What Is an Integrable Mapping?

1991 ◽  
pp. 251-272 ◽  
Author(s):  
A. P. Veselov
Keyword(s):  
Integrable Mapping

Integrable mapping, perturbations and attractors

1992 ◽  
Vol 46 (1) ◽  
pp. 9-13
Author(s):  
M Debnath ◽  
A RoyChowdhury
Keyword(s):  
Integrable Mapping

scholarly journals Analytical Approximation of the Dissipative Standard Map

1999 ◽  
Vol 172 ◽  
pp. 451-452 ◽  
Author(s):  
Alessandra Celletti ◽  
Gabriella Della Penna ◽  
Claude Froeschlé
Keyword(s):  
Approximate Solutions ◽  
Analytical Approximation ◽  
Real Parameter ◽  
Dissipative Systems ◽  
Standard Map ◽  
Dissipative Coefficient ◽  
Standard Mapping ◽  
Integrable Mapping

We investigate the dynamics of a dissipative standard mapping defined by the equationswhere y ∈ R, x ∈ T and ε is a real parameter, we refer to 0 < α < 1 as the “dissipative parameter” and to ψ as the “dissipative coefficient” (ε = α = 0 provides an integrable mapping). Notice that the dynamics is contractive, since the jacobian of the above mapping equals to 1 − α. In particular, we want to compare (see Celletti et al., 1997) the solutions associated to the conservative map (i.e., α = 0) with that related to (1) (α ≠ 0). For simplicity, we consider the case when α = ε2 and construct explicit approximate solutions to the conservative and dissipative systems, using a suitable parametrization like in (Celletti and Chierchia, 1988).

scholarly journals Products of a C-Measure and a Locally Integrable Mapping

1957 ◽  
Vol 9 ◽  
pp. 475-486 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Integral Representation ◽  
Topological Space ◽  
Orlicz Spaces ◽  
Complex Numbers ◽  
Bounded Operators ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Special Cases ◽  
Integrable Mapping

Let C be the field of complex numbers and E a locally compact topological space. The authors' theory of C-bimeasures Λ and their Λ-integrals in (1; 2) leads to integral representation of bounded operators from A to B' where A and B are MT-spaces as defined in (3). These MT-spaces include the -spaces and Orlicz spaces as special cases.

scholarly journals Fine structure of matrix Darboux-Toda integrable mapping

1998 ◽  
Vol 242 (1-2) ◽  
pp. 31-35 ◽  
Author(s):  
A.N Leznov ◽  
E.A Yuzbashyan
Keyword(s):  
Fine Structure ◽  
Integrable Mapping

scholarly journals Deautonomization by singularity confinement: an algebro-geometric justification

2015 ◽  
Vol 471 (2183) ◽  
pp. 20140956 ◽  
Author(s):  
T. Mase ◽  
R. Willox ◽  
B. Grammaticos ◽  
A. Ramani
Keyword(s):  
Linear Transformation ◽  
Blow Up ◽  
Picard Group ◽  
Geometrical Analysis ◽  
Standard Practice ◽  
Singularity Confinement ◽  
Discrete Painlevé Equations ◽  
Independent Variable ◽  
Free Parameters ◽  
Integrable Mapping

The ‘deautonomization’ of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage leads to a non-integrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painlevé equations, how their regularization through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.

scholarly journals A fixed point theorem for mappings satisfying a general contractive condition of integral type

2002 ◽  
Vol 29 (9) ◽  
pp. 531-536 ◽  
Author(s):  
A. Branciari
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Real Number ◽  
Fixed Points ◽  
Compact Subset ◽  
Metric Spaces ◽  
Contractive Condition ◽  
Integral Type ◽  
Complete Metric Spaces ◽  
Integrable Mapping

We analyze the existence of fixed points for mappings defined on complete metric spaces(X,d)satisfying a general contractive inequality of integral type. This condition is analogous to Banach-Caccioppoli's one; in short, we study mappingsf:X→Xfor which there exists a real numberc∈]0,1[, such that for eachx,y∈Xwe have∫0d(fx,fy)φ(t)dt≤c∫0d(x,y)φ(t)dt, whereφ:[0,+∞[→[0,+∞]is a Lebesgue-integrable mapping which is summable on each compact subset of[0,+∞[, nonnegative and such that for eachε>0,∫0εφ(t)dt>0.

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