scholarly journals Scaling Properties of a Hybrid Fermi-Ulam-Bouncer Model

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Diego F. M. Oliveira ◽  
Rafael A. Bizão ◽  
Edson D. Leonel
Keyword(s):  
Critical Exponents ◽  
Average Velocity ◽  
Chaotic Regime ◽  
Two Dimensional ◽  
Scaling Properties ◽  
One Dimensional ◽  
Dynamical Properties ◽  
Scaling Invariant ◽  
Area Preserving

Some dynamical properties for a one-dimensional hybrid Fermi-Ulam-bouncer model are studied under the framework of scaling description. The model is described by using a two-dimensional nonlinear area preserving mapping. Our results show that the chaotic regime below the lowest energy invariant spanning curve is scaling invariant and the obtained critical exponents are used to find a universal plot for the second momenta of the average velocity.

Bifurcations of one- and two-dimensional maps

1984 ◽  
Vol 311 (1515) ◽  
pp. 43-102 ◽  
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Hénon Map ◽  
Stable And Unstable Manifolds ◽  
Henon Map ◽  
One Dimensional ◽  
The Family ◽  
Area Preserving

We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C 3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y 2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^ e is the Henon map. The maps F e have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ ue , for small e . Moreover, we are able to extend these results to the area preserving family F/u. 1 , thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the ( /u, e ) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F /u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.

SOME PECULIAR PROPERTIES OF THE DARWIN’S LAGRANGIAN

1995 ◽  
Vol 09 (23) ◽  
pp. 3069-3083 ◽  
Author(s):  
I.P. PAVLOTSKY ◽  
M. STRIANESE
Keyword(s):  
Phase Space ◽  
Dimensional Case ◽  
Minimal Distance ◽  
Rectilinear Motion ◽  
Two Dimensional ◽  
Singular Surfaces ◽  
One Dimensional ◽  
Local Singularity ◽  
Dynamical Properties

In the post-Galilean approximation the Lagrangians are singular on a submanifold of the phase space. It is a local singularity, which differs from the ones considered by Dirac. The dynamical properties are essentially peculiar on the studied singular surfaces. In the preceding publications,1,2,3 two models of singular relativistic Lagrangians and the rectilinear motion of two electrons, determined by Darwin’s Lagrangian, were examined. In the present paper we study the peculiar dynamical properties of the two-dimensional Darwin’s Lagrangian. In particular, it is shown that the minimal distance between two electrons (the so called “radius of electron”) appears in the two-dimensional motion as well as in one-dimensional case. Some new peculiar properties are discovered.

scholarly journals Can Drag Force Suppress Fermi Acceleration in a Bouncer Model?

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Francys Andrews de Souza ◽  
Lucas Eduardo Azevedo Simões ◽  
Mário Roberto da Silva ◽  
Edson D. Leonel
Keyword(s):  
Drag Force ◽  
Average Velocity ◽  
Classical Particle ◽  
Nonlinear Mapping ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Control Parameters ◽  
Fermi Acceleration ◽  
Moving Wall ◽  
Dynamical Properties

Some dynamical properties for a bouncer model—a classical particle of massmfalling in the presence of a constant gravitational fieldgand hitting elastically a periodically moving wall—in the presence of drag force that is assumed to be proportional to the particle's velocity are studied. The dynamics of the model is described in terms of a two-dimensional nonlinear mapping obtained via solution of the second Newton's law of motion. We characterize the behavior of the average velocity of the particle as function of the control parameters as well as the time. Our results show that the average velocity starts growing at first and then bends towards a regime of constant value, thus confirming that the introduction of drag force is a sufficient condition to suppress Fermi acceleration in the model.

CRITICAL EXPONENTS AND SCALING PROPERTIES FOR THE CHAOTIC DYNAMICS OF A PARTICLE IN A TIME-DEPENDENT POTENTIAL BARRIER

2012 ◽  
Vol 22 (10) ◽  
pp. 1250250 ◽  
Author(s):  
DIOGO RICARDO DA COSTA ◽  
ANDRÉ LUÍS PRANDO LIVORATI ◽  
EDSON D. LEONEL
Keyword(s):  
Critical Exponents ◽  
Chaotic Dynamics ◽  
Time Varying ◽  
Control Parameters ◽  
Mixed Structure ◽  
Scaling Properties ◽  
Scaling Invariance ◽  
Dependent Potential ◽  
Area Preserving ◽  
Number Of Iterations

