Effects of noise on periodic orbits of the logistic map

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Feng-guo Li
Keyword(s):  
Periodic Orbit ◽  
White Noise ◽  
Probability Density ◽  
Periodic Orbits ◽  
Gaussian Noise ◽  
Dynamical Behavior ◽  
Gaussian White Noise ◽  
Logistic Map ◽  
Period Doubling ◽  
Colored Gaussian Noise

AbstractNoise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.

scholarly journals Description of the macroscopic dynamics of populations of phase elements with white non-Gaussian noise based on the circular cumulant approach

2021 ◽  
pp. 05-12
Author(s):  
A. V. Dolmatova ◽  
◽  
I. V. Tiulkina ◽  
D. S. Goldobin ◽  
◽  
...  
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Macroscopic Description ◽  
Reduced Equations ◽  
Non Gaussian ◽  
Dynamics Of Populations ◽  
Stable Noise ◽  
Good Agreement

We use the method of circular cumulants, which allows us to construct a low-mode macroscopic description of the dynamics of populations of phase elements subject to non-Gaussian white noise. In this work, we have obtained two-cumulant reduced equations for alpha-stable noise. The application of the approach is demonstrated for the case of the Kuramoto ensemble with non-Gaussian noise. The results of numerical calculations for the ensemble of N = 1500 elements, the numericalsimulation of the chain of equations for the Kuramoto–Daido order parameters (Fourier modes of the probability density) with 200 terms (in the thermodynamic limit of an infinitely large ensemble) and the theoretical solution on the basis of the two-cumulant approximation are in good agreement with each other.

Noise and Contact in Dynamic AFM Operations

2011 ◽  
Author(s):  
Ishita Chakraborty ◽  
Balakumar Balachandran
Keyword(s):  
White Noise ◽  
Evolution Equations ◽  
Stochastic Dynamics ◽  
Gaussian White Noise ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Dynamic Mode ◽  
Grazing Impacts ◽  
Representative System

In this article, the authors study the effects of Gaussian white noise on the dynamics of an atomic force microscope (AFM) cantilever operating in a dynamic mode by using a combination of numerical and analytical efforts. As a representative system, a combination of Si cantilever and HOPG sample is used. The focus of this study is on understanding the stochastic dynamics of a micro-cantilever, when the excitation frequencies are away from the first natural frequency of the system. In the previous efforts of the authors, period-doubling bifurcations close to grazing impacts have been reported for micro-cantilevers when the excitation frequency is in between the first and the second natural frequencies of the system. In the present study, it is observed that the addition of Gaussian white noise along with a harmonic excitation produces a near-grazing contact, when there was previously no contact between the tip and the sample with only the harmonic excitation. Moment evolution equations derived from a Fokker-Planck system are used to obtain numerical results, which support the statement that the addition of noise facilitates contact between the tip and the sample.

scholarly journals EMD OF GAUSSIAN WHITE NOISE: EFFECTS OF SIGNAL LENGTH AND SIFTING NUMBER ON THE STATISTICAL PROPERTIES OF INTRINSIC MODE FUNCTIONS

2009 ◽  
Vol 01 (04) ◽  
pp. 517-527 ◽  
Author(s):  
GASTÓN SCHLOTTHAUER ◽  
MARÍA EUGENIA TORRES ◽  
HUGO L. RUFINER ◽  
PATRICK FLANDRIN
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Kernel Smoothing ◽  
Gaussian White Noise ◽  
Large Data ◽  
Intrinsic Mode Functions ◽  
Mode Decomposition ◽  
Non Gaussian ◽  
Data Length ◽  
Number Of Iterations

This work presents a discussion on the probability density function of Intrinsic Mode Functions (IMFs) provided by the Empirical Mode Decomposition of Gaussian white noise, based on experimental simulations. The influence on the probability density functions of the data length and of the maximum allowed number of iterations is analyzed by means of kernel smoothing density estimations. The obtained results are confirmed by statistical normality tests indicating that the IMFs have non-Gaussian distributions. Our study also indicates that large data length and high number of iterations produce multimodal distributions in all modes.

