# The Dual Nature of Chaos and Order in the Atmosphere

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

“Weather possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons”.

## 2. Analysis and Discussion

#### 2.1. An Analogy for Monostability and Multistability Using Skiing and Kayaking

#### 2.2. Single-Types of Attractors, SDIC, and Monostability within the L63 Model

#### 2.3. Coexisting Attractors and Multistability within the GLM

#### 2.4. Time Varying Multistability and Recurrent Slowly Varying Solutions

#### 2.5. Onset of Emerging Solutions

#### 2.6. Various Types of Solutions within the L69 Model

- The L69 model is a closure-based, physically multiscale, mathematically linear, and numerically ill-conditioned system.
- The L69 multiscale model has been used for revealing energy transfer and scale interaction.
- The L69 linear model cannot produce chaos.
- Since it possesses both positive and negative eigenvalues with large variances, yielding a large condition number (e.g., Figure 4 and Figure 5 of [17]), the L69 model produces a different kind of sensitivity, as compared to SDIC within the L63 model.
- The model permits the occurrence of linearly stable and unstable solutions as well as oscillatory solutions. However, only unstable solutions have been a focus in predictability studies.

#### 2.7. Distinct Predictability within Lorenz Models

- The L63 nonlinear model with monostability is effective for revealing the chaotic nature of weather, suggesting finite intrinsic predictability within the chaotic regime of the system (i.e., the atmosphere).
- The L69 linear model with ill-conditioning easily captures unstable modes and, thus, is effective for revealing the practical finite predictability of the model.
- The GLM with multistability suggests both limited and unlimited (i.e., up to a system’s lifetime) intrinsic predictability for chaotic and non-chaotic solutions, respectively.

