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March  2020, 25(3): 1097-1108. doi: 10.3934/dcdsb.2019210

Homoclinic orbits and chaos in the generalized Lorenz system

Ting Yang

School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China

Received  October 2018 Revised  May 2019 Published  March 2020 Early access  September 2019

This paper investigates the homoclinic orbits and chaos in the generalized Lorenz system. Using center manifold theory and Lyapunov functions, we get non-existence conditions of homoclinic orbits associated with the origin. The existence conditions of the homoclinic orbits are obtained by Fishing Principle. Therefore, sufficient and necessary conditions of existence of homoclinic orbits associated with the origin are given. Furthermore, with the broken of the homoclinic orbits, we show that the chaos is in the sense generalized Shil'nikov homoclinic criterion.

Keywords: Generalized Lorenz system, homoclinic orbits, chaos, fishing principle, Lyapunov functions.
Mathematics Subject Classification: Primary: 34C37; Secondary: 34C28.
Citation: Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210
References:
[1]

A. Ashraf and A. Abdulnasser, On the design of chaos-based secure communication systems, Commun. Nonlin. Sci., 16 (2011), 3721-3737.  doi: 10.1016/j.cnsns.2010.12.032.  Google Scholar

[2]

J. Bao and Q. Yang, A new method to find homoclinic and heteroclinic orbits, Appl. Math. Comput., 217 (2011), 6526-6540.  doi: 10.1016/j.amc.2011.01.032.  Google Scholar

[3]

S. Čelikovský and G. Chen, On a generalized Lorenz canonical form of chaotic systems, Int. J. Bifurcat. Chaos, 12 (2002), 1789-1812.  doi: 10.1142/S0218127402005467.  Google Scholar

[4]

S. Čelikovský and G. Chen, Secure synchronization of a class of chaotic systems from a nonlinear observer approach, IEEE Trans. Automat. Contr., 50 (2005), 76-82.  doi: 10.1109/TAC.2004.841135.  Google Scholar

[5]

S. Čelikovský and G. Chen, On the generalized Lorenz canonical form, Chaos Solitons Fractals, 26 (2005), 1271-1276.  doi: 10.1016/j.chaos.2005.02.040.  Google Scholar

[6]

X. Chen, Lorenz equations part Ⅰ: Existence and nonexistence of homoclinic orbits, SIAM J. Math. Anal., 27 (1996), 1057-1069.  doi: 10.1137/S0036141094264414.  Google Scholar

[7]

L. O. Chua, M. Komuro and T. Matsumoto, The double scroll family, IEEE Trans. Circuits Syst., 33 (1986), 1072-1118.   Google Scholar

[8]

B. A. Coomes, H. Koçak and K. J. Palmer, A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics, J. Dyn. Differ. Equ., 28 (2016), 1081–1114. doi: 10.1007/s10884-015-9437-y.  Google Scholar

[9]

S. P. Hastings and W. C. Troy, A proof that the Lorenz equations have a homoclinic orbit, J. Differ. Equ., 113 (1994), 166-188.  doi: 10.1006/jdeq.1994.1119.  Google Scholar

[10] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third edition. Elsevier/Academic Press, Amsterdam, 2013.   Google Scholar
[11]

G. A. Leonov, Attractors, limit cycles and homoclinic orbits of low dimensional quadratic systems. analytical methods, Can. Appl. Math. Q., 17 (2009), 121-159.   Google Scholar

[12]

G. A. Leonov, General existence conditions of homoclinic trajectories in dissipative systems, Lorenz, Shimizu-Morioka, Lu and Chen systems, Phys. Lett. A, 376 (2012), 3045-3050.  doi: 10.1016/j.physleta.2012.07.003.  Google Scholar

[13]

G. A. Leonov, Fishing principle for homoclinic and heteroclinic trajectories, Nonlinear Dyn., 78 (2014), 2751-2758.  doi: 10.1007/s11071-014-1622-8.  Google Scholar

[14]

G. A. Leonov, Existence Conditions of Homoclinic Trajectories in Tigan System, Int. J. Bifurcat. Chaos, 25 (2015), 1550175. doi: 10.1142/S0218127415501758.  Google Scholar

[15]

E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141.   Google Scholar

[16]

P. Namayanja, Chaotic dynamics in a transport equation on a network, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3415-3426.  doi: 10.3934/dcdsb.2018283.  Google Scholar

[17] J. Palis, J. P. Júnior and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics, Cambridge University Press, Cambridge, 1995.   Google Scholar
[18]

K. Rajagopal, A. Akgul, S. Jafari and B. Aricioglu, A chaotic memcapacitor oscillator with two unstable equilibriums and its fractional form with engineering applications., Nonlinear Dynam., 91 (2018), 957-974.   Google Scholar

[19]

L. P. Shil'nikov, A. L. Shil'nikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific, Singapore, 2001. doi: 10.1142/4221.  Google Scholar

