Skip to main content
SpringerLink
Log in

A Family of Periodic Motions to Chaos with Infinite Homoclinic Orbits in the Lorenz System

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, the bifurcation dynamics of a family of \((n_{1},1,n_{2})\)-periodic motions to chaos with infinite homoclinic orbits \((\min{\{n_{1},n_{2}\}}=1\), \(\max{\{n_{1},n_{2}}\}=1,2,{\ldots},)\) in the Lorenz system is studied through the discrete mapping method. The bifurcation trees of \((n_{1},1,n_{2})\)-periodic motion to chaos are presented through discrete nodes and harmonic amplitudes. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The bifurcation scenarios of \((n_{1},1,n_{2})\)-period-1 motions to chaos are similar each other. The critical values for existence of \((n_{1},1,n_{2})\)-related periodic motions are determined for saddle-node and period-doubling bifurcations. The homoclinic orbits are associated with unstable periodic motions on the bifurcation trees of the \((n_{1},1,n_{2})\)-periodic motions to chaos. The homoclinic obits and periodic motions are illustrated from the bifurcation trees of the \((n_{1},1,n_{2})\)-periodic motions to chaos. The numerical and analytical trajectories of unstable periodic motions were presented for comparison. If the numerical simulations did not have any computational errors, the numerical and analytical solutions of unstable periodic motions in the Lorenz system should be identical. Thus, one observed so-called strange attractors in the Lorenz system through numerical simulations, which are not real strange attractors. This paper is specially dedicated to the good friend and colleague in memory of Gennady A. Leonov for his contributions on nonlinear dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 9
Fig. 9
Fig. 9
Fig. 10
Fig. 10
Fig. 10
Fig. 10
Fig. 11
Fig. 11
Fig. 11
Fig. 11
Fig. 12
Fig. 12
Fig. 13
Fig. 13
Fig. 14
Fig. 14
Fig. 15
Fig. 16
Fig. 16
Fig. 16
Fig. 17
Fig. 17
Fig. 18
Fig. 18
Fig. 19
Fig. 19
Fig. 20
Fig. 20
Fig. 21
Fig. 21
Fig. 22
Fig. 22
Fig. 23
Fig. 23
Fig. 24
Fig. 25
Fig. 26

Similar content being viewed by others

REFERENCES

  1. S. Guo and A. C. J. Luo, ‘‘On infinite homoclinic orbits induced by unstable periodic orbits in the Lorenz system,’’ Chaos 31, 043106 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Guo and A. C. J. Luo, ‘‘Bifurcation trees of (1:2)-asymmetric periodic motions with corresponding infinite homoclinic orbits in the Lorenz system,’’ J. Vibrat. Test. Syst. Dyn. 5, 373–406 (2021).

    Article  Google Scholar 

  3. E. N. Lorenz, ‘‘Deterministic nonperiodic flow,’’ J. Atmos. Sci. 20, 130–141 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  4. J. H. Curry, ‘‘A generalized Lorenz system,’’ Commun. Math. Phys. 60, 193–204 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Tucker, ‘‘The Lorenz attractor exists,’’ C.R. Acad. Sci., Ser. I: Math. 328, 1197–1202 (1999).

    MathSciNet  MATH  Google Scholar 

  6. L. P. Shilnikov, ‘‘On a Poincaré–Birkhoff problem,’’ Mat. Sb. 116, 378–397 (1967).

    Google Scholar 

  7. S. P. Hastings and W. C. Troy, ‘‘A proof that the Lorenz equations have a homoclinic orbit,’’ J. Differ. Equat. 113, 166–188 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  8. N. K. Gavrilov and L. P. Shilnikov, ‘‘On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. II,’’ Mat. Sb. 90, 139–156 (1973).

    Article  MathSciNet  Google Scholar 

  9. V. S. Afraimovich, V. V. Bykov, and L. P. Shilnikov, ‘‘Origin and structure of the Lorenz attractor,’’ Dokl. Akad. Nauk SSSR 234, 336–339 (1977).

