Behavior of a Logistic Map Driven by White Noise

Yang, Zhengling; Gao, Yang; Yong-Tao, Gao; Zhang, Jun

doi:10.1088/0256-307x/26/6/060506

Cited by 8 publications

(6 citation statements)

References 20 publications

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“…Zheng at al. 20 suggested that it is only after the four-cycle begins when noise can excite chaos. Figure 1(b) shows the return maps for the logistic map for a parameter value of 3.6.…”

Section: Analysis Of Noisy Logistic Mapmentioning

confidence: 99%

“…It hides the local dynamical changes and the higher periodic orbits. As four different periods are visible when a very low level of noise is added, this level of noise may induce chaos 20 .…”

Section: Analysis Of Nnd Model For Si Enginesmentioning

confidence: 99%

“…[17][18][19][20][21][22][23] Noise can induce chaos in a nonlinear dynamic map. 20,22,23 But noise can also be induced in a dynamical system to check the robustness of the deterministic dynamical system against the fluctuations. In the present study, both these effects related to noise are examined for a low-dimensional model.…”

Section: Introductionmentioning

confidence: 99%

“…Due to its mathematical simplicity and related complexity, a Logistic map influenced by noise is one of the most studied maps. [17][18][19][20][21][22][23] Noise can induce chaos in a nonlinear dynamic map. 20,22,23 But noise can also be induced in a dynamical system to check the robustness of the deterministic dynamical system against the fluctuations.…”

Section: Introductionmentioning

confidence: 99%

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Application of delay embedding and recurrence analysis to a noisy nonlinear map for cyclic combustion dynamics of SI engines

Singh

Maurya

2022

International Journal of Engine Research

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A noisy nonlinear dynamical (NND) model available for cyclic variations in SI engines is examined using nonlinear time series methods. Heat release time series computed from this model is analyzed by employing bifurcation diagrams, return maps, recurrence plots, and recurrence quantitative analysis techniques. The analysis methodology is at first applied to a noisy Logistic map to optimize the necessary parameters, and later the same is used for the NND map. The presence of noise hides the local patterns of the dynamics and higher periodic orbits, making the bifurcation diagrams look fuzzy for both maps. An increase in the noise level and uncombusted residuals in the NND map advance the onset point of bifurcations toward a higher equivalence ratio. The application of the NND model is extended to represent the SI engine combustion dynamics for a higher amount of internal EGR as it is favored for cycle-resolved control due to its feedback period of one combustion cycle. The combined effect of lean charge and internal EGR leads to a more complex dynamical nature. The addition of a higher amount of internal EGR adds some complex chaotic-like regions in the bifurcation diagrams. For a low level of fluctuations in the NND map, more than four noisy periodic orbits are observed for such operating conditions. The geometry of phase space trajectories for the low-level noisy model data with a high level of internal EGR resembles a chaotic system. Bifurcation diagrams, recurrence plots, and quantitative recurrence analysis indicate an intermittent route to chaos at a highly lean and diluted charge. The presence of deterministic chaotic characteristics in combustion dynamics, along with effort-saving techniques of data acquisition and estimating combustion phasing, depicts the possibilities of practical application in active onboard control of engine systems.

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“…Zheng at al. 20 suggested that it is only after the four-cycle begins when noise can excite chaos. Figure 1(b) shows the return maps for the logistic map for a parameter value of 3.6.…”

Section: Analysis Of Noisy Logistic Mapmentioning

confidence: 99%

Section: Analysis Of Nnd Model For Si Enginesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Application of delay embedding and recurrence analysis to a noisy nonlinear map for cyclic combustion dynamics of SI engines

Singh

Maurya

2022

International Journal of Engine Research

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“…There are little literatures in the study of second kind of error causation. Our previous research showed that the forecasting accuracy can be improved more than 0.3% ~ 1% by a better SNR of the input data for the forecasting of short term expressway traffic flows and wind speeds [9][10][11].…”

Section: Imentioning

confidence: 99%

Reduce forecasting errors by low order nonlinear transformations for complex time series

Yang

Shu-Chun

Zhang

et al. 2011

2011 International Conference on Electronics, Communications and Control (ICECC)

Self Cite

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Besides the forecasting errors by a model with not entirely describing the characteristics of the complex systems in the real world, a digital computer can produce additive errors due to its finite digits, such as the singularity in matrix inversion, if there are large white noise in the forecasted series. Low order nonlinear transformations can make the inputted data of forecasting models to an appropriate signal to noise ratio, decrease the bad influence of the "trend", and make a high order forecasting model equivalently into a linear model, even a lower order model than a linear one. All these advantages reduce the forecasting errors. The numerical experiments support the theoretical analyses of a low order nonlinear transformation.

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Removing Forecasting Errors with White Gaussian Noise after Square Root Transformation

Yang

Liu

Zhu

et al. 2016

Journal of Forecasting

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An analytical model has been developed in the present paper based on a square root transformation of white Gaussian noise. The mathematical expectation and variance of the new asymmetric distribution generated by white Gaussian noise after a square root transformation are analytically deduced from the preceding four terms of the Taylor expansion. The model was first evaluated against numerical experiments and a good agreement was obtained. The model was then used to predict time series of wind speeds and highway traffic flows. The simulation results from the new model indicate that the prediction accuracy could be improved by 0.1–1% by removing the mean errors. Further improvement could be obtained for non‐stationary time series, which had large trends. Copyright © 2016 John Wiley & Sons, Ltd.

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Behavior of a Logistic Map Driven by White Noise

Cited by 8 publications

References 20 publications

Application of delay embedding and recurrence analysis to a noisy nonlinear map for cyclic combustion dynamics of SI engines

Application of delay embedding and recurrence analysis to a noisy nonlinear map for cyclic combustion dynamics of SI engines

Reduce forecasting errors by low order nonlinear transformations for complex time series

Removing Forecasting Errors with White Gaussian Noise after Square Root Transformation

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