Shen, B.-W., R. A. Pielke Sr., and X. Zeng 2023b: 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Special Issue on Multistability, Multiscale Predictability, and Sensitivity in Numerical Models. [Editorial] Atmosphere 2023, 14(8), 1279; https://doi.org/10.3390/atmos14081279 (22 journal pages)
Shen, B.-W.* 2023: Attractor Coexistence, Butterfly Effects, and Chaos Theory (ABC): A Review of Lorenz Models and a Generalized Lorenz Model. https://doi.org/10.13140/RG.2.2.30961.35685. (draft). (accepted, Sep. 13, 2023)
Shen, B.-W., 2023: A Review of Lorenz's Models from 1960 to 2008. International Journal of Bifurcation and Chaos. Vol. 33, No. 10, 2330024 (2023). https://doi.org/10.1142/S0218127423300240.
Shen, B.-W., R. A. Pielke Sr., X. Zeng, and X. Zeng, 2023: Lorenz’s View on the Predictability Limit. Encyclopedia 2023, 3(3), 887-899; https://doi.org/10.3390/encyclopedia3030063
Paxson, W#. and B.-W. Shen*, 2023: A KdV-SIR Equation and Its Analytical Solutions: An Application for COVID-19 Data Analysis. Chaos, Solitons & Fractals. https://doi.org/10.1016/j.chaos.2023.113610
Shen, B.-W., R. A. Pielke Sr., X. Zeng, J. Cui^#, S. Faghih-Naini^#, W. Paxson^#, A. Kesarkar, X. Zeng, R. Atlas, 2022c: The Dual Nature of Chaos and Order in the Atmosphere. Atmosphere 13, no. 11: 1892. https://doi.org/10.3390/atmos13111892.
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J. Cui, S. Faghih-Naini, W. Paxson, R. Atlas, 2022b: Three Kinds of Butterfly Effects Within Lorenz Models. Encyclopedia 2, no. 3: 1250-1259. https://doi.org/10.3390/encyclopedia2030084
Paxson, W#. and B.-W. Shen, 2022: A KdV-SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves. International Journal of Bifurcation and Chaos. Vol 13, No. 23, https://doi.org/10.1142/S0218127422501991
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, 2022a: One Saddle Point and Two Types of Sensitivities Within the Lorenz 1963 and 1969 Models. Atmosphere 13, no. 5: 753. https://doi.org/10.3390/atmos13050753
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini#, J. Cui#, and R. Atlas, 2021: Is Weather Chaotic? Coexisting Attractors and Multistability. 2021 Conference onWeather Analysis and Forecasting. Central Weather Bureau. October 26-28, 2021. https://doi.org/10.13140/RG.2.2.11229.95205 (invited)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini#, J. Cui#, and R. Atlas, 2021: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. AOGS. August 2, 2021. (pdf) https://bit.ly/2WsCtia (invited)
Cui, J.^# and B.-W. Shen^*, 2021: A Kernel Principal Component Analysis of Coexisting Attractors within a Generalized Lorenz Model. Chaos, Solitons & Fractals, 146. https://doi.org/10.1016/j.chaos.2021.110865.
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini#, J. Cui#, and R. Atlas, 2021: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. Bulletin of the American Meteorological Society, 102(1), E148-E158. Retrieved Jan 29, 2021, from https://journals.ametsoc.org/view/journals/bams/102/1/BAMS-D-19-0165.1.xml
Shen, B.-W.^*, 2021: Solitary Waves, Homoclinic Orbits, and Nonlinear Oscillations within the non-dissipative Lorenz Model, the inviscid Pedlosky Model, and the KdV Equation. In: Christos H. Skiadas, Yiannis Dimotikalis (eds) The 13th Chaos International Conference CHAOS 2020. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-70795-8_58 (proofs available from pdf)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini, J. Cui, R. Atlas, T.A. Reyes, 2021: Is Weather Chaotic? Coexisting Chaotic and Non-Chaotic Attractors within Lorenz Models. In: Christos H. Skiadas, Yiannis Dimotikalis (eds) The 13th Chaos International Conference CHAOS 2020. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-70795-8_57 (proofs available from pdf)
Shen, B.-W.^*, 2020: Homoclinic Orbits and Solitary Waves within the non-dissipative Lorenz Model and KdV Equation. International Journal of Bifurcation and Chaos. 30. 2050257-1-15. DOI:10.1142/S0218127420502570.
