June  2021, 26(6): 3357-3380. doi: 10.3934/dcdsb.2020235

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

1. 

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization, and Big Data Processing, Yulin Normal University, Yulin 537000, China

2. 

College of Science, Guangxi University for Nationalities, Guangxi, 530006, China

3. 

School of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan, Hubei 430074, China

4. 

Zhejiang Institute, China University of Geosciences, Hangzhou, Zhejiang 311305, China

* Corresponding author: weizhouchao@163.com

Received  January 2020 Revised  May 2020 Published  June 2021 Early access  August 2020

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant No. 11961074), Natural Science Foundation of Guangxi Province (Grant Nos. 2018GXNSFDA281028, 2017GXNSFAA198234), the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No. [2018] 35), and the Science Technology Program of Yulin Normal University (Grant No. 2017YJKY28). The second author is supported by the Postgraduate Innovation Program of Guangxi University for Nationalities (Grant No. GXUN-CHXZS2018042). The third author is supported by National Natural Science Foundation of China (Grant No. 11772306), Zhejiang Provincial Natural Science Foundation of China under Grant (No.LY20A020001), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (CUGGC05)

The present work is devoted to giving new insights into a chaotic system with two stable node-foci, which is named Yang-Chen system. Firstly, based on the global view of the influence of equilibrium point on the complexity of the system, the dynamic behavior of the system at infinity is analyzed. Secondly, the Jacobi stability of the trajectories for the system is discussed from the viewpoint of Kosambi-Cartan-Chern theory (KCC-theory). The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The obtained results show that in the sense of Jacobi stability, all equilibrium points of the system, including those of the two linear stable node-foci, are Jacobi unstable. These studies show that one might witness chaotic behavior of the system trajectories before they enter in a neighborhood of equilibrium point or periodic orbit. There exists a sort of stability artifact that cannot be found without using the powerful method of Jacobi stability analysis.

Citation: Yongjian Liu, Qiujian Huang, Zhouchao Wei. Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3357-3380. doi: 10.3934/dcdsb.2020235
References:
[1]

H. Abolghasem, Jacobi stability of circular orbits in a central force, Journal of Dynamical Systems and Geometric Theories, 10 (2012), 197-214.  doi: 10.1080/1726037X.2012.10698621.

[2]

H. Abolghasem, Liapunov stability versus Jacobi stability, Journal of Dynamical Systems and Geometric Theories, 10 (2012), 13-32.  doi: 10.1080/1726037X.2012.10698604.

[3]

H. Abolghasem, Jacobi stability of Hamiltonian system, International Journal of Pure and Applied Mathematics, 87 (2013), 181-194.  doi: 10.12732/ijpam.v87i1.11.

[4]

P. L. AntonelliS. F. Rutz and V. S. Sabau, A transient-state analysis of Tyson's model for the cell division cycle by means of KCC-theory, Open Systems and Information Dynamics, 9 (2002), 223-238.  doi: 10.1023/A:1019752327311.

[5]

P. L. Antonelli and I. Bucataru, Volterra-Hamilton production models with discounting: General theory and worked examples, Nonlinear Anal. RWA, 2 (2001), 337-356.  doi: 10.1016/S0362-546X(00)00101-2.

[6]

P. L. AntonelliS. F. Rutz and C. E. Hirakawa, The mathematical theory of endosymbiosis I, Nonlinear Anal. RWA, 12 (2011), 3238-3251.  doi: 10.1016/j.nonrwa.2011.05.023.

[7]

B. C. BaoH. BaoN. WangM. Chen and Q. Xu, Hidden extreme multistability in memristive hyperchaotic system, Chaos Solitons & Fractals, 94 (2017), 102-111.  doi: 10.1016/j.chaos.2016.11.016.

[8]

C. G. BoehmerT. Harko and S. V. Sabau, Jacobi stability analysis of dynamical systems-applications in gravitation and cosmology, Advances in Theoretical and Mathematical Physics, 16 (2012), 1145-1196.  doi: 10.4310/ATMP.2012.v16.n4.a2.

[9]

C. G. Bohmer and T. Harko, Nonlinear stability analysis of the Emden-Fowler equation, Journal of Nonlinear Mathematical Physics, 17 (2010), 503-516.  doi: 10.1142/S1402925110001100.

[10]

E. Cartan and D. D. Kosambi, Observations sur le mémoire précédent, Mathematische Zeitschrift, 37 (1933), 619-622.  doi: 10.1007/BF01474603.

[11]

G. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurc. Chaos, 9 (1999), 1465-1466.  doi: 10.1142/S0218127499001024.

