Stochasticity in the Josephson map

1996 ◽  
Vol 56 (3) ◽  
pp. 493-506 ◽  
Author(s):  
Y. Nomura ◽  
Y. H. Ichikawa ◽  
A. T. Filipov
Keyword(s):  
Nonlinear Dynamics ◽  
Diffusion Coefficient ◽  
Phase Space ◽  
Characteristic Function ◽  
Numerical Investigation ◽  
Stochastic Diffusion ◽  
Standard Map ◽  
External Bias ◽  
Stochastic Parameter ◽  
Phase Space Portrait

The Josephson map describes the nonlinear dynamics of systems characterized by the standard map with a uniform external bias superposed. The intricate structures of the phase-space portrait of the Josephson map are examined here on the basis of the associated tangent map. A numerical investigation of stochastic diffusion in the Josephson map is compared with the renormalized diffusion coefficient calculated using the characteristic function. The global stochasticity of the Josephson map occurs at far smaller values of the stochastic parameter than is the case of the standard map.

Chaos dynamique des particules relativistes accélérées dans un paquet d'ondes électrostatiques

2004 ◽  
Vol 82 (6) ◽  
pp. 467-479
Author(s):  
A Raouak ◽  
D Saifaoui ◽  
A Dezairi
Keyword(s):  
Numerical Simulation ◽  
Electromagnetic Wave ◽  
Wave Packet ◽  
Numerical Results ◽  
Theoretical Computation ◽  
Stochastic Diffusion ◽  
Standard Map ◽  
Stochastic Parameter ◽  
Accelerated Particles

In this work, we study the diffusion of particle accelerated in an electromagnetic wave packet, through a numerical simulation of the relativistic standard map. We contribute to the field of stochastic diffusion of accelerated particles as a function of the stochastic parameter K, specially the transition between partial and global stochasticity, and we also compare our theoretical computation of the diffusion with numerical results.

scholarly journals Stochastic Diffusion of Energetic Ions in Wendelstein-Type Stellarators

2018 ◽  
Vol 63 (6) ◽  
pp. 495
Author(s):  
A. V. Tykhyy
Keyword(s):  
Diffusion Coefficient ◽  
Adiabatic Invariant ◽  
Stochastic Diffusion ◽  
Energetic Ions ◽  
Separatrix Crossing

The collisionless stochastic diffusion of energetic ions in optimized stellarators of the Wendelstein type has been considered. The phenomenon concerned was predicted earlier in the framework of a simplified theory describing the separatrix crossing by ions. The jumps of the adiabatic invariant in magnetic configurations of a stellarator are calculated. The analysis of the results obtained confirms the importance of the stochastic diffusion and demonstrates that the diffusion coefficient can considerably exceed the available result.

Statistical properties for compositions of standard maps with increasing coefficient

2020 ◽  
pp. 1-44
Author(s):  
ALEX BLUMENTHAL
Keyword(s):  
Phase Space ◽  
Chaotic Behavior ◽  
Positive Lebesgue Measure ◽  
Large Set ◽  
Rigorous Analysis ◽  
Strong Law ◽  
Standard Map ◽  
Positive Volume ◽  
Open Question ◽  
Volume Preserving

The Chirikov standard map is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Rigorous analysis is notoriously difficult and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any parameter value. Here we study a problem of intermediate difficulty: compositions of standard maps with increasing coefficient. When the coefficients increase to infinity at a sufficiently fast polynomial rate, we obtain a strong law, a central limit theorem, and quantitative mixing estimates for Holder observables. The methods used are not specific to the standard map and apply to a class of compositions of ‘prototypical’ two-dimensional maps with hyperbolicity on ‘most’ of phase space.

Response Scenario and Nonsmooth Features in the Nonlinear Dynamics of an Impacting Inverted Pendulum

2005 ◽  
Vol 1 (1) ◽  
pp. 56-64 ◽  
Author(s):  
Stefano Lenci ◽  
Lucio Demeio ◽  
Milena Petrini
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Investigation ◽  
Inverted Pendulum ◽  
Numerical Results ◽  
Dynamical Response ◽  
Global Bifurcations ◽  
Bifurcation Diagrams ◽  
Local And Global Bifurcations ◽  
And Behavior

In this work, we perform a systematic numerical investigation of the nonlinear dynamics of an inverted pendulum between lateral rebounding barriers. Three different families of considerably variable attractors—periodic, chaotic, and rest positions with subsequent chattering—are found. All of them are investigated, in detail, and the response scenario is determined by both bifurcation diagrams and behavior charts of single attractors, and overall maps. Attention is focused on local and global bifurcations that lead to the attractor-basin metamorphoses. Numerical results show the extreme richness of the dynamical response of the system, which is deemed to be of interest also in view of prospective mechanical applications.

scholarly journals Tutorial Notes on One-Party and Two-Party Gaussian States

2003 ◽  
Vol 01 (02) ◽  
pp. 153-188 ◽  
Author(s):  
Berthold-Georg Englert ◽  
Krzysztof Wódkiewicz
Keyword(s):  
Quantum Information ◽  
Phase Space ◽  
Characteristic Function ◽  
Information Science ◽  
Operator Method ◽  
Two Dimensional ◽  
Quantum Information Science ◽  
Gaussian States ◽  
One Dimensional ◽  
Practical Applications

Gaussian states — or, more generally, Gaussian operators — play an important role in Quantum Optics and Quantum Information Science, both in discussions about conceptual issues and in practical applications. We describe, in a tutorial manner, a systematic operator method for first characterizing such states and then investigating their properties. The central numerical quantities are the covariance matrix that specifies the characteristic function of the state, and the closely related matrices associated with Wigner's and Glauber's phase space functions. For pedagogical reasons, we restrict the discussion to one-dimensional and two-dimensional Gaussian states, for which we provide illustrating and instructive examples.

The phase-space hydrodynamic model for the quantum standard map

1991 ◽  
Vol 63 (1-3) ◽  
pp. 279-305 ◽  
Author(s):  
Alejandro Spina ◽  
Rex.T. Skodje
Keyword(s):  
Phase Space ◽  
Hydrodynamic Model ◽  
Standard Map

scholarly journals Chaotic Saddles in a Generalized Lorenz Model of Magnetoconvection

2020 ◽  
Vol 30 (12) ◽  
pp. 2030034
Author(s):  
Francis F. Franco ◽  
Erico L. Rempel
Keyword(s):  
Nonlinear Dynamics ◽  
Fractal Dimension ◽  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Lorenz Model ◽  
Fractal Basin Boundaries ◽  
Chaotic Transients ◽  
Basin Boundaries

The nonlinear dynamics of a recently derived generalized Lorenz model [ Macek & Strumik, 2010 ] of magnetoconvection is studied. A bifurcation diagram is constructed as a function of the Rayleigh number where attractors and nonattracting chaotic sets coexist inside a periodic window. The nonattracting chaotic sets, also called chaotic saddles, are responsible for fractal basin boundaries with a fractal dimension near the dimension of the phase space, which causes the presence of very long chaotic transients. It is shown that the chaotic saddles can be used to infer properties of chaotic attractors outside the periodic window, such as their maximum Lyapunov exponent.

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