scholarly journals Linearly Coupled Anharmonic Oscillators and Integrability

1987 ◽  
Vol 40 (5) ◽  
pp. 587
Author(s):  
W-H Steeb ◽  
JA Louw ◽  
CM Villet
Keyword(s):  
Lyapunov Exponent ◽  
Anharmonic Oscillator ◽  
First Integrals ◽  
Chaotic Behaviour ◽  
Painlevé Test ◽  
Anharmonic Oscillators ◽  
One Dimensional

The Painleve test for a linearly coupled anharmonic oscillator is performed. We show that this system does not pass the Painleve test. This suggests that this system is not integrable. Moreover, we apply Ziglin's (1983) theorem which provides a criterion for non-existence of first integrals besides the Hamiltonian. Calculating numerically the maximal one-dimensional Lyapunov exponent, we find regions with positive exponents. Thus, the system can show chaotic behaviour. Finally we compare our results with the quartic coupled anharmonic oscillator.

scholarly journals Chaos in an anharmonic oscillator

1995 ◽  
Vol 37 (2) ◽  
pp. 186-207 ◽  
Author(s):  
J. R. Christie ◽  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous System ◽  
Anharmonic Oscillator ◽  
Homoclinic Orbits ◽  
Chaotic Behaviour ◽  
Differential Systems ◽  
Melnikov's Method ◽  
One Dimensional ◽  
Melnikov’S Method

AbstractUsing Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

Perturbation and variational-perturbation method for the free energy of anharmonic oscillators

2007 ◽  
Vol 85 (1) ◽  
pp. 13-30 ◽  
Author(s):  
K Vlachos ◽  
V Papatheou ◽  
A Okopińska
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Perturbation Method ◽  
Perturbation Methods ◽  
Anharmonic Oscillator ◽  
Numerical Calculations ◽  
Anharmonic Oscillators ◽  
One Dimensional ◽  
Variational Perturbation ◽  
Fifth Order

The perturbation and the variational-perturbation methods are applied for calculating the partition function of one-dimensional oscillators with anharmonicity x2n. New formally simple expressions for the free energy and for the Rayleigh–Schrodinger energy corrections are derived. It is shown that the variational-perturbation method overcomes all the deficiencies of the conventional perturbation method. The results of fifth-order numerical calculations for the free energy of the quartic, quartic–sextic, and octic anharmonic oscillator are highly accurate in the whole range of temperatures. PACS Nos.: 03.65.–w, 05.30.–d

scholarly journals Externally Driven Nonlinear Oscillator, PainlevéTest, First Integrals and Lie Symmetries

1993 ◽  
Vol 48 (8-9) ◽  
pp. 943-946
Author(s):  
W.-H. Steeb ◽  
N. Euler
Keyword(s):  
Vector Field ◽  
Closed Form ◽  
Nonlinear Oscillator ◽  
Anharmonic Oscillator ◽  
Lie Symmetry ◽  
First Integrals ◽  
Lie Symmetries ◽  
Painlevé Test ◽  
Smooth Functions

Abstract For arbitrary constants c1 , c2 and an arbitrary smooth functions f the driven anharmonic oscillator d2u/dt2+c1du/dt + c2u + u3 =f(t) cannot be solved in closed form. We apply the Painleve test to obtain the constraint on the constants c1 , c2 and the function f for which the equation passes the test. We also give the Lie symmetry vector field and first integrals for this equation

Non-Linearity and Chaotic Behaviour in Cyprus Stock Market

2015 ◽  
Vol 5 (4) ◽  
Author(s):  
Athina Bougioukou
Keyword(s):  
Lyapunov Exponent ◽  
Stock Market ◽  
Chaos Theory ◽  
Linear Dependence ◽  
Chaotic Behavior ◽  
Efficient Market Hypothesis ◽  
Chaotic Behaviour ◽  
Maximum Lyapunov Exponent ◽  
Efficient Market ◽  
General Index

The intention of this research is to investigate the aspect of non-linearity and chaotic behavior of the Cyprus stock market. For this purpose, we use non-linearity and chaos theory. We perform BDS, Hinich-Bispectral tests and compute Lyapunov exponent of the Cyprus General index. The results show that existence of non-linear dependence and chaotic features as the maximum Lyapunov exponent was found to be positive. This study is important because chaos and efficient market hypothesis are mutually exclusive aspects. The efficient market hypothesis which requires returns to be independent and identically distributed (i.i.d.) cannot be accepted.

A Chaotification model based on sine and cosecant functions for enhancing chaos

2021 ◽  
Author(s):  
Yuqing Li ◽  
Xing He ◽  
Dawen Xia
Keyword(s):  
Fixed Point ◽  
Lyapunov Exponent ◽  
Chaotic Maps ◽  
Dynamic Properties ◽  
Chaotic Map ◽  
Sample Entropy ◽  
One Dimensional ◽  
Model Based ◽  
Chaotic Region ◽  
Definition Of

Chaotic maps with higher chaotic complexity are urgently needed in many application scenarios. This paper proposes a chaotification model based on sine and cosecant functions (CMSC) to improve the dynamic properties of existing chaotic maps. CMSC can generate a new map with higher chaotic complexity by using the existing one-dimensional (1D) chaotic map as a seed map. To discuss the performance of CMSC, the chaos properties of CMSC are analyzed based on the mathematical definition of the Lyapunov exponent (LE). Then, three new maps are generated by applying three classical 1D chaotic maps to CMSC respectively, and the dynamic behaviors of the new maps are analyzed in terms of fixed point, bifurcation diagram, sample entropy (SE), etc. The results of the analysis demonstrate that the new maps have a larger chaotic region and excellent chaotic characteristics.

scholarly journals Energy levels of one-dimensional anharmonic oscillator via neural networks

2019 ◽  
Vol 34 (12) ◽  
pp. 1950088 ◽  
Author(s):  
Halil Mutuk
Keyword(s):  
Neural Network ◽  
Field Theory ◽  
Anharmonic Oscillator ◽  
Energy Levels ◽  
High Accuracy ◽  
Network System ◽  
Quantum Field ◽  
One Dimensional ◽  
Neural Network System ◽  
Good Agreement

In this work, we obtained energy levels of one-dimensional quartic anharmonic oscillator by using neural network system. Quartic anharmonic oscillator is a very important tool in quantum mechanics and also in the quantum field theory. Our results are in good agreement in high accuracy with the reference studies.

V—The Brownian Movement of a One-Dimensional Non-Harmonic Oscillator

1968 ◽  
Vol 68 (1) ◽  
pp. 70-82
Author(s):  
A. J. Allnutt
Keyword(s):  
Approximate Solution ◽  
Simple Model ◽  
Harmonic Oscillator ◽  
Langevin Equation ◽  
Iterative Procedure ◽  
Anharmonic Oscillator ◽  
Crystalline Solid ◽  
Brownian Movement ◽  
One Dimensional ◽  
Numerical Example

SynopsisThe Langevin equation for the harmonic oscillator is solved by a different method from that normally used. The approximate solution for the case of the slightly anharmonic oscillator is then obtained by an iterative procedure and the results are illustrated by a numerical example based on a simple model of a crystalline solid.

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