[12]

Y. Chen and Z. B. Yin, The Jacobi stability of a Lorenz-type multistable hyperchaotic system with a curve of equilibria, Int. J. Bifurc. Chaos, 29 (2019), 1950062, 10 pp. doi: 10.1142/S0218127419500627.

[13]

S.-S. Chern, Sur la geometrie d'un systeme d'equations differentielles du second ordre, Bulletin des Sciences Mathematiques, 63 (1939), 206-212. 

[14]

A. Cima and J. Llibre, Bounded polynomial vector fields, Trans. Am. Math. Soc., 318 (1990), 557-579.  doi: 10.1090/S0002-9947-1990-0998352-5.

[15]

B. Danila, T. Harko, M. K. Mak, P. Pantaragphong and S. V. Sabau, Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background, Advances in High Energy Physics, (2016), Article ID 7521464 26 pp.

[16]

C. Feng, Q. Huang and Y. Liu, Jacobi analysis for an unusual 3D autonomous system, International Journal of Geometric Methods in Modern Physics, 17 (2020), 2050062, 20 pp. doi: 10.1142/S0219887820500620.

[17]

M. K. Gupta and C. K. Yadav, KCC theory and its application in a tumor growth model, Mathematical Methods in the Applied Sciences., 40 (2017), 7470-7487.  doi: 10.1002/mma.4542.

[18]

M. K. Gupta and C. K. Yadav, Jacobi stability analysis of Rössler system, Int. J. Bifurc. Chaos, 27 (2017), 63-76.  doi: 10.1142/S0218127417500560.

[19]

M. K. Gupta and C. K. Yadav, Jacobi stability analysis of modified Chua circuit system, International Journal of Geometric Methods in Modern Physics, 14 (2017), 121-142.  doi: 10.1142/S021988781750089X.

[20]

M. K. Gupta and C. K. Yadav, Jacobi stability analysis of Rikitake system, International Journal of Geometric Methods in Modern Physics, 13 (2016), 1650098. doi: 10.1142/S0219887816500985.

[21]

M. K. Gupta and C. K. Yadav, Rabinovich-Fabrikant system in view point of KCC theory in Finsler geometry, Journal of Interdisciplinary Mathematics, 22 (2019), 219-241.  doi: 10.1080/09720502.2019.1614249.

[22]

T. Harko and V. S. Sabau, Jacobi stability of the vacuum in the static spherically symmetric brane world models, Physical Review D, 77 (2008), 104009. doi: 10.1103/PhysRevD.77.104009.

[23]

T. Harko, C. Y. Ho, C. S. Leung and S. Yip, Jacobi stability analysis of the Lorenz system, International Journal of Geometric Methods in Modern Physics, 12 (2015), 1550081. doi: 10.1142/S0219887815500814.

[24]

Q. Huang, A. Liu and Y. Liu, Jacobi stability analysis of the Chen system, Int. J. Bifurc. Chaos, 29 (2019), 1950139. doi: 10.1142/S0218127419501396.

[25]

D. D. Kosambi, Parallelism and path-space, Mathematische Zeitschrift, 37 (1933), 608-618.  doi: 10.1007/BF01474602.

[26]

M. Kumar, T. N. Mishra and B. Tiwari, Stability analysis of Navier-Stokes system, International Journal of Geometric Methods in Modern Physics, 16 (2019), 1950157. doi: 10.1142/S0219887819501573.

[27]

G. A. Leonov, Lyapunov functions in the global analysis of chaotic systems, Ukrainian Mathematical Journal, 70 (2018), 42-66.  doi: 10.1007/s11253-018-1487-y.

[28]

C.-L. Li and Y.-B. Zhao, A unified Lorenz-like system and its tracking control, Communications in Theoretical Physics, 63 (2015), 317-324.  doi: 10.1088/0253-6102/63/3/317.

[29]

X. LiaoG. ZhouQ. YangY. Fu and G. Chen, Constructive proof of Lagrange stability and sufficient-Necessary conditions of Lyapunov stability for Yang-Chen chaotic system, Appl. Math. Comput., 309 (2017), 205-221.  doi: 10.1016/j.amc.2017.03.033.

[30]

Y. Liu and Q. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal. Real World Appl., 11 (2010), 2563-2572.  doi: 10.1016/j.nonrwa.2009.09.001.

[31]

Y. LiuS. Pang and D. Chen, An unusual chaotic system and its control, Mathematical and Computer Modelling, 57 (2013), 2473-2493.  doi: 10.1016/j.mcm.2012.12.006.