Shen, B.-W.^*, 2019a: Aggregated Negative Feedback in a Generalized Lorenz Model. International Journal of Bifurcation and Chaos, Vol. 29, No. 3 (2019) 1950037 (20 pages). https://doi.org/10.1142/S0218127419500378
Shen, B.-W.^*, 2019b: On the Predictability of 30-day Global Mesoscale Simulations of Multiple African Easterly Waves during Summer 2006: A View with a Generalized Lorenz Model. Geosciences 2019, 9(7), 281; https://doi.org/10.3390/geosciences9070281
Shen, B.-W.^*, T. Reyes^#, and S. Faghih-Naini^#, 2018: Coexistence of Chaotic and Non-Chaotic Orbits in a New Nine-Dimensional Lorenz Model. In: Skiadas C., Lubashevsky I. (eds) 11th Chaotic Modeling and Simulation International Conference. CHAOS 2018. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-15297-0_22 (slides) (pdf) (google ebook)
Reyes, T.^# and B.-W. Shen^*, 2019a: A Recurrence Analysis of Chaotic and Non-Chaotic Solutions within a Generalized Nine-Dimensional Lorenz Model. Chaos, Solitons & Fractals. 125 (2019), 1-12. https://doi.org/10.1016/j.chaos.2019.05.003
Reyes, T.^# and B.-W. Shen^*, 2019b: A Recurrence Analysis of of Multiple African Easterly Waves during Summer 2006. (accepted, May 16, 2019; published)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, S. Faghih-Naini^#, C.-L. Shie, R. Atlas, J.-J. Baik, and T. A. L. Reyes^#, 2018: Butterfly Effects of the First and Second Kinds: New Insights Revealed by High-dimensional Lorenz Models. The 11th Chaos International Conference (CHAOS2018), Rome, Italy, June 5-8, 2018. (slides) (pdf)
Shen, B.-W.^*, 2018: On periodic solutions in the non-dissipative Lorenz model: the role of the nonlinear feedback loop. Tellus A: 2018, 70, 1471912, https://doi.org/10.1080/16000870.2018.1471912. (pdf)
Faghih-Naini, S.^# and B.-W. Shen^*, 2018: Quasi-periodic orbits in the five-dimensional non-dissipative Lorenz model: the role of the extended nonlinear feedback loop. International Journal of Bifurcation and Chaos, Vol. 28, No. 6 (2018) 1850072 (20 pages). https://doi.org/10.1142/S0218127418500724 (IJBC is ranked 13rd out of 116 in the Multidisciplinary category) (selected as one of IJBC featured articles of 2018) (gallery) (pdf)

Papers under review/in preparation

Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, and X. Zeng, 2023c: Exploring the Origin of the Two-Week Predictability Limit: A Revisit of Lorenz’s Predictability Studies in the 1960s. Available from ResearchGate: https://doi.org/10.13140/RG.2.2.13760.30727 (submitted July 21, 2023; revised November 12, 2023)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, 2022: Comments on “Mesoscale Predictability in Moist Midlatitude Cyclones Is Not Sensitive to the Slope of the Background Kinetic Energy Spectrum” by Lloveras, Tierney, and Durran (2022). (submitted to JAS, April 21, 2022). https://doi.org/10.13140/RG.2.2.10944.20484 (re-written, yielding the following draft)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, 2023: Revisiting Lorenz’s and Lilly’s Empirical Formulas for Predictability Estimates. https://doi.org/10.13140/RG.2.2.32941.15849 (resubmitted, Sep. 2023)
Pielke, R.A., B.-W. Shen, X. Zeng, 2023: Can a butterfly in Brazil cause a tornado in Texas? (Submitted, November 28, 2023)
Shen, B.-W., R. Pielke, Sr., and X. Zeng, 2022: Responses (II) to Reviewers' Comments on "A Note on Lorenz’s and Lilly’s Empirical Formulas for Predictability Estimates." Available from ResearchGate: https://doi.org/10.13140/RG.2.2.33511.73120
Shen, B.-W.^*, et al., 2021: Is Weather Chaotic? Coexisting Attractors, Multistability, and Predictability. https://doi.org/10.13140/RG.2.2.19105.74081 (Part 1 published in Shen et al. 2022c)
Shen, B.-W., 2023: A Minimal Set of Dissipative Terms in the Lorenz 1963 Model for Chaos. https://doi.org/10.13140/RG.2.2.30583.04003 (under revision)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, S. Faghih-Naini^#, and J. Cui, 2021: On the Intrinsic and Practical Predictability of the Turning Point of an Expanding Epidemic. (Submitted to Proceedings of the National Academy of Sciences, Jan. 12, 2021; Under review, Jan. 26, 2021; Part 1 published in Paxson and Shen 2022).