    MathSciNet  Google Scholar 

  10. D. Rand, ‘‘The topological classification of Lorenz attractors,’’ Math. Proc. Cambridge Phil. Soc. 83, 451–460 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Guckenheimer and R. F. Williams, ‘‘Structural stability of Lorenz attractors,’’ Publ. Math. Inst. Hautes Etudes Sci. 50, 59–72 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  12. R. F. Williams, ‘‘The structure of Lorenz attractors,’’ Publ. Math. Inst. Hautes Etudes Sci. 50, 73–99 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  13. K. A. Robbins, ‘‘Periodic solutions and bifurcation structure at high R in the Lorenz model,’’ SIAM J. Appl. Math. 36, 457–472 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  14. D. V. Lyubimov and M. A. Zaks, ‘‘Two mechanisms of the transition to chaos in finite-dimensional models of convection,’’ Phys. D (Amsterdam, Neth.) 9, 52–64 (1983).

  15. J. Frøyland and K. H. Alfsen, ‘‘Lyapunov-exponent spectra for the Lorenz model,’’ Phys. Rev. A 29, 2928–2931 (1984).

    Article  Google Scholar 

  16. T. Shimizu and N. Morioka, ‘‘Chaos and limit cycles in the Lorenz model,’’ Phys. Lett. A 66, 182–184 (1978).

    Article  MathSciNet  Google Scholar 

  17. J. L. Kaplan and J. A. Yorke, ‘‘Preturbulence: A regime observed in a fluid flow model of Lorenz,’’ Commun. Math. Phys. 67 (2), 93–108 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  18. E. A. Jackson, ‘‘The Lorenz system: I. The global structure of its stable manifolds,’’ Phys. Scr. 32, 469–475 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  19. E. A. Jackson, ‘‘The Lorenz system: II. The homoclinic convolution of the stable manifolds,’’ Phys. Scr. 32, 476–481 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  20. J. A. Yorke and E. D. Yorke, ‘‘Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model,’’ J. Stat. Phys. 21, 263–277 (1979).

    Article  MathSciNet  Google Scholar 

  21. C. Sparrow, ‘‘An introduction to the Lorenz equations,’’ IEEE Trans. Circuits Syst. 30, 533–542 (1983).

    Article  MathSciNet  Google Scholar 

  22. A. F. Vakakis and M. F. A. Azeez, ‘‘Analytic approximation of the homoclinic orbits of the Lorenz system at \(\sigma=10\), \(b=8/3\) and \(\rho=13.926\),’’ Nonlin. Dyn. 15, 245–257 (1998).

    MathSciNet  MATH  Google Scholar 

  23. G. A. Leonov, ‘‘Estimation of loop-bifurcation parameters for a saddle-point separatrix of a Lorenz system,’’ Differ. Equat. 24, 634–638 (1988).

    MATH  Google Scholar 

  24. G. A. Leonov, ‘‘On homoclinic bifurcation in the Lorenz system,’’ Vestn. SPb. Univ., Math. 32 (1), 13–15 (1999).

    Google Scholar 

  25. G. A. Leonov, ‘‘Bounds for attractors and the existence of homoclinic orbits in the Lorenz system,’’ J. Appl. Math. Mech. 65, 19–32 (2001).

    Article  MathSciNet  Google Scholar 

  26. G. A. Leonov, ‘‘The Tricomi problem on the existence of homoclinic orbits in dissipative systems,’’ J. Appl. Math. Mech. 77, 296–304 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Saha, D. C. Saha, A. Ray, and A. R. Chowdhury, ‘‘On some properties of memristive Lorenz equation–theory and experiment,’’ J. Appl. Nonlin. Dyn. 7, 413–423 (2018).

    Article  MathSciNet  Google Scholar 

  28. X. Liu and L. Hong, ‘‘The adaptive synchronization of the stochastic fractional-order complex Lorenz system,’’ J. Appl. Nonlin. Dyn. 4, 267–279 (2015).

    Article  MATH  Google Scholar 

  29. S. S. Hassan, ‘‘Computational complex dynamics of the discrete Lorenz system,’’ J. Appl. Nonlin. Dyn. 8, 345–366 (2016).

