We also prove that in the case where $\mu>0$ and with $1<q<p<\infty$ the results are completely different from those for the usua lSteklov eigenvalue problem involving the $p$-Laplacian with indefinite weight, which is retrieved when $\mu=0$. Precisely, we show that when $\mu>0$ there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.

scholarly journals Parabolic fixed points, invariant curves and action-angle variables

1990 ◽  
Vol 10 (2) ◽  
pp. 231-245 ◽  
Author(s):  
Dov Aharonov ◽  
Uri Elias
Keyword(s):  
Fixed Point ◽  
Hamiltonian System ◽  
Fixed Points ◽  
Phase Flow ◽  
Invariant Curves ◽  
Elliptic Case ◽  
System A ◽  
Twist Mapping ◽  
Area Preserving ◽  
Parabolic Fixed Point

AbstractA fixed point of an area-preserving mapping of the plane is called elliptic if the eigenvalues of its linearization are of unit modulus but not ±1; it is parabolic if both eigenvalues are 1 or −1. The elliptic case is well understood by Moser's theory. Here we study when is a parabolic fixed point surrounded by closed invariant curves. We approximate our mapping T by the phase flow of an Hamiltonian system. A pair of variables, closely related to the action-angle variables, is used to reduce T into a twist mapping. The conditions for T to have closed invariant curves are stated in terms of the Hamiltonian.

scholarly journals Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Chiara Zanini ◽  
Fabio Zanolin
Keyword(s):  
Complex Dynamics ◽  
Neumann Boundary ◽  
Step Function ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
Existence And Multiplicity ◽  
Twist Maps ◽  
Topological Horseshoes ◽  
Recent Developments ◽  
The One

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

scholarly journals Research on the Complex Dynamic Characteristics and RLS Estimation’s Influence Based on Price and Service Game

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Junhai Ma ◽  
Zhanbing Guo
Keyword(s):  
Least Squares ◽  
Fixed Points ◽  
Dynamic Characteristics ◽  
Complex Dynamics ◽  
Recursive Least Squares ◽  
Game Model ◽  
Complex Dynamic ◽  
Service Market ◽  
The Real ◽  
Price Game

Considering that the real competitions in service market contain two important factors, price and service, we build a dynamical price and service game model and study the complex dynamics of this bivariate game. Some special properties about the adjustment of service are noted by comparing our innovative bivariate game model with previous univariate game model. Besides, we discuss the stabilities of fixed points and compare the price and service game with price game. What is more, the recursive least-squares (RLS) estimation is introduced to substitute naive estimation; then the impacts of RLS estimation are studied by comparing it with naive estimation.

On the Existence of Chaos in a Discontinuous Area-Preserving Map Arising in Financial Markets

2018 ◽  
Vol 28 (14) ◽  
pp. 1850177 ◽  
Author(s):  
En-Guo Gu
Keyword(s):  
Financial Markets ◽  
Complex Dynamics ◽  
Piecewise Linear ◽  
Market Model ◽  
Parameter Setting ◽  
Smale Horseshoe ◽  
Model Dynamics ◽  
Internal Change ◽  
Area Preserving

In this paper, we further study a discontinuous piecewise-linear financial market model established in our previous paper. The model dynamics is driven by a two-dimensional discontinuous area-preserving map. We exploit its complex dynamics from the viewpoint of open conservative systems. We first give the classification of fixed points, and then theoretically present the existence of periodic saddle orbit and a homoclinic chaos for some parameter setting. We finally apply the Conley–Moser conditions to verify the existence of Smale horseshoe-like dynamics and chaos for another parameter setting. The result is helpful for understanding the internal change rule of the finance market.

scholarly journals Multiplicity results for asymmetric boundary value problems with indefinite weights

2004 ◽  
Vol 2004 (11) ◽  
pp. 957-979
Author(s):  
Francesca Dalbono
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Asymptotically Linear ◽  
Topological Methods ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Rotation Numbers ◽  
Indefinite Weights ◽  
Nodal Properties

We prove existence and multiplicity of solutions, with prescribed nodal properties, to a boundary value problem of the formu″+f(t,u)=0,u(0)=u(T)=0. The nonlinearity is supposed to satisfy asymmetric, asymptotically linear assumptions involving indefinite weights. We first study some auxiliary half-linear, two-weighted problems for which an eigenvalue theory holds. Multiplicity is ensured by assumptions expressed in terms of weighted eigenvalues. The proof is developed in the framework of topological methods and is based on some relations between rotation numbers and weighted eigenvalues.

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