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, S. Faghih-Naini^#, C.-L. Shie, Atlas, J.-J. Baik, T.A.L. Reyes^#, 2018: Butterfly Effects of the First and Second Kinds in Lorenz Models. https://dx.doi.org/10.13140/RG.2.2.26390.45124 (Part I published as a BAMS InBox article on 28 Sep 2020; Part II published in Chaos 2020 conference proceedings in 2021; Part III with three kinds of butterfly effects, published in 2022; Part 4 submitted by Pielke Sr., Shen, and Xubin 2023).
Shen, B.-W.^*, R. A. Pielke Sr., et al., 2020: a proposal for the JAS review article entitled "Butterfly Effects of the First and Second Kinds: A Review Based on Original and Generalized Lorenz Models." http://doi.org/10.13140/RG.2.2.32236.80002 (under revisions)
Shen, B.-W.^*, R. A. Pielke Sr., et al., 2020: Butterfly Effects of the First and Second Kinds: A Review Based on Original and Generalized Lorenz Models. http://doi.org/10.13140/RG.2.2.12952.42245 (under revisions).
Shen, B.-W.^*, C.-D. J. Lin, and J. Cui^#, 2019: Reveal Scale Dependence in the Chaotic Solutions of High-dimensional Lorenz models using Factor Analysis (to be submitted).
Shen, B.-W.^*, 2020: Computational Chaos in High-dimensional Lorenz Models. (invited, to be submitted)

Recent Presentations:

Paxson, W. and Shen, B.-W.^*,2023: A KdV-SIR Equation and Analytical Solutions:Relationship to the Non-dissipative Lorenz Model and an Application for COVID-19 Data Analysis. https://doi.org/10.13140/RG.2.2.23816.80648
Shen, B.-W.^*, et al, 2022: Attractor Coexistence, Butterfly Effects, and Chaos Theory (ABC): A Review of Lorenz Models and a Generalized Lorenz Model. https://doi.org/10.13140/RG.2.2.30961.35685
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, 2022: Finite Predictability and Two Types of Sensitivities Within the Lorenz 1963 and 1969 Models. (abstract accepted)
Shen, B.-W.^*, 2022: Coexisting Chaotic and Non-chaotic Attractors, Multistability, Multiscale Instability, and Predictability within Lorenz Models. Feb. 2020. https://doi.org/10.13140/RG.2.2.35120.02562 (invited)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, 2022: One Saddle Point and Two Types of Sensitivities Within the Lorenz 1963 and 1969 Models. EGU General Assembly 2022. Vienna | Austria, 23–27 May 2022, EGU22-10890, 2022. (abstract accepted)
Paxson, W. and B.-W. Shen^*, 2022: A KdV-S.I.R.-type Model Under a Weak Outbreak: Analytical Solutions of Solitary Epidemic Waves and Homoclinic Orbits. June 2022. https://doi.org/10.13140/RG.2.2.31803.90408 (abstract accepted)
Shen, B.-W., R. A. Pielke Sr., X. Zeng, S. Faghih-Naini#, J. Cui#, and R. Atlas, 2021: Is Weather Chaotic? Coexisting Attractors and Multistability. 2021 Conference on Weather Analysis and Forecasting. Central Weather Bureau. October 26-28, 202. Slides: https://doi.org/10.13140/RG.2.2.13244.77440 (invited) (recorded video)
Shen, B.-W.^*, 2021: Collaboration with Tao on Multiscale Simulations and Multiscale Modeling. In Celebration of Wei-Kuo Tao’s Retirement. NASA Goddard Space Flight Center 18 December 2021. 10.13140/RG.2.2.31861.09440 (video)
Shen, B.-W.^*, 2021: Is Weather Chaotic? Multistability, Multiscale Instability, and Predictability within Lorenz Models. Oxford University. 11 October 2021. https://doi.org/10.13140/RG.2.2.13991.88485 (recorded video) (invited)
Shen, B.-W.^*, 2021: Is Weather Chaotic? Coexisting Attractors and Multistability within a Generalized Lorenz Model. NCU. August, 2021.