[20]

C. P. Silva, Shil'nikov's theorem-a tutorial, IEEE Trans. Circuits Syst., 40 (1993), 675-682.  doi: 10.1109/81.246142.  Google Scholar

[21] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, CRC Press, 2018.   Google Scholar
[22]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer-Verlag, New York, 2003.  Google Scholar

show all references

References:
[1]

A. Ashraf and A. Abdulnasser, On the design of chaos-based secure communication systems, Commun. Nonlin. Sci., 16 (2011), 3721-3737.  doi: 10.1016/j.cnsns.2010.12.032.  Google Scholar

[2]

J. Bao and Q. Yang, A new method to find homoclinic and heteroclinic orbits, Appl. Math. Comput., 217 (2011), 6526-6540.  doi: 10.1016/j.amc.2011.01.032.  Google Scholar

[3]

S. Čelikovský and G. Chen, On a generalized Lorenz canonical form of chaotic systems, Int. J. Bifurcat. Chaos, 12 (2002), 1789-1812.  doi: 10.1142/S0218127402005467.  Google Scholar

[4]

S. Čelikovský and G. Chen, Secure synchronization of a class of chaotic systems from a nonlinear observer approach, IEEE Trans. Automat. Contr., 50 (2005), 76-82.  doi: 10.1109/TAC.2004.841135.  Google Scholar

[5]

S. Čelikovský and G. Chen, On the generalized Lorenz canonical form, Chaos Solitons Fractals, 26 (2005), 1271-1276.  doi: 10.1016/j.chaos.2005.02.040.  Google Scholar

[6]

X. Chen, Lorenz equations part Ⅰ: Existence and nonexistence of homoclinic orbits, SIAM J. Math. Anal., 27 (1996), 1057-1069.  doi: 10.1137/S0036141094264414.  Google Scholar

[7]

L. O. Chua, M. Komuro and T. Matsumoto, The double scroll family, IEEE Trans. Circuits Syst., 33 (1986), 1072-1118.   Google Scholar

[8]

B. A. Coomes, H. Koçak and K. J. Palmer, A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics, J. Dyn. Differ. Equ., 28 (2016), 1081–1114. doi: 10.1007/s10884-015-9437-y.  Google Scholar

[9]

S. P. Hastings and W. C. Troy, A proof that the Lorenz equations have a homoclinic orbit, J. Differ. Equ., 113 (1994), 166-188.  doi: 10.1006/jdeq.1994.1119.  Google Scholar

[10] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third edition. Elsevier/Academic Press, Amsterdam, 2013.   Google Scholar
[11]

G. A. Leonov, Attractors, limit cycles and homoclinic orbits of low dimensional quadratic systems. analytical methods, Can. Appl. Math. Q., 17 (2009), 121-159.   Google Scholar

[12]

G. A. Leonov, General existence conditions of homoclinic trajectories in dissipative systems, Lorenz, Shimizu-Morioka, Lu and Chen systems, Phys. Lett. A, 376 (2012), 3045-3050.  doi: 10.1016/j.physleta.2012.07.003.  Google Scholar

[13]

G. A. Leonov, Fishing principle for homoclinic and heteroclinic trajectories, Nonlinear Dyn., 78 (2014), 2751-2758.  doi: 10.1007/s11071-014-1622-8.  Google Scholar

[14]

G. A. Leonov, Existence Conditions of Homoclinic Trajectories in Tigan System, Int. J. Bifurcat. Chaos, 25 (2015), 1550175. doi: 10.1142/S0218127415501758.  Google Scholar

[15]

E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141.   Google Scholar

[16]

P. Namayanja, Chaotic dynamics in a transport equation on a network, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3415-3426.  doi: 10.3934/dcdsb.2018283.  Google Scholar

[17] J. Palis, J. P. Júnior and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics, Cambridge University Press, Cambridge, 1995.   Google Scholar
[18]

K. Rajagopal, A. Akgul, S. Jafari and B. Aricioglu, A chaotic memcapacitor oscillator with two unstable equilibriums and its fractional form with engineering applications., Nonlinear Dynam., 91 (2018), 957-974.   Google Scholar

[19]

L. P. Shil'nikov, A. L. Shil'nikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific, Singapore, 2001. doi: 10.1142/4221.  Google Scholar

[20]

C. P. Silva, Shil'nikov's theorem-a tutorial, IEEE Trans. Circuits Syst., 40 (1993), 675-682.  doi: 10.1109/81.246142.  Google Scholar

[21] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, CRC Press, 2018.   Google Scholar
[22]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer-Verlag, New York, 2003.  Google Scholar

Figure 1.  Two symmetrical homoclinic orbits of system (1)
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Figure 2.  Chaotic attractor $ \mathcal{A} $ of system (1) with $ s_1 = 5 $, $ s_2 = 4 $, $ d = 1.5 $, $ q = 2 $, $ R = 3.08435 $
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