#### 2.8. Non-Chaotic Weather Systems

#### 2.9. Suggested Future Tasks

## 3. Concluding Remarks

“The atmosphere possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons”,

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] - Lorenz, E.N. Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas? In Proceedings of the 139th Meeting of AAAS Section on Environmental Sciences, New Approaches to Global Weather, GARP, AAAS, Cambridge, MA, USA, 29 December 1972. [Google Scholar]
- Lorenz, E.N. The Essence of Chaos; University of Washington Press: Seattle, WA, USA, 1993; 227p. [Google Scholar]
- Gleick, J. Chaos: Making a New Science; Penguin: New York, NY, USA, 1987; 360p. [Google Scholar]
- The Nobel Committee for Physics. Scientific Background on the Nobel Prize in Physics 2021 For Groundbreaking Contributions to Our Understanding of Complex Physical Systems; The Nobel Committee for Physics: Stockholm, Sweden, 2021; Available online: https://www.nobelprize.org/prizes/physics/2021/popular-information/ (accessed on 28 June 2022).
- Maxwell, J.C. Matter and Motion; Dover: Mineola, NY, USA, 1952. [Google Scholar]
- Poincaré, H. Sur le problème des trois corps et les équations de la dynamique. Acta Math.
**1890**, 13, 1–270. [Google Scholar] - Poincaré, H. Science et Méthode. Flammarion; Maitland, F., Translator; Science and Method 1908; Thomas Nelson and Sons: London, UK, 1914. [Google Scholar]
- Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X.; Baik, J.-J.; Faghih-Naini, S.; Cui, J.; Atlas, R. Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. Bull. Am. Meteorol. Soc.
**2021**, 2, E148–E158. [Google Scholar] [CrossRef] - Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X.; Baik, J.-J.; Faghih-Naini, S.; Cui, J.; Atlas, R.; Reyes, T.A. Is Weather Chaotic? Coexisting Chaotic and Non-Chaotic Attractors within Lorenz Models. In Proceedings of the 13th Chaos International Conference CHAOS 2020, Florence, Italy, 9–12 June 2020; Skiadas, C.H., Dimotikalis, Y., Eds.; Springer Proceedings in Complexity. Springer: Cham, Switzerland, 2021. [Google Scholar] [CrossRef]
- Lorenz, E.N. The predictability of a flow which possesses many scales of motion. Tellus
**1969**, 21, 289–307. [Google Scholar] [CrossRef] - Lorenz, E.N. Investigating the Predictability of Turbulent Motion. In Statistical Models and Turbulence, Proceedings of the symposium held at the University of California, San Diego, CA, USA, 15–21 July 1971; Springer: Berlin/Heidelberg, Germany, 1972; pp. 195–204. [Google Scholar]
- Lorenz, E.N. Low-order models representing realizations of turbulence. J. Fluid Mech.
**1972**, 55, 545–563. [Google Scholar] [CrossRef] - Lorenz, E.N. Irregularity: A fundamental property of the atmosphere. Tellus
**1984**, 36A, 98–110. [Google Scholar] [CrossRef] - Lorenz, E.N. Predictability—A Problem Partly Solved. In Proceedings of the Seminar on Predictability, Reading, UK, 4–8 September 1995; ECMWF: Reading, UK, 1996; Volume 1. [Google Scholar]
- Lorenz, E.N. Designing Chaotic Models. J. Atmos. Sci.
**2005**, 62, 1574–1587. [Google Scholar] [CrossRef] - Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X. One Saddle Point and Two Types of Sensitivities Within the Lorenz 1963 and 1969 Models. Atmosphere
**2022**, 13(5), 753. [Google Scholar] [CrossRef] - Shen, B.-W. Aggregated Negative Feedback in a Generalized Lorenz Model. Int. J. Bifurc. Chaos
**2019**, 29, 1950037. [Google Scholar] [CrossRef] - Shen, B.-W. On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model. Geosciences
**2019**, 9, 281. [Google Scholar] [CrossRef] - Shen, B.-W.; Reyes, T.; Faghih-Naini, S. Coexistence of Chaotic and Non-Chaotic Orbits in a New Nine-Dimensional Lorenz Model. In Proceedings of the 11th Chaotic Modeling and Simulation International Conference, CHAOS 2018, Rome, Italy, 5–8 June 2018; Skiadas, C., Lubashevsky, I., Eds.; Springer Proceedings in Complexity. Springer: Cham, Switzerland, 2019. [Google Scholar] [CrossRef]
- Pedlosky, J. Finite-amplitude baroclinic waves with small dissipation. J. Atmos. Sci.