Shen, B.-W.*, 2021: Is Weather Chaotic? Coexisting Chaotic and Non-Chaotic Attractors and Time Varying Multistability within a Generalized Lorenz Model. 4 August 2021. HFIP. http://doi.org/10.13140/RG.2.2.14271.02720 (invited)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini#, J. Cui#, and R. Atlas, 2021: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model.AGOS. August 2, 2021. http://doi.org/DOI:%2010.13140/RG.2.2.14991.10407 (AOGS, invited).
Shen, B.-W.*, 2021: An Insightful Analysis of the Lorenz 1969 Model: A Perspective of Dynamical Systems Theory. NTU. May 18, 2021. http://doi.org/DOI:%2010.13140/RG.2.2.26735.15520. (invited)
Shen, B.-W.*, 2021: On the Dual Nature of Chaos and Order in Weather and Climate: Understanding and Detecting Coexisting Attractors. NASA Goddard Earth Sciences Data and Information Services Center (GES DISC). May 4, 2021. http://doi.org/10.13140/RG.2.2.24054.73281 (invited)
Shen, B.-W.^*, 2021: Lorenz Models, Butterfly Effect, and Predictability. The University of Arizona. http://doi.org/10.13140/RG.2.2.28674.30402. March 18, 2021 (invited) (recorded video)
Shen, B.-W.^*, 2021: A Kernel Principal Component Analysis of Coexisting Attractors within a Generalized Lorenz Model. BDA 600. San Diego State University. March 4, 2021. http://doi.org/10.13140/RG.2.2.26008.93440 (invited)
Shen, B.-W.^*et al., 2021: Butterfly Effects of the First and Second Kinds: A Review Based on Original and Generalized Lorenz Models.The 14th Chaos International Conference CHAOS 2021. (June)
Shen, B.-W. et al. 2021: Epidemic Waves, Solitary Waves, and Homoclinic Orbits Within the Epidemic SIR, the KdV, and Non-Dissipative Lorenz Systems. (an abstract submitted, Jan. 6, 2021)
Cui, J.^# and B.-W. Shen^*, 2020: Applying a Kernel PCA Method to Reveal Coexisting Attractors within a Generalized Lorenz Model. Chaotic Modeling & Simulation Web Conference. 22-24 October 2020. http://doi.org/10.13140/RG.2.2.28342.11843. (a plenary talk)
Shen, B.-W.*, 2020: A Lighting Talk entitled "Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model".http://doi.org/10.13140/RG.2.2.27349.86241. 26 June 2020.
Shen, B.-W.*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, T.A.L. Reyes^#, S. Faghih-Naini^#, R. Atlas, and J. Cui^#, 2020: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. The 13th Chaos International Conference (CHAOS2020). 9-12 June 2020. (virtual conference) https://doi.org/10.13140/RG.2.2.35867.90404
Shen, B.-W.*, 2020: Homoclinic Orbits and Solitary Waves within the non-dissipative Lorenz Model and KdV Equation. The 13th Chaos International Conference (CHAOS2020). 9-12 June 2020. https://doi.org/10.13140/RG.2.2.34190.18244
Shen, B.-W.*, On the Dual Nature of Chaos and Order in Weather and Climate: New Insights and Opportunities Within a Generalized Lorenz Model. https://doi.org/10.13140/RG.2.2.12468.19843. Computational Science Research Center. 6 November 2020. (recorded presentation)
Shen, B.-W.*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, T.A.L. Reyes^#, S. Faghih-Naini^#, R. Atlas, and J. Cui^#, 2020: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model.100th AMS meeting. Boston, MA, 12-16 January 2020. http://doi.org/10.13140/RG.2.2.21811.07204 (slides) (recorded presentation)
Shen, B.-W.*, 2020: On the Predictability of 30-day Global Mesoscale Simulations of Multiple African Easterly Waves during Summer 2006: A View with a Generalized Lorenz Model. 100th AMS meeting. Boston, MA, 12-16 January 2020. (slides)
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, T.A.L. Reyes^#, S. Faghih-Naini^#, R. Atlas, and J. Cui^#, 2019: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. Workshop 2 “Big data, data assimilation, and uncertainty quantification” Institut Henry Poincare, Paris (France), 12-15 November 2019. Available from ResearchGate http://doi.org/10.13140/RG.2.2.21811.07204 (slides) (recorded talk)
Shen, B.-W.*, 2019: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. Presented for Math542. San Diego, 17 October, 2019. Available from Research Gate: http://bit.ly/2Br3yEO
Shen, B.-W.^*, 2019: Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. Research Center for Environmental Changes, Academia Sinica. Available from ResearchGate http://doi.org/10.13140/RG.2.2.14720.07681/1
Shen, B.-W.^*, 2019: Butterfly Effects and Chaos within a Generalized Lorenz Model: New Insights and Opportunities. http://doi.org/10.13140/RG.2.2.24721.28000 NOAA/AOML/HRD (Hurricane Research Division), Feb. 25, 2019.
