
-
Previous Article
The fast signal diffusion limit in nonlinear chemotaxis systems
- DCDS-B Home
- This Issue
-
Next Article
Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain
Homoclinic orbits and chaos in the generalized Lorenz system
School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China |
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.
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. Chua, M. 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. Hirsch, S. 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. 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. |
[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.
![]() |
[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. |
[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. Chua, M. 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. Hirsch, S. 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. 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. |
[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.
![]() |
[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. |
[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. |


[1] |
Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020378 |
[2] |
Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020 doi: 10.3934/jcd.2021006 |
[3] |
Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 |
[4] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[5] |
Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144 |
[6] |
Junchao Zhou, Yunge Xu, Lisha Wang, Nian Li. Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020121 |
[7] |
Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021 doi: 10.3934/jgm.2021002 |
[8] |
Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004 |
[9] |
Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020409 |
[10] |
Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 |
[11] |
Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020381 |
[12] |
Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020048 |
[13] |
Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021010 |
[14] |
Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069 |
[15] |
Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314 |
[16] |
Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 |
[17] |
Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020346 |
[18] |
Jie Shen, Nan Zheng. Efficient and accurate sav schemes for the generalized Zakharov systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 645-666. doi: 10.3934/dcdsb.2020262 |
[19] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[20] |
Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020169 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]