**1971**, 28, 587–597. [Google Scholar] [CrossRef] - Pedlosky, J. Limit cycles and unstable baroclinic waves. J. Atmos. Sci.
**1972**, 29, 53–63. [Google Scholar] [CrossRef] - Pedlosky, J. Geophysical Fluid Dynamics, 2nd ed.; Springer: New York, NY, USA, 1987; 710p. [Google Scholar]
- Pedlosky, J. The Effect of Beta on the Downstream Development of Unstable, Chaotic BaroclinicWaves. J. Phys. Oceanogr.
**2019**, 49, 2337–2343. [Google Scholar] [CrossRef] - Shen, B.-W. On periodic solutions in the non-dissipative Lorenz model: The role of the nonlinear feedback loop. Tellus A
**2018**, 70, 1471912. [Google Scholar] [CrossRef] - Faghih-Naini, S.; Shen, B.-W. Quasi-periodic orbits in the five-dimensional non-dissipative Lorenz model: The role of the extended nonlinear feedback loop. Int. J. Bifurc. Chaos
**2018**, 28, 1850072. [Google Scholar] [CrossRef] - Shen, B.-W. Homoclinic Orbits and Solitary Waves within the non-dissipative Lorenz Model and KdV Equation. Int. J. Bifurc. Chaos
**2020**, 30, 2050257. [Google Scholar] [CrossRef] - Shen, B.-W. Solitary Waves, Homoclinic Orbits, and Nonlinear Oscillations within the non-dissipative Lorenz Model, the inviscid Pedlosky Model, and the KdV Equation. In Proceedings of the 13th Chaos International Conference CHAOS 2020, Florence, Italy, 9–12 June 2020; Skiadas, C.H., Dimotikalis, Y., Eds.; Springer Proceedings in Complexity. Springer: Cham, Switzerland, 2021. [Google Scholar]
- Paxson, W.; Shen, B.-W. 2022: A KdV-SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves. Int. J. Bifurc. Chaos
**2022**, 32, 2250199. [Google Scholar] [CrossRef] - Yorke, J.; Yorke, E. Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model. J. Stat. Phys.
**1979**, 21, 263–277. [Google Scholar] [CrossRef] - Sprott, J.C.; Wang, X.; Chen, G. Coexistence of Point, periodic and Strange attractors. Int. J. Bifurc. Chaos
**2013**, 23, 1350093. [Google Scholar] [CrossRef] - Jordan, D.W.; Smith, P. Nonlinear Ordinary Differential Equations. In An Introduction for Scientists and Engineers, 4th ed.; Oxford University Press: Oxford, UK, 2007; p. 560. [Google Scholar]
- Thompson, J.M.T.; Stewart, H.B. Nonlinear Dynamics and Chaos, 2nd ed.; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2002; p. 437. [Google Scholar]
- Grebogi, C.; McDonald, S.W.; Ott, E.; Yorke, J.A. Final state sensitivity: An obstruction to predictability. Phys. Lett. A
**1983**, 99, 415–418. [Google Scholar] [CrossRef] - Leonov, G.A.; Kuznetsov, N.V. Hidden attractors in dynamical systems. from hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos
**2014**, 23, 1330002. [Google Scholar] [CrossRef] - Lorenz, E.N. Can chaos and intransitivity lead to interannual variability? Tellus
**1990**, 42A, 378–389. [Google Scholar] [CrossRef] - Lorenz, E.N. The predictability of hydrodynamic flow. Trans. N. Y. Acad. Sci.
**1963**, 25, 409–432. [Google Scholar] [CrossRef] - Lai, Q.; Chen, S. Coexisting attractors generated from a new 4D smooth chaotic system. Int. J. Contr. Autom. Syst.
**2016**, 14, 1124–1131. [Google Scholar] [CrossRef] - Jafari, S.; Rajagopal, K.; Hayat, T.; Alsaedi, A.; Pham, V.-T. Simplest Megastable Chaotic Oscillator. Int. J. Bifurc. Chaos
**2019**, 29, 1950187. [Google Scholar] [CrossRef] - Hilborn, R.C. Chaos and Nonlinear Dynamics. In An Introduction for Scientists and Engineers, 2nd ed.; Oxford University Press: Oxford, UK, 2000; p. 650. [Google Scholar]
- Shen, B.-W. Nonlinear feedback in a five-dimensional Lorenz model. J. Atmos. Sci.