Shen, B.-W.^*, 2019: Butterfly Effects and Chaos: New Insights Revealed by a Generalized Lorenz Model. National Taiwan University. Jan. 3 2019 (invited) http://doi.org/10.13140/RG.2.2.15765.99040
Shen, B.-W.^*, 2019: Butterfly Effects and Chaos: New Insights Revealed by a Generalized Lorenz Model. National Central University. Jan. 4 2019 (invited) http://doi.org/10.13140/RG.2.2.15765.99040
Shen, B.-W., 2018: A Generalized Lorenz Model. Prepared for Math542, computational ODEs (first released on October 2; updated on December 23). http://dx.doi.org/10.13140/RG.2.2.19713.71520
Reyes, T.^# and B.-W. Shen^*, 2018: Applying Recurrence Analysis to Illustrate the Co-existence of Chaotic and Non-Chaotic Solutions within a Generalized Lorenz Model. AGU 2018 Fall Meeting. Washington, D.C., December 10-14, 2018. https://dx.doi.org/10.13140/RG.2.2.20197.29925
Cui, J.^# and B.-W. Shen^*, 2018: Hierarchical Scale Dependence and Co-existence of Chaotic and Non-Chaotic Processes within a Generalized Lorenz Model: A Study using Kernel PCA and SVM Methods. AGU 2018 Fall Meeting. Washington, D.C., December 10-14, 2018. https://goo.gl/JDE3Fq
Shen, B.-W.^*, S. Cheung, J.-L. F. Li, T.A.L. Reyes^#, J. Cui^#, S. Faghih-Naini^#: 2018: Reveal the Role of Butterfly Effects and Multiscale Processes in Predictability using Advanced Concurrent Visualization and Multiscale Analysis (PEEMD) Methods. AGU 2018 Fall Meeting. Washington, D.C., December 10-14, 2018. (invited) https://goo.gl/2nG4hg
Shen, B.-W.^*, S. Cheung, J.-L. F. Li, T.A.L. Reyes^#, J. Cui^#, S. Faghih-Naini^#: On the Predictability of Short-term Climate Simulations of African Easterly Waves within a Global Mesoscale Model: A View with a Generalized Lorenz Model. AGU 2018 Fall Meeting. Washington, D.C., December 10-14, 2018. https://dx.doi.org/10.13140/RG.2.2.15995.36648
Shen, B.-W.^*, 2018:Understanding Butterfly Effects and Predictability. http://dx.doi.org/10.13140/RG.2.2.27266.58569. Technical report as supplemental materials for the manuscript of Shen et al. (2018) to be submitted to BAMS.
Shen, B.-W.^*, 2018: Concurrent Visualization (CV) and Parallel Ensemble Empirical Mode Decomposition (PEEMD) for Big Earth Science Data Analysis. March, 2, 2018. (invited) (slides)
Shen, B.-W.^*, Y.-L. Wu^@, and S. Cheung, 2018: Exploring the Role of Large-Scale Environmental Flow in Tropical Cyclone Genesis: 10-year Data Analysis using the PEEMD. AOGS 15th Annual Meeting. Honolulu, Hawaii, June 03-08, 2018. https://doi.org/10.13140/RG.2.2.12243.68640
Shen, B.-W.^*, 2018: Understanding the Predictability of Short-term Climate Simulations of African Easterly Waves using a Global Mesoscale Model and an Idealized Lorenz Model. AOGS 15th Annual Meeting. Honolulu, Hawaii, June 03-08, 2018. http://doi.org/10.13140/RG.2.2.31530.93129
Shen, B.-W.^*, R. A. Pielke Sr., X. Zeng, I. A. Santos, S. Faghih-Naini^#, J. Buchmann, C.-L. Shie, and R. Atlas, 2018: Butterfly Effects of the First and Second Kinds in Lorenz Models. AMS 2018 annual meeting. January 7-11, 2018. https://doi.org/10.13140/rg.2.2.36540.74881 (pdf) (slides)
Faghih-Naini, S.^# and B.-W. Shen^*, 2018:High-dimensional Lorenz Modeling in Python: Chaotic, Limit Cycle and Quasi-Periodic Solutions. AMS 2018 annual meeting. January 7-11, 2018. (pdf) (slides)