    Article  MATH  Google Scholar 

  30. A. C. J. Luo, Discretization and Implicit Mapping Dynamics (Springer, Berlin, 2015).

    Book  MATH  Google Scholar 

  31. A. C. J. Luo, ‘‘Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems,’’ Int. J. Bifurc. Chaos 25, 1550044 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  32. Y. Guo and A. C. J. Luo, ‘‘On complex periodic motions and bifurcations in a periodically forced, damped, hardening Duffing oscillator,’’ Chaos, Solitons Fractals 81, 378–399 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  33. A. C. J. Luo and S. Xing, ‘‘Analytical predictions of period-1 motions to chaos in a periodically driven quadratic nonlinear oscillator with a time-delay,’’ Math. Model. Nat. Phenom. 11 (2), 75–88 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  34. Y. Xu, Z. Chen, and A. C. J. Luo, ‘‘An independent period-3 motion to chaos in a nonlinear flexible rotor system,’’ Int. J. Dyn. Control 8, 337–351 (2020).

    Article  Google Scholar 

  35. C. Guo and A. C. J. Luo, ‘‘Period-3 motions to chaos in an inverted pendulum with a periodic base movement,’’ Int. J. Dyn. Control 10, 663–680 (2021).

    MATH  Google Scholar 

  36. A. N. Pchelintsev, ‘‘Numerical and physical modeling of the dynamics of the Lorenz system,’’ Numer. Anal. Appl. 7, 159–167 (2014).

    Article  Google Scholar 

  37. A. N. Pchelintsev, ‘‘An accurate numerical method and algorithm for constructing solutions of chaotic systems,’’ J. Appl. Nonlin. Dyn. 9, 207–211 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  38. A. N. Pchelintsev, ‘‘A numerical-analytical method for constructing periodic solutions of the Lorenz system,’’ Differ. Equat. Control Proces. 2020 (4), 59–76 (2020).

    MathSciNet  MATH  Google Scholar 

  39. A. C. J. Luo, Toward Analytical Chaos in Nonlinear Systems (Wiley, New York, 2013).

    Google Scholar 

  40. A. C. J. Luo, Continuous Dynamical Systems (HEP/L&H Scientific, Beijing/Glen Carbon, 2012).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, IL 62026-1805, Edwardsville, USA

    Siyu Guo & Albert C. J. Luo

Authors
  1. Siyu Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Albert C. J. Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Albert C. J. Luo.

Additional information

(Submitted by S. Yu. Pilyugin)

Appendix

Appendix

From Eqs. (4) and (5), the partial derivative matrices for the mapping \(P_{k}(k=1,2,\cdots,mN)\) are

$$\frac{\partial{\mathbf{g}}_{k}}{\partial{\mathbf{x}}_{k-1}}=\left[\begin{matrix}{\frac{\partial g_{1,k}}{\partial x_{1,k-1}}}&{\frac{\partial g_{1,k}}{\partial x_{2,k-1}}}&{\frac{\partial g_{1,k}}{\partial x_{3,k-1}}}\\ {\frac{\partial g_{2,k}}{\partial x_{1,k-1}}}&{\frac{\partial g_{2,k}}{\partial x_{2,k-1}}}&{\frac{\partial g_{2,k}}{\partial x_{3,k-1}}}\\ {\frac{\partial g_{3,k}}{\partial x_{1,k-1}}}&{\frac{\partial g_{3,k}}{\partial x_{2,k-1}}}&{\frac{\partial g_{3,k}}{\partial x_{3,k-1}}}\end{matrix}\right],\quad\frac{\partial{\mathbf{g}}_{k}}{\partial{\mathbf{x}}_{k}}=\left[\begin{matrix}{\frac{\partial g_{1,k}}{\partial x_{1,k}}}&{\frac{\partial g_{1,k}}{\partial x_{2,k}}}&{\frac{\partial g_{1,k}}{\partial x_{3,k}}}\\ {\frac{\partial g_{2,k}}{\partial x_{1,k}}}&{\frac{\partial g_{2,k}}{\partial x_{2,k}}}&{\frac{\partial g_{2,k}}{\partial x_{3,k}}}\\ {\frac{\partial g_{3,k}}{\partial x_{1,k}}}&{\frac{\partial g_{3,k}}{\partial x_{2,k}}}&{\frac{\partial g_{3,k}}{\partial x_{3,k}}}\end{matrix}\right].$$
(A1)

