Deterministic nonperiodic flow.

*(English)*Zbl 1417.37129Summary: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic.

The feasibility of very-long-range weather prediction is examined in the light of these results.

A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic.

The feasibility of very-long-range weather prediction is examined in the light of these results.

##### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |

86A10 | Meteorology and atmospheric physics |

PDF
BibTeX
XML
Cite

\textit{E. N. Lorenz}, J. Atmos. Sci. 20, No. 2, 130--141 (1963; Zbl 1417.37129)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Lorenz Z N. Deterministic non-periodic flow. J Atoms Sci, 1963, 20: 130-141 · Zbl 1417.37129 |

[2] | Lorenz E N. The Essence of Chaos. Washington: University of Washington Press, 1993 · Zbl 0835.58001 |

[3] | Sparrow C. The Lorenz Equations: Bifurcation, Chaos and Strange Attractors. Berlin-Heidelberg, New York: Springer-Verlag, 1976 |

[4] | Stwart I. The Lorenz attractor exists. Nature, 2002, 406: 948-949 |

[5] | Chen G R, Lv J H. Dynamics Analysis, Control and Synchronization of Lorenz System (in Chinese). Beijing: Science Press, 2003 |

[6] | Yang W L, Wang T N. Theory and Application of Nonlinear Dynamics (in Chinese). Beijing: National Defense Industry Press, 2007 |

[7] | Warwick T. A rigorous ODE solver and Smale’s 14th problem. Found Comput Math, 2002, 2: 53-117 · Zbl 1047.37012 |

[8] | Leonov G A, Bunin A L, Kokxh N. Attractor localization of the Lorenz system. ZAMM, 1987, 67: 649-656 · Zbl 0653.34040 |

[9] | Leonov G A. Bound for attractors and the existence of homoclinic orbits in the Lorenz system. J Appl Math Mech, 2001, 65: 19-32 |

[10] | Liao X X, Fu Y L, Xie S L. On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization. Sci China Ser F-Inf Sci, 2005, 48: 304-321 · Zbl 1187.37047 |

[11] | Li D M, Lu J A, Wa X Q, et al. Estimating the bounded for the Lorenz family of chaotic systems. Chaos Solut Fract, 2005, 23: 529-534 · Zbl 1061.93506 |

[12] | Yu P, Liao X X. New estimations for globally attractive and positive invariant set of the family of the Lorenz systems. Int J Bifur Chaos, 2006, 16: 3383-3390 · Zbl 1116.37026 |

[13] | Liao X X, Fu Y L, Xie S L, et al. Globally exponentially attractive sets of family of Lorenz systems. Sci China Ser F-Inf Sci, 2008, 51: 283-292 · Zbl 1148.37025 |

[14] | Li Y X. Research on Anticontrol of Hyperchaos for Continuous-time Systems (in Chinese). Ph.d. Dissertation, Gruangzhou: Guangdong University of Technology, 2005 |

[15] | Yang Q, Chen G. A chaotic system with one saddle and two stable node-foci. Int J Bifur Chaos, 2008, 18: 1393-1414 · Zbl 1147.34306 |

[16] | Liao X X. Talking on the theory, methods and application of Lyapunov stability (in Chinese). J Nanjing Univ Inf Sci Tech: Nat Sci Ed, 2009, 1: 1-15 |

[17] | Liu Y G. Global stabilization by output feedback for a class of nonlinear systems with uncertain control coefficients and unmeasured states dependent growth. Sci China Ser F-Inf Sci, 2008, 51: 1508-1520 · Zbl 1147.93341 |

[18] | Luo Q, Deng F Q, Mao X R, et al. Theory and application of stability for stochastic reaction diffusion systems. Sci China Ser F-Inf Sci, 2008, 51: 158-170 · Zbl 1148.35106 |

[19] | Chen Y Y, Luo Q. Global exponential stability in Lagrange sense for a class of neural networks (in Chinese). J Nanjing Univ Inf Sci Tech: Nat Sci Ed, 2009, 1: 50-58 · Zbl 1212.93238 |

[20] | Luo Y P, Xia W H, Liu G R, et al. \(W_{1,2}\)(Ω)-and \(X_{1,2}\)(Ω)-stability of reaction-diffusion cellular neural networks with delay. Sci China Ser F-Inf Sci, 2008, 51: 1980-1991 · Zbl 1291.35113 |

[21] | Liao X X. Theory and Application of Stability. Wuhan: Huazhong Normal University Press, 2001 |

[22] | Liao X X. Theory, |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.