Discrete chaotic maps obtained by symmetric integration

Butusov, Denis N.; Каримов, А.; Pyko, Nikita S.; Pyko, Svetlana A.; Bogachev, Mikhail I.

doi:10.1016/j.physa.2018.06.100

Cited by 22 publications

(11 citation statements)

References 53 publications

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“…We have numerically integrated the equations of motion ( 13 ) and ( 14 ) by means of the standard fourth-order Runge-Kutta algorithm with an integration step

. It is well known that numerical methods, like the one used herein, present an energy drift for long-time simulations [ 48 , 49 , 50 ]; using the integration step above, we have checked that the total energy was preserved within a relative precision

.We have considered two different kinds of initial conditions. In the first one we take random sets of angular momenta

, with components drawn from a symmetric uniform distribution and then redefined in order to yield zero total momentum.…”

Section: Resultsmentioning

confidence: 99%

“…As mentioned in the beginning of this section, we have used the standard fourth-order Runge-Kutta algorithm for the integration of the equations of motion, by monitoring that the total energy was preserved within a relative precision

(at least). However, further simulations using specially designed symplectic methods for Hamiltonian problems [ 48 , 49 , 50 ] are welcome.…”

Section: Resultsmentioning

confidence: 99%

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d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

Rodríguez

Nobre

Tsallis

2019

Entropy

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We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model ( d = 1 , 2 , 3 ) with interactions decaying with the distance r i j as 1 / r i j α ( α ≥ 0 ), where the limit α = 0 ( α → ∞ ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α / d > 1 ( 0 ≤ α / d ≤ 1 ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ ∼ N − κ , where κ ( α / d ) depends only on the ratio α / d ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α / d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α / d makes this model similar to the α -XY and α -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.

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“…We have numerically integrated the equations of motion ( 13 ) and ( 14 ) by means of the standard fourth-order Runge-Kutta algorithm with an integration step

.We have considered two different kinds of initial conditions. In the first one we take random sets of angular momenta

, with components drawn from a symmetric uniform distribution and then redefined in order to yield zero total momentum.…”

Section: Resultsmentioning

confidence: 99%

(at least). However, further simulations using specially designed symplectic methods for Hamiltonian problems [ 48 , 49 , 50 ] are welcome.…”

Section: Resultsmentioning

confidence: 99%

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

Rodríguez

Nobre

Tsallis

2019

Entropy

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show abstract

“…As we mentioned above, the sociocultural discourse should be considered as an external factor of this system. One of the possible scenarios to introduce this factor can be the recently discovered adaptive symmetry phenomenon [29].…”

Section: Competitive Lotka-volterra Modelmentioning

confidence: 99%

“…These factors are external to the dynamic system and cannot be directly introduced as system parameters or state variables without affecting the basic principles of its dynamics. Recently, some authors proposed the concept of adaptive symmetry as a technique to control the dynamics of discrete nonlinear maps in an almost linear way [29,30]. Based on the idea of semi-implicit [31] and semi-explicit integration [32], this approach allows one to introduce external control law into the discrete model dynamics.…”

Section: Introductionmentioning

confidence: 99%

Discrete Competitive Lotka–Volterra Model with Controllable Phase Volume

Ворошилова

Wafubwa

2020

Systems

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The simulation of population dynamics and social processes is of great interest in nonlinear systems. Recently, many scholars have paid attention to the possible applications of population dynamics models, such as the competitive Lotka–Volterra equation, in economic, demographic and social sciences. It was found that these models can describe some complex behavioral phenomena such as marital behavior, the stable marriage problem and other demographic processes, possessing chaotic dynamics under certain conditions. However, the introduction of external factors directly into the continuous system can influence its dynamic properties and requires a reformulation of the whole model. Nowadays most of the simulations are performed on digital computers. Thus, it is possible to use special numerical techniques and discrete effects to introduce additional features to the digital models of continuous systems. In this paper we propose a discrete model with controllable phase-space volume based on the competitive Lotka–Volterra equations. This model is obtained through the application of semi-implicit numerical methods with controllable symmetry to the continuous competitive Lotka–Volterra model. The proposed model provides almost linear control of the phase-space volume and, consequently, the quantitative characteristics of simulated behavior, by shifting the symmetry of the underlying finite-difference scheme. We explicitly show the possibility of introducing almost arbitrary law to control the phase-space volume and entropy of the system. The proposed approach is verified through bifurcation, time domain and phase-space volume analysis. Several possible applications of the developed model to the social and demographic problems’ simulation are discussed. The developed discrete model can be broadly used in modern behavioral, demographic and social studies.

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“…Symmetry is always investigated as a geometric property or algebraic structure in nonlinear science. For instance, the property analysis and control of symmetric and asymmetric chaotic systems can be found in [16][17][18]. Early studies on fractals analyzed the symmetry property of the planar M-set generated from map (1) [2] and some generalized maps [19].…”

Section: Introductionmentioning

confidence: 99%

The Symmetry in the Noise-Perturbed Mandelbrot Set

Sun

2019

Symmetry

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This paper investigates the destruction of the symmetrical structure of the noise-perturbed Mandelbrot set (M-set). By applying the “symmetry criterion” method, we quantitatively compare the damages to the symmetry of the noise-perturbed Mandelbrot set resulting from additive and multiplicative noises. Because of the uneven distribution between the core positions and the edge positions of the noise-perturbed Mandelbrot set, the comparison results reveal a paradox between the visual sense and quantified result. Thus, we propose a new “visual symmetry criterion” method that is more suitable for the measurement of visual asymmetry.

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Discrete chaotic maps obtained by symmetric integration

Cited by 22 publications

References 53 publications

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

Discrete Competitive Lotka–Volterra Model with Controllable Phase Volume

The Symmetry in the Noise-Perturbed Mandelbrot Set

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