**2014**, 71, 1701–1723. [Google Scholar] [CrossRef] - Shen, B.-W. Nonlinear feedback in a six-dimensional Lorenz Model: Impact of an additional heating term. Nonlin. Processes Geophys.
**2015**, 22, 749–764. [Google Scholar] [CrossRef] - Shen, B.-W. Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model. Nonlin. Processes Geophys.
**2016**, 23, 189–203. [Google Scholar] [CrossRef] - Shen, B.-W. On an extension of the nonlinear feedback loop in a nine-dimensional Lorenz model. Chaotic Modeling Simul.
**2017**, 2, 147–157. [Google Scholar] - Reyes, T.; Shen, B.-W. A Recurrence Analysis of Chaotic and Non-Chaotic Solutions within a Generalized Nine-Dimensional Lorenz Model. Chaos Solitons Fractals
**2019**, 125, 1–12. [Google Scholar] [CrossRef] - Cui, J.; Shen, B.-W. A Kernel Principal Component Analysis of Coexisting Attractors within a Generalized Lorenz Model. Chaos Solitons Fractals
**2021**, 146, 110865. [Google Scholar] [CrossRef] - Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X.; Cui, J.; Faghih-Naini, S.; Paxson, W.; Atlas, R. Three Kinds of Butterfly Effects within Lorenz Models. Encyclopedia
**2022**, 2, 1250–1259. [Google Scholar] [CrossRef] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom.
**1985**, 16, 285–317. [Google Scholar] [CrossRef] - Eckhardt, B.; Yao, D. Local Lyapunov exponents in chaotic systems. Phys. D Nonlinear Phenom.
**1993**, 65, 100–108. [Google Scholar] [CrossRef] - Nese, J.M. Quantifying local predictability in phase space. Phys. D Nonlinear Phenom.
**1989**, 35, 237–250. [Google Scholar] [CrossRef] - Slingo, J.; Palmer, T. Uncertainty in weather and climate prediction. Philos. Trans. R. Soc. A
**2011**, 369A, 4751–4767. [Google Scholar] [CrossRef] - Lewis, J. Roots of ensemble forecasting. Mon. Weather. Rev.
**2005**, 133, 1865–1885. [Google Scholar] [CrossRef] - Orszag, S.A. Analytical theories of turbulence. J. Fluid Mech.
**1970**, 41, 363–386. [Google Scholar] [CrossRef] - Orszag, S.A. Fluid Dynamics; Balian, R., Peuble, J.L., Eds.; Gordon and Breach: London, UK, 1977. [Google Scholar]
- Leith, C.E. Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci.
**1971**, 28, 145–161. [Google Scholar] [CrossRef] - Leith, C.E.; Kraichnan, R.H. Predictability of turbulent flows. J. Atmos. Sci.
**1972**, 29, 1041–1058. [Google Scholar] [CrossRef] - Lilly, D.K. Numerical simulation of two-dimensional turbulence. Phys. Fluids
**1969**, 12 (Suppl. 2), 240–249. [Google Scholar] [CrossRef] - Lilly, D.K. Numerical simulation studies of two-dimensional turbulence: II. Stability and predictability studies. Geophys. Fluid Dyn.
**1972**, 4, 1–28. [Google Scholar] [CrossRef] - Lilly, K.D. Lectures in Sub-Synoptic Scales of Motions and Two-Dimensional Turbulence. In Dynamic Meteorology; Morel, P., Ed.; Reidel: Boston, MA, USA, 1973; pp. 353–418. [Google Scholar]
- Vallis, G. Atmospheric and Oceanic Fluid Dynamics; Cambridge University Press: Cambridge, UK, 2006; p. 745. [Google Scholar]
- Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X. A Note on Lorenz’s and Lilly’s Empirical Formulas for Predictability Estimates. ResearchGate
**2022**, preprint. [Google Scholar] [CrossRef] - Ghil, M.; Read, P.; Smith, L. Geophysical flows as dynamical systems: The influence of Hide’s experiments. Astron. Geophys.
**2010**, 51, 4.28–4.35. [Google Scholar] [CrossRef] - Read, P. Application of Chaos to Meteorology and Climate. In The Nature of Chaos; Mullin, T., Ed.; Clarendo Press: Oxford, UK, 1993; pp. 222–260. [Google Scholar]
- Legras, B.; Ghil, M. Persistent anomalies, blocking, and variations in atmospheric predictability. J. Atmos. Sci.