Response Statistics and Reliability Analysis of a Mistuned and Frictionally Damped Bladed Disk Assembly Subjected to White Noise Excitation

2010 ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Gaussian White Noise ◽  
Random Excitation ◽  
Forced Response ◽  
Computationally Efficient ◽  
Bladed Disk ◽  
White Noise Excitation ◽  
Bladed Disk Assembly ◽  
Mean Square Response

In the design of gas turbine engines, the analysis of nonlinear vibrations of mistuned and frictionally damped blade-disk assembly subjected to random excitation is highly complex. The transitional probability density function (PDF) for the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and backward Kolmogorov equations. This paper presents important improvement and extensions to a computationally efficient higher order, finite difference (FD) technique for the solution of higher dimensional FP equation corresponding to a two degree of freedom nonlinear system representative of vibration of tip shrouded frictionally damped bladed disk assembly subjected to Gaussian white noise excitation. Effects of friction damping on the mean square response of a blade are investigated. The friction coefficient of the damper is assumed to be a function of the sliding velocity of the contact surface. The effects of stiffness and damping mistuning on the forced response of frictionally damped bladed disk are investigated. Numerical studies are presented for a pair of mistuned blades of cyclic assemblies. The response and reliability of a blade subjected to random excitation is also obtained. With time averaged probability density as an invariant measure, the probability of large excursion in case of damping mistuning is also presented. The results of the FD method are validated by comparing with Monte Carlo Simulation (MCS) results.

On the Fundamental Issues in Nonstationary Dynamics and Chaos

1993 ◽  
Author(s):  
Ross M. Evan-Iwanowski ◽  
Chu Ho Lu ◽  
Germain L. Ostiguy
Keyword(s):  
Duffing Oscillator ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Period Doubling ◽  
Codimension One ◽  
Extended Value ◽  
Limit Motion ◽  
Short Time ◽  
Nonstationary Dynamics ◽  
Considerable Complexity

Abstract In nonstationary (NS) systems some control parameters (CPs) have the following forms: CP(t) = CP0 + ψ(t), where CP0 = const, and ψ(t) are arbitrary functions of t. Other arbitrary functions which play a pivotal role in the NS systems are the parameter functions ϕ(CP) = 0, CP = {CP1, CP2...CPn}. While the functions ψ(t) determine the time directions of the NS dynamical behavior, the functions ϕ(CP) = 0 determine the paths for the CPs to follow. The NS processes are permanently transient due to the functions ψ(t) and/or ϕ(CP) = 0, and for that reason, they can be extremely complex. Clearly then, it is essential to address the fundamental problems of cohesion and definitness of these processes. Using select examples, these issues have been studied in this presentation and have been resolved in positive. Specifically, the following have been demonstrated: (1) convergence (definitivness) of the NS logistic map and the softening Duffing oscillator to an NS limit motion (2) the appearance of a sequence of similar attractors for different NS bifurcations (3) the effects of different parameter paths, ϕ(CP) = 0, in the period doubling region of the Duffing oscillator (4) the effects of linear and cyclic paths in transition through the Ueda bifurcation regions. The results obtained show considerable complexity of the NS dynamic and chaotic responses (5) for exponential ψ(t), the Lorenz “weather” three-term model exhibit a periodic “window” in the chaotic range for an extended value of t. (6) the effects of different ψ(t) in the typical codimension one bifurcations (7) the ST chaos may be created or annihilated by injection of NS inputs (8) an efficient and fast stabilization, i.e., reduction of ST vibration to near the static equilibrium in a short time, can be accomplished by NS changes of the parameters of the system.

scholarly journals GENERALIZED WIENER PROCESS AND KOLMOGOROV'S EQUATION FOR DIFFUSION INDUCED BY NON-GAUSSIAN NOISE SOURCE

2005 ◽  
Vol 05 (02) ◽  
pp. L267-L274 ◽  
Author(s):  
ALEXANDER DUBKOV ◽  
BERNARDO SPAGNOLO
Keyword(s):  
White Noise ◽  
Wiener Process ◽  
Anomalous Diffusion ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Non Gaussian ◽  
Generalized Wiener Process ◽  
Fokker Planck

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

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