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

Homoclinic orbits and chaos in the generalized Lorenz system

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.

Citation: Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete and 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

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

[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.

[13]

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

[14]

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

[15]

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

[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.

[17] J. PalisJ. 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. 
[18]

K. RajagopalA. AkgulS. 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. 

[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.

[20]

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

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

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

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

[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.

[13]

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

[14]

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

[15]

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

[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.

[17] J. PalisJ. 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. 
[18]

K. RajagopalA. AkgulS. 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. 

[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.

[20]

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

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

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

Figure 1.  Two symmetrical homoclinic orbits of system (1)
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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