**1985**, 42, 433–471. [Google Scholar] [CrossRef] - Patil, D.J.; Hunt, B.R.; Kalnay, E.; Yorke, J.A.; Ott, E. Local low-dimensionality of atmospheric dynamics. Phys. Rev. Lett.
**2001**, 86, 5878–5881. [Google Scholar] [CrossRef] - Oczkowski, M.; Szunyogh, I.; Patil, D.J. Mechanisms for the Development of Locally Low-Dimensional Atmospheric Dynamics. J. Atmos. Sci.
**2005**, 62, 1135–1156. [Google Scholar] [CrossRef] - Ott, E.; Hunt, B.R.; Szunyogh, I.; Corazza, M.; Kalnay, E.; Patil, D.J.; Yorke, J. Exploiting Local Low Dimensionality of the Atmospheric Dynamics for Efficient Ensemble Kalman Filtering. 2002. Available online: https://doi.org/10.48550/arXiv.physics/0203058 (accessed on 1 November 2022).
- Zeng, X.; Pielke, R.A., Sr.; Eykholt, R. Chaos theory and its applications to the atmosphere. Bull. Am. Meteorol. Soc.
**1993**, 74, 631–644. [Google Scholar] [CrossRef] - Ghil, M. A Century of Nonlinearity in the Geosciences. Earth Space Sci.
**2019**, 6, 1007–1042. [Google Scholar] [CrossRef] - Charney, J.G.; DeVore, J.G. Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci.
**1979**, 36, 1205–1216. [Google Scholar] [CrossRef] - Crommelin, D.T.; Opsteegh, J.D.; Verhulst, F. A mechanism for atmospheric regime behavior. J. Atmos. Sci.
**2004**, 61, 1406–1419. [Google Scholar] [CrossRef] - Ghil, M.; Robertson, A.W. “Waves” vs. “particles” in the atmosphere’s phase space: A pathway to long-range forecasting? Proc. Natl. Acad. Sci. USA
**2002**, 99 (Suppl. 1), 2493–2500. [Google Scholar] [CrossRef] - Renaud, A.; Nadeau, L.-P.; Venaille, A. Periodicity Disruption of a Model Quasibiennial Oscillation of Equatorial Winds. Phys. Rev. Lett.
**2019**, 122, 214504. [Google Scholar] [CrossRef] - Ramesh, K.; Murua, J.; Gopalarathnam, A. Limit-cycle oscillations in unsteady flows dominated by intermittent leading-edge vortex shedding. J. Fluids Struct.
**2015**, 55, 84–105. [Google Scholar] [CrossRef] - Goler, R.A.; Reeder, M.J. The generation of the morning glory. J. Atmos. Sci.
**2004**, 61, 1360–1376. [Google Scholar] [CrossRef] - Wu, Y.-L.; Shen, B.-W. An evaluation of the parallel ensemble empirical mode decomposition method in revealing the role of downscaling processes associated with African easterly waves in tropical cyclone genesis. J. Atmos. Ocean. Technol.
**2016**, 33, 1611–1628. [Google Scholar] [CrossRef] - Shen, B.-W.; Cheung, S.; Wu, Y.; Li, F.; Kao, D. Parallel Implementation of the Ensemble Empirical Mode Decomposition (PEEMD) and Its Application for Earth Science Data Analysis. Comput. Sci. Eng.
**2017**, 19, 49–57. [Google Scholar] [CrossRef] - Shilnikov, L.P. On a new type of bifurcation of multi-dimensional dynamical systems. Dokl. Akad. Nauk SSSR
**1969**, 10, 1368–1371. [Google Scholar] - Gonchenko, S.V.; Turaev, D.V.; Shilnikov, L.P. Dynamical phenomena in multidimensional systems with a structurally unstable homoclinic Poincar6 curve. Russ. Acad. Sci. Dokl. Mat.
**1993**, 47, 410–415. [Google Scholar] - Belhaq, M.; Houssni, M.; Freire, E.; Rodriguez-Luis, A.J. Asymptotics of Homoclinic Bifurcation in a Three-Dimensional System. Nonlinear Dyn.
**2000**, 21, 135–155. [Google Scholar] [CrossRef] - Shimizu, T.; Morioka, N. On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Phys. Lett. A
**1980**, 76, 201–204. [Google Scholar] [CrossRef] - Shil’nilov, A.L. On bifurcations of a Lorenz-like attractor in the Shimizu-Morioka system. Phys. D Nonlinear Phenom.
**1992**, 62, 332–346. [Google Scholar] - Shil’nikov, A.L.; Shil’nikov, L.P.; Turaev, D.V. Normal Forms and Lorenz Attractors. Int. J. Bifurc. Chaos
**1993**, 3, 1123–1139. [Google Scholar] [CrossRef] - Gonchenko, S.; Kazakov, A.; Turaev, D.; Shilnikov, A.L. Leonid Shilnikov and mathematical theory of dynamical chaos. Chaos
**2022**, 32, 010402. [Google Scholar] [CrossRef] [PubMed] - Simonnet, E.; Ghil, M.; Dijkstra, H. Homoclinic bifurcation in the quasi-geostrophic double-gyre circulation. J. Mar. Res.
**2005**, 63, 931–956. [Google Scholar] [CrossRef]