Substitution of Eq. (A.1) into Eq. (10) yields the corresponding Jacobian matrix, i.e.,

$$DP_{k}=-\left[\begin{matrix}{\frac{\partial g_{1,k}}{\partial x_{1,k}}}&{\frac{\partial g_{1,k}}{\partial x_{2,k}}}&{\frac{\partial g_{1,k}}{\partial x_{3,k}}}\\ {\frac{\partial g_{2,k}}{\partial x_{1,k}}}&{\frac{\partial g_{2,k}}{\partial x_{2,k}}}&{\frac{\partial g_{2,k}}{\partial x_{3,k}}}\\ {\frac{\partial g_{3,k}}{\partial x_{1,k}}}&{\frac{\partial g_{3,k}}{\partial x_{2,k}}}&{\frac{\partial g_{3,k}}{\partial x_{3,k}}}\end{matrix}\right]^{-1}\left[\begin{matrix}{\frac{\partial g_{1,k}}{\partial x_{1,k-1}}}&{\frac{\partial g_{1,k}}{\partial x_{2,k-1}}}&{\frac{\partial g_{1,k}}{\partial x_{3,k-1}}}\\ {\frac{\partial g_{2,k}}{\partial x_{1,k-1}}}&{\frac{\partial g_{2,k}}{\partial x_{2,k-1}}}&{\frac{\partial g_{2,k}}{\partial x_{3,k-1}}}\\ {\frac{\partial g_{3,k}}{\partial x_{1,k-1}}}&{\frac{\partial g_{3,k}}{\partial x_{2,k-1}}}&{\frac{\partial g_{3,k}}{\partial x_{3,k-1}}}\end{matrix}\right],$$

where

$$\frac{\partial g_{1,k}}{\partial x_{1,k-1}}=-1+\textstyle{1\over 2}h\sigma,\quad\dfrac{\partial g_{1,k}}{\partial x_{2,k-1}}=-\textstyle{1\over 2}h\sigma,\quad\dfrac{\partial g_{1,k}}{\partial x_{3,k-1}}=0,$$
$$\dfrac{\partial g_{2,k}}{\partial x_{1,k-1}}=-\textstyle{1\over 2}h(\rho-\bar{x}_{3,k}),\quad\dfrac{\partial g_{2,k}}{\partial x_{2,k-1}}=-1+\textstyle{1\over 2}h,\quad\dfrac{\partial g_{2,k}}{\partial x_{3,k-1}}=\textstyle{1\over 2}h\bar{x}_{1,k},$$
$$\dfrac{\partial g_{3,k}}{\partial x_{1,k-1}}=-\textstyle{1\over 2}h\bar{x}_{2,k},\quad\dfrac{\partial g_{3,k}}{\partial x_{2,k-1}}=-\textstyle{1\over 2}h\bar{x}_{1,k},\quad\dfrac{\partial g_{3,k}}{\partial x_{3,k-1}}=-1+\textstyle{1\over 2}h\beta;$$

and

$$\dfrac{\partial g_{1,k}}{\partial x_{1,k}}=1+\textstyle{1\over 2}h\sigma,\quad\dfrac{\partial g_{1,k}}{\partial x_{2,k}}=-\textstyle{1\over 2}h\sigma,\quad\dfrac{\partial g_{1,k}}{\partial x_{3,k}}=0,$$
$$\dfrac{\partial g_{2,k}}{\partial x_{1,k}}=-\textstyle{1\over 2}h(\rho-\bar{x}_{3,k}),\quad\dfrac{\partial g_{2,k}}{\partial x_{2,k}}=1+\textstyle{1\over 2}h,\quad\dfrac{\partial g_{2,k}}{\partial x_{3,k}}=\textstyle{1\over 2}h\bar{x}_{1,k},$$
$$\dfrac{\partial g_{3,k}}{\partial x_{1,k}}=-\textstyle{1\over 2}h\bar{x}_{2,k},\quad\dfrac{\partial g_{3,k}}{\partial x_{2,k}}=-\textstyle{1\over 2}h\bar{x}_{1,k},\quad\dfrac{\partial g_{3,k}}{\partial x_{3,k}}=1+\textstyle{1\over 2}h\beta.$$

The expressions of mid points \(\bar{x}_{1,k}\), \(\bar{x}_{2,k}\) and \(\bar{x}_{3,k}\) are given in Eq. (2).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, S., Luo, A.C. A Family of Periodic Motions to Chaos with Infinite Homoclinic Orbits in the Lorenz System. Lobachevskii J Math 42, 3382–3437 (2021). https://doi.org/10.1134/S1995080222020093

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080222020093

Keywords:

Search

Navigation