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Tellus A: Dynamic Meteorology and Oceanography Volume 61, 2008 - Issue 4

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Original Articles

On the reliability of computed chaotic solutions of non-linear differential equations

Shijun LiaoState Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, ChinaCorrespondence[email protected]

Pages 550-564 | Received 14 Oct 2008, Accepted 05 Mar 2009, Published online: 15 Dec 2016

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https://doi.org/10.1111/j.1600-0870.2009.00402.x

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Citations (44)

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Articles from other publishers (44)

Bo Zhang & Shijun Liao. (2023) Ultra-chaos in a meandering jet flow. Physica D: Nonlinear Phenomena 455, pages 133886.
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Tianli Hu & Shijun Liao. (2023) Ultra-chaotic motion in the hexagonal Beltrami flow. AIP Advances 13:8.
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Marek Lampart & Jaroslav Zapoměl. (2023) The disturbance influence on vibration of a belt device driven by a crank mechanism. Chaos, Solitons & Fractals 173, pages 113634.
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Yu Yang, Shijie Qin & Shijun Liao. (2023) Ultra-chaos of a mobile robot: A higher disorder than normal-chaos. Chaos, Solitons & Fractals 167, pages 113037.
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Ivan Hristov, Radoslava Hristova, Stefka Dimova, Petar Armyanov, Nikolay Shegunov, Igor Puzynin, Taisia Puzynina, Zarif Sharipov & Zafar Tukhliev. 2023. Advanced Computing in Industrial Mathematics. Advanced Computing in Industrial Mathematics 56 66 .

Shijie Qin & Shijun Liao. (2022) Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence. Journal of Fluid Mechanics 948.
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Shijun Liao, Xiaoming Li & Yu Yang. (2022) Three-body problem — From Newton to supercomputer plus machine learning. New Astronomy 96, pages 101850.
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Ke Ding & Arne J. Pearlstein. (2022) Free Response of a Rotational Nonlinear Energy Sink Coupled to a Linear Oscillator: Fractality, Riddling, and Initial-Condition Sensitivity at Large Initial Displacements. Journal of Applied Mechanics 89:6.
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Ke Ding & Arne J. Pearlstein. (2021) Free Response of a Rotational Nonlinear Energy Sink Coupled to a Linear Oscillator: Fractality, Riddling, and Initial-Condition Sensitivity at Intermediate Initial Displacements. Journal of Applied Mechanics 88:12.
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Xiaoming Li, Xiaochen Li, Linghui He & Shijun Liao. (2021) Triple collision orbits in the free-fall three-body system without binary collisions. Celestial Mechanics and Dynamical Astronomy 133:10.
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Wieslaw Marszalek, Jan Sadecki & Maciej Walczak. (2021) Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams. Entropy 23:7, pages 876.
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Tianzhuang Xu, Jing Li, Zhihui Li & Shijun Liao. (2021) Accurate predictions of chaotic motion of a free fall disk. Physics of Fluids 33:3.
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XiaoMing Li, XiaoChen Li & ShiJun Liao. (2020) One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems. Science China Physics, Mechanics & Astronomy 64:1.
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Shijie Qin & Shijun Liao. (2020) Influence of numerical noises on computer-generated simulation of spatio-temporal chaos. Chaos, Solitons & Fractals 136, pages 109790.
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Shijun Liao & Xiaoming Li. (2019) On the periodic solutions of the three-body problem. National Science Review 6:6, pages 1070-1071.
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Xiaoming Li & Shijun Liao. (2019) Collisionless periodic orbits in the free-fall three-body problem. New Astronomy 70, pages 22-26.
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Erivelton G Nepomuceno & Matjaž Perc. (2019) Computational chaos in complex networks. Journal of Complex Networks.
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Wanting Xu, Michael L. Stein & Ian Wisher. (2019) Modeling and Predicting Chaotic Circuit Data. SIAM/ASA Journal on Uncertainty Quantification 7:1, pages 31-52.
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Ashraf A. Zaher. (2018) On the Discretization of Continuous-Time Chaotic Systems for Digital Implementations. Journal of Physics: Conference Series 1141, pages 012116.
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Xiaoming Li & Shijun Liao. (2018) Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems. Applied Mathematics and Mechanics 39:11, pages 1529-1546.
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Xiaoming Li, Yipeng Jing & Shijun Liao. (2018) Over a thousand new periodic orbits of a planar three-body system with unequal masses. Publications of the Astronomical Society of Japan 70:4.
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Jifeng Cui, Wenyu Zhang, Zeng Liu & Jianglong Sun. (2017) On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator. Numerical Algorithms 78:4, pages 1217-1231.
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Erivelton G. Nepomuceno, Samir A.M. Martins, Bruno C. Silva, Gleison F.V. Amaral & Matjaž Perc. (2018) Detecting unreliable computer simulations of recursive functions with interval extensions. Applied Mathematics and Computation 329, pages 408-419.
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Xiaoming Li, Xiaochen Li & Shijun Liao. (2018) The effect of numerical noises on statistics of a chaotic dynamical system. Journal of Physics: Conference Series 1039, pages 012017.
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Lu Li, Zhiliang Lin & Yan Hao. 2018. Computational Science – ICCS 2018. Computational Science – ICCS 2018 722 735 .