Some scaling properties of the chaotic sea for a particle confined inside an infinitely deep potential box containing a time varying barrier are studied. The dynamics of the particle is described by a two-dimensional, nonlinear and area-preserving mapping for the variables energy of the particle and time. The phase space of the model exhibits a mixed structure with Kolmogorov–Arnold–Moser islands, chaotic seas and invariant spanning curves limiting the chaotic orbits. Average properties of the chaotic sea including the first momenta and the deviation of the second momenta are obtained as a function of: (i) number of iterations (n), and (ii) time (t). By the use of scaling arguments, critical exponents for the ensemble average of the first momenta are obtained and compared for both cases (i) and (ii). Scaling invariance of the average properties for the chaotic sea is obtained as a function of the control parameters that describe the model.

scholarly journals On the notion of the dimension with respect to a dynamical system

1984 ◽  
Vol 4 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Invariant Sets ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Common Features ◽  
Dynamical Properties ◽  
The One ◽  
Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

SCALING PROPERTIES OF DECONSTRUCTION INTERFACES IN DISORDERED MEDIA

2001 ◽  
Vol 12 (01) ◽  
pp. 71-78 ◽  
Author(s):  
JUAN R. SANCHEZ
Keyword(s):  
Universality Class ◽  
Quenched Disorder ◽  
Disordered Media ◽  
Rough Interface ◽  
Scaling Exponents ◽  
Scaling Properties ◽  
One Dimensional ◽  
Depinning Transition ◽  
Dynamical Properties ◽  
Dimensional Surface

The scaling properties of interfaces generated by a disaggregation model in 1+1 dimensions are studied by numerical simulations. The model presented here for the disaggregation process takes into account the possibility of having quenched disorder in the bulk under deconstruction. The disorder can be considered to model several types of irregularities appearing in real materials (dislocations, impurities). The presence of irregularities makes the intensity of the attack to be not uniform. In order to include this effect, the computational bulk is considered to be composed by two types of particles: those particles which can be easily detached and other particles that are not sensible to the etching attack. As the detachment of particles proceeds in time, the dynamical properties of the rough interface are studied. The resulting one-dimensional surface show self-affine properties and the values of the scaling exponents are reported when the interface is still moving near the depinning transition. According to the scaling exponents presented here, the model must be considered to belong to a new universality class.

Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist

1997 ◽  
Vol 17 (3) ◽  
pp. 575-591 ◽  
Author(s):  
H. ERIK DOEFF
Keyword(s):  
Fixed Point ◽  
Rotation Number ◽  
Rational Number ◽  
Dehn Twist ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Rotation Vectors ◽  
Rotation Set ◽  
Area Preserving

We extend the theory of rotation vectors to homeomorphisms of the two-dimensional torus that are homotopic to a Dehn twist. We define a one-dimensional rotation number and recreate the theory of the homotopic case to the identity case. We prove that if such a map is area preserving and has mean rotation number zero, then it must have a fixed point. We prove that the rotation set is a compact interval, and that if the rotation interval contains two distinct numbers, then for any rational number in the rotation set there exists a periodic point with that rotation number. Finally, we prove that any interval with rational endpoints can be realized as the rotation set of a map homotopic to a Dehn twist.

ON BEHAVIORS OF TWO-DIMENSIONAL ENDOMORPHISMS: ROLE OF THE CRITICAL CURVES

1993 ◽  
Vol 03 (01) ◽  
pp. 187-194 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
TONY NARAYANINSAMY
Keyword(s):  
Critical Points ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
Dynamical Properties

Critical curves are the natural two-dimensional extension of the notion of critical points in one-dimensional endomorphisms. They play a fundamental role in determining the dynamical properties and their bifurcations. This letter demonstrates such a role for two new behaviors.

ESTIMATING THE DIMENSIONAL CHARACTERISTICS OF TWO-DIMENSIONAL PATTERNS

1995 ◽  
Vol 05 (01) ◽  
pp. 109-121 ◽  
Author(s):  
A.L. ZHELEZNYAK ◽  
L.O. CHUA
Keyword(s):  
Fractal Dimensions ◽  
Computer Experiments ◽  
Two Dimensional ◽  
One Dimensional ◽  
Theoretical Predictions ◽  
The Matrix ◽  
Spatially Extended ◽  
Spatially Extended Systems ◽  
Dynamical Properties ◽  
Generalized Dimensions

Dynamical properties of two-dimensional patterns generated by spatially extended systems can be described via the characteristics of attractors in the matrix phase space of the associated translation (or translational-evolution) dynamical systems. Questions regarding the possibility of estimating the fractal dimensions of two-dimensional patterns from the fractal dimensions of one-dimensional observables scanning the patterns along a chosen path are investigated. The presented proofs state that the generalized dimensions of the scanning observables are lower bounds for estimating the corresponding generalized dimensions of two-dimensional patterns. Spatial field distributions defined as superposition of planar waves and different spatiotemporal patterns produced by cellular neural networks made of Chua’s circuits are studied numerically. The results of computer experiments confirm the theoretical predictions presented in this paper.

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