**Figure 1.**Skiing as used to reveal monostability (left and middle, Lorenz 1993 [3]) and kayaking as used to indicate multistability (right, courtesy of Shutterstock-Carol Mellema https://www.shutterstock.com/image-photo/kayaker-enjoys-whitewater-sinks-smoky-mountains-649533271 (accessed 1 November 2022)). A stagnant area is outlined with a white box.

**Figure 2.**Three types of solutions within the Lorenz 1963 model. Steady-state (

**a**,

**d**), chaotic (

**b**,

**e**), and limit cycle (

**c**,

**f**) solutions appear at small, moderate, and large normalized Rayleigh parameters (i.e., r = 20, 28, and 350), respectively. Control and parallel runs are shown in red and blue, respectively. SDIC is indicated by visible blue and red curves in panel (

**b**), where the first and second green horizonal lines indicate CDIC and SDIC, respectively. (

**a**–

**c**) depict the time evolution of Y. (

**d**–

**f**) show orbits within the X–Y space, appearing as a point attractor (

**a**,

**d**), a chaotic attractor (

**b**,

**e**), and a periodic attractor (

**c**,

**f**), respectively (after Shen et al., 2021 [10]). The other two parameters are kept as constants: σ = 10 and b = 8/3. The initial conditions of $\left(X,Y,Z\right)$ for the control and parallel runs are $\left(0,1,0\right)$ and $\left(0,1+\u03f5,0\right),$ respectively.

**Figure 3.**Three types of solutions within the X–Y–Z phase space obtained from the Lorenz 1963 model. Panels (

**a**–

**c**) display a steady-state solution, a chaotic solution, and a limit cycle with small, medium, and large heating parameters, respectively. While panels (

**a**,

**b**) show the solution for τ $\in $ [0, 30] panel, to reveal its isolated feature, panel (

**c**) displays the limit cycle solution for τ $\in $ [10, 30]. Values of parameters are the same as those in the control run in Figure 2.

**Figure 4.**Two kinds of attractor coexistence using the GLM with 9 modes. Each panel displays orbits from 128 runs with different ICs for τ $\in $ [0.625, 5]. Curves in different colors indicate orbits with different initial conditions. (

**a**) displays the coexistence of chaotic and steady-state solutions with r = 680. Stable critical points are shown with large blue dots. (

**b**) displays the coexistence of the limit cycle and steady-state solutions with r = 1600.

**Figure 5.**Two kinds of attractor coexistence revealed by three trajectories using a time varying heating parameter (i.e., Rayleigh parameter), r = 1200 + 520 sin (τ/5), within a GLM (Shen, 2019 [18]). The green, blue, and red lines represent the solutions of the control and two parallel runs. The parallel runs include an initial tiny perturbation, $\u03f5={10}^{-8}$ or $\u03f5=-{10}^{-8}$. The heating function is indicated by an orange line. From top to bottom, panels (

**a**,

**b**) display the three orbits and the heating parameters for τ $\in $ [0, 35π], respectively. Panel (

**c**) for τ $\in $ [19, 21] displays diverged trajectories, showing SDIC. The first kind of attractor coexistence (i.e., coexisting chaotic and steady-state solutions) is shown in panel (

**d**) for τ $\in $ [29, 31]. The green line, indeed, represents a steady state solution. The second kind of attractor coexistence (i.e., coexisting regular oscillations and steady-state solutions) is shown in panel (

**e**) for τ $\in $ [39, 41]. Panel (

**f**) displays a nearly steady-state solution (2Y/3) and the heating function for τ $\in $ [30, 110]. The three vertical lines in panel (

**b**) indicate the starting time for the analysis in Figure 6. (After Shen et al., 2021 [9]).

**Figure 6.**Panels (

**a**–

**c**) display the same trajectory during three different time intervals of τ $\in $ [Ts, Ts + π], with the starting time Ts equal to 10π, 20π, and 30π, respectively. The orange line in each panel represents the half value of the heating function.