XiaoMing Li & ShiJun Liao. (2017) More than six hundred new families of Newtonian periodic planar collisionless three-body orbits. Science China Physics, Mechanics & Astronomy 60:12.
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Pengfei Wang. (2017) A high-order spatiotemporal precision-matching Taylor–Li scheme for time-dependent problems. Advances in Atmospheric Sciences 34:12, pages 1461-1471.
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L. De Micco, M. Antonelli & H.A. Larrondo. (2017) Stochastic degradation of the fixed-point version of 2 D -chaotic maps. Chaos, Solitons & Fractals 104, pages 477-484.
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Shi-jun Liao. (2017) On the clean numerical simulation (CNS) of chaotic dynamic systems. Journal of Hydrodynamics 29:5, pages 729-747.
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Erivelton Geraldo Nepomuceno & Eduardo M.A.M. Mendes. (2017) On the analysis of pseudo-orbits of continuous chaotic nonlinear systems simulated using discretization schemes in a digital computer. Chaos, Solitons & Fractals 95, pages 21-32.
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ZhiLiang Lin, LiPo Wang & ShiJun Liao. (2016) On the origin of intrinsic randomness of Rayleigh-Bénard turbulence. Science China Physics, Mechanics & Astronomy 60:1.
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Pengfei Wang. (2016) Forward Period Analysis Method of the Periodic Hamiltonian System. PLOS ONE 11:10, pages e0163303.
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Shijun Liao & Xiaoming Li. (2015) On the Inherent Self-Excited Macroscopic Randomness of Chaotic Three-Body Systems. International Journal of Bifurcation and Chaos 25:09, pages 1530023.
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Yong Liu, Pengfei Wang & Gang Huang. (2014) Study on the reliable computation time of the numerical model using the sliding temporal correlation method. Theoretical and Applied Climatology 119:3-4, pages 539-550.
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Pengfei Wang, Yong Liu & Jianping Li. (2014) Clean numerical simulation for some chaotic systems using the parallel multiple-precision Taylor scheme. Chinese Science Bulletin 59:33, pages 4465-4472.
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XiaoMing Li & ShiJun Liao. (2014) On the stability of the three classes of Newtonian three-body planar periodic orbits. Science China Physics, Mechanics & Astronomy 57:11, pages 2121-2126.
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Shijun Liao. (2014) Can We Obtain a Reliable Convergent Chaotic Solution in any Given Finite Interval of Time?. International Journal of Bifurcation and Chaos 24:09, pages 1450119.
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Shijun Liao. (2014) Physical limit of prediction for chaotic motion of three-body problem. Communications in Nonlinear Science and Numerical Simulation 19:3, pages 601-616.
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ShiJun Liao & PengFei Wang. (2014) On the mathematically reliable long-term simulation of chaotic solutions of Lorenz equation in the interval [0,10000]. Science China Physics, Mechanics and Astronomy 57:2, pages 330-335.
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Shijun Liao. (2013) On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems. Chaos, Solitons & Fractals 47, pages 1-12.
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Shijun Liao. (2012) Chaos: A bridge from microscopic uncertainty to macroscopic randomness. Communications in Nonlinear Science and Numerical Simulation 17:6, pages 2564-2569.
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Pengfei Wang, Jianping Li & Qian Li. (2011) Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of Lorenz equations. Numerical Algorithms 59:1, pages 147-159.
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Shijun LiaoShijun Liao. 2012. Homotopy Analysis Method in Nonlinear Differential Equations. Homotopy Analysis Method in Nonlinear Differential Equations 239 284 .

Shijun LiaoShijun Liao. 2012. Homotopy Analysis Method in Nonlinear Differential Equations. Homotopy Analysis Method in Nonlinear Differential Equations 3 14 .

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