DYNAMICS OF SOME RATIONAL DISCRETE DYNAMICAL SYSTEMS VIA INVARIANTS

2006 ◽  
Vol 16 (03) ◽  
pp. 631-645 ◽  
Author(s):  
ANNA CIMA ◽  
ARMENGOL GASULL ◽  
VÍCTOR MAÑOSA
Keyword(s):  
Dynamical Systems ◽  
Discrete Dynamical Systems ◽  
Third Order ◽  
Möbius Transformations ◽  
The Third ◽  
Mobius Transformations

We consider several discrete dynamical systems for which some invariants can be found. Our study includes complex Möbius transformations as well as the third-order Lyness recurrence.

scholarly journals SIMPLEST ODE EQUIVALENTS OF CHUA'S EQUATIONS

2000 ◽  
Vol 10 (01) ◽  
pp. 1-23 ◽  
Author(s):  
JIŘÍ POSPÍŠIL ◽  
ZDENĚK KOLKA ◽  
JANA HORSKÁ ◽  
JAROMÍR BRZOBOHATÝ
Keyword(s):  
Dynamical Systems ◽  
Piecewise Linear ◽  
Order System ◽  
Topological Conjugacy ◽  
Third Order ◽  
The Third ◽  
Canonical State ◽  
Stereoscopic View ◽  
State Models ◽  
Mutual Relations

The so-called elementary canonical state models of the third-order piecewise-linear (PWL) dynamical systems, as the simplest ODE equivalents of Chua's equations, are presented. Their mutual relations using the linear topological conjugacy are demonstrated in order to show in detail that Chua's equations and their canonical ODE equivalents represent various forms of qualitatively equivalent models of third-order dynamical systems. New geometrical aspects of the corresponding transformations together with examples of typical chaotic attractors in the stereoscopic view, give the possibility of a deeper insight into the third-order system dynamics.

ON THE STUDY OF NONLINEAR INTEGRABLE SYSTEMS IN (2+1) DIMENSIONS BY DRINFELD-SOKOLOV METHOD

1995 ◽  
Vol 10 (37) ◽  
pp. 2843-2852
Author(s):  
I. MUKHOPADHYAY ◽  
A. ROYCHOWDHURY
Keyword(s):  
Dynamical Systems ◽  
Poisson Structure ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Complete Integrability ◽  
Kp Equation ◽  
Third Order ◽  
Integrable Equations ◽  
The Third ◽  
Integrable Dynamical Systems

The Drinfeld-Sokolov formalism is extended to the case of operator-valued affine Lie algebra to derive nonlinear integrable dynamical systems in (2+1) dimensions. The Poisson structure of these integrable equations are also worked out. While from the first- and second-order flows we get some new integrable equations in (2+1) dimensions, the KP equation is seen to result from the third-order flow. Complete integrability of such equations and the existence of the bi-Hamiltonian structure are demonstrated.

scholarly journals Repeated compositions of Möbius transformations

2018 ◽  
Vol 40 (2) ◽  
pp. 437-452
Author(s):  
MATTHEW JACQUES ◽  
IAN SHORT
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Limit Point ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Central Importance ◽  
Möbius Transformations ◽  
Necessary And Sufficient ◽  
Point Type ◽  
Mobius Transformations

We consider a class of dynamical systems generated by finite sets of Möbius transformations acting on the unit disc. Compositions of such Möbius transformations give rise to sequences of transformations that are used in the theory of continued fractions. In that theory, the distinction between sequences of limit-point type and sequences of limit-disc type is of central importance. We prove that sequences of limit-disc type only arise in exceptional circumstances, and we give necessary and sufficient conditions for a sequence to be of limit-disc type. We also calculate the Hausdorff dimension of the set of sequences of limit-disc type in some significant cases. Finally, we obtain strong and complete results on the convergence of these dynamical systems.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

scholarly journals Ultrafast Third-Order Nonlinear Optical Response Excited by fs Laser Pulses at 1550 nm in GaN Crystals

Materials ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3194
Author(s):  
Adrian Petris ◽  
Petronela Gheorghe ◽  
Tudor Braniste ◽  
Ion Tiginyanu
Keyword(s):  
Harmonic Generation ◽  
Nonlinear Optical ◽  
Laser Pulses ◽  
Third Harmonic Generation ◽  
High Repetition Rate ◽  
Third Order ◽  
Third Harmonic ◽  
The Third ◽  
High Repetition ◽  
Gan Crystal

The ultrafast third-order optical nonlinearity of c-plane GaN crystal, excited by ultrashort (fs) high-repetition-rate laser pulses at 1550 nm, wavelength important for optical communications, is investigated for the first time by optical third-harmonic generation in non-phase-matching conditions. As the thermo-optic effect that can arise in the sample by cumulative thermal effects induced by high-repetition-rate laser pulses cannot be responsible for the third-harmonic generation, the ultrafast nonlinear optical effect of solely electronic origin is the only one involved in this process. The third-order nonlinear optical susceptibility of GaN crystal responsible for the third-harmonic generation process, an important indicative parameter for the potential use of this material in ultrafast photonic functionalities, is determined.

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