**Figure 7.**Panels (

**a**,

**b**) display the same trajectory during three different time intervals of τ $\in $ [Ts, Ts +π], with the starting time Ts equal to 12π, 22π, and 32π, respectively. These three time intervals are referred to as Epoch-1, Epoch-2, and Epoch-3, respectively. In panels (

**c**,

**d**), to adjust the phase differences between two solutions curves, a time lag is added into Epoch-1.

Name | Definitions | Recommendations |
---|---|---|

1st kind of attractor coexistence | The coexistence of chaotic and steady-state solutions | [9,10,28] |

2nd kind of attractor coexistence | The coexistence of nonlinear oscillatory and steady-state solutions | [9,10] |

attractor | The smallest attracting point set that, itself, cannot be decomposed into two or more subsets with distinct basins of attraction. | [31] |

autonomous | A system of ODEs is autonomous if time does not explicitly appear within the equations. | [32] |

bifurcation | It occurs when the structure of a system’s solution significantly changes as a control parameter varies. | [32,33] |

butterfly effect | The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration. | [3] |

basin of attraction | As time advances, orbits initialized within a basin tend asymptotically to the attractor lying within the basin. The set of initial conditions leading to a given attractor. | [33] |

chaos | Bounded aperiodic orbits exhibit a sensitive dependence on ICs. | [3] |

final state sensitivity | Nearby orbits settle to one of multiple attractors for a finite but arbitrarily long time. | [34] |

hidden attractor | An attractor is called a hidden attractor if its basin of attraction does not intersect with small neighborhoods of equilibria. | [35] |

intransitivity | A specific type of solution lasts forever. | [36] |

intrinsic predictability | Predictability that is only dependent on flow itself. | [9,37] |

limit cycle | A nonlinear oscillatory solution; an isolated closed orbit | [32] |

monostability | The appearance of single-type solutions | [9,10,17] |

multistability | A system with multistability contains more than one bounded attractor that depends only on initial conditions. For example, the coexistence of two types of solutions. | [9,10,17,38,39] |

non-autonomous | Variable time ($\tau )$ appears on the right-hand side of the equations. | [32] |

phase space | Within a system of the first-order ODEs, a phase space or state space can be constructed using time-dependent variables as coordinates. | [40] |

practical predictability | Predictability that is limited by imperfect initial conditions and/or (mathematical) formulas. | [9,37] |

recurrence | Defined when a trajectory returns back to the neighborhood of a previously visited state. Recurrence may be viewed as a generalization of “periodicity” that braces quasi-periodicity with multiple frequencies and chaos. | [33] |

sensitive dependence | The property characterizing an orbit if most other orbits that pass close to it at some point do not remain close to it as time advances. | [3] |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shen, B.-W.; Pielke, R., Sr.; Zeng, X.; Cui, J.; Faghih-Naini, S.; Paxson, W.; Kesarkar, A.; Zeng, X.; Atlas, R.
The Dual Nature of Chaos and Order in the Atmosphere. *Atmosphere* **2022**, *13*, 1892.
https://doi.org/10.3390/atmos13111892

**AMA Style**

Shen B-W, Pielke R Sr., Zeng X, Cui J, Faghih-Naini S, Paxson W, Kesarkar A, Zeng X, Atlas R.
The Dual Nature of Chaos and Order in the Atmosphere. *Atmosphere*. 2022; 13(11):1892.
https://doi.org/10.3390/atmos13111892

**Chicago/Turabian Style**

Shen, Bo-Wen, Roger Pielke, Sr., Xubin Zeng, Jialin Cui, Sara Faghih-Naini, Wei Paxson, Amit Kesarkar, Xiping Zeng, and Robert Atlas.
2022. "The Dual Nature of Chaos and Order in the Atmosphere" *Atmosphere* 13, no. 11: 1892.
https://doi.org/10.3390/atmos13111892