[32]

Y. Liu and Q. Yang, Dynamics of the Lü system on the invariant algebraic surface and at infinity, Int. J. Bifurc. Chaos, 21 (2011), 2559-2582.  doi: 10.1142/S0218127411029938.

[33]

Y. Liu, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the conjugate Lorenz-type system, Nonlinear Anal. Real World Appl., 13 (2012), 2466-2475.  doi: 10.1016/j.nonrwa.2012.02.011.

[34]

Y. Liu, Analysis of global dynamics in an unusual 3D chaotic system, Nonlinear Dyn., 70 (2012), 2203-2212.  doi: 10.1007/s11071-012-0610-0.

[35]

J. Llibre and M. Messias, Global dynamics of the Rikitake system, Physica D, 238 (2009), 241-252.  doi: 10.1016/j.physd.2008.10.011.

[36]

J. Llibre, M. Messias and P. R. da Silva, On the global dynamics of the Rabinovich system, J. Phys. A, Math. Theor., 41 (2008), 275210. doi: 10.1088/1751-8113/41/27/275210.

[37]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[38]

J. Lü and G. Chen, A new chaotic attractor coined, Int. J. Bifurc. Chaos, 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.

[39]

Q. LuoX. Liao and Z. Zeng, Sufficient and necessary conditions for Lyapunov stability of Lorenz system and their application, Science China Information Sciences, 53 (2010), 1574-1583.  doi: 10.1007/s11432-010-4032-7.

[40]

M. Messias, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system, J. Phys. A, Math. Theor., 42 (2009), 115101. doi: 10.1088/1751-8113/42/11/115101.

[41]

M. Messias, Dynamics at infinity of a cubic Chua's system, Int. J. Bifurc. Chaos, 21 (2011), 333-340.  doi: 10.1142/S0218127411028453.

[42]

S. Oiwa and T. Yajima, Jacobi stability analysis and chaotic behavior of nonlinear double pendulum, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750176. doi: 10.1142/S0219887817501766.

[43]

O. E. Rössler, An equation for continuous chaos, Phys. Lett. A, 57 (1976), 397-398.  doi: 10.1016/0375-9601(76)90101-8.

[44]

V. S. Sabau, Systems biology and deviation curvature tensor, Nonlinear Anal. RWA, 6 (2005), 563-587.  doi: 10.1016/j.nonrwa.2004.12.012.

[45]

G. van der Schrier and L. R. Maas, The diffusionless Lorenz equations; Silnikov bifurcations and reduction to an explicit map, Physica D, 141 (2000), 19-36.  doi: 10.1016/S0167-2789(00)00033-6.

[46]

A. Vanevcek and S. Celikovský, Control Systems: From linear analysis to synthesis of chaos, Prentice Hall International (UK) Ltd., (1996), 238-257. 

[47]

T. Yajima and K. Yamasaki, Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system, International Journal of Geometric Methods in Modern Physics, 13 (2016), 1650045. doi: 10.1142/S0219887816500456.

[48]

K. Yamasaki and T. Yajima, Lotka–Volterra system and KCC theory: Differential geometric structure of competitions and predations, Nonlinear Analysis: Real World Applications, 14 (2013), 1845-1853.  doi: 10.1016/j.nonrwa.2012.11.015.

[49]

K. Yamasaki and T. Yajima, Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory, Journal of Dynamical Systems and Geomatric Theories, 14 (2016), 137-153.  doi: 10.1080/1726037X.2016.1250500.

[50]

K. Yamasaki and T. Yajima, KCC analysis of the normal form of typical bifurcations in one-dimensional dynamical systems: Geometrical invariants of saddle-node, transcritical, and pitchfork bifurcations, Int. J. Bifurc. Chaos, 27 (2017), 1750145. doi: 10.1142/S0218127417501450.

[51]

Q. YangG. Chen and T. Zhou, A unified Lorenz-type system and its canonical form, Int. J. Bifurc. Chaos, 16 (2006), 2855-2871.  doi: 10.1142/S0218127406016501.

[52]

Q. Yang and G. Chen, A chaotic system with one saddle and two stable node-foci, Int. J. Bifurc. Chaos, 18 (2008), 1393-1414.  doi: 10.1142/S0218127408021063.

[53]

Q. Yang and Y. Liu, A hyperchaotic system from a chaotic system with one saddle and two stable node-foci, Journal of Mathematical Analysis and Applications, 360 (2009), 293-306.  doi: 10.1016/j.jmaa.2009.06.051.

[54]

Z. Zhang, T. Ding, W. Huang et al., Qualitative Theory of Differential Equations, Science Press, Beijing, in Chinese (1985).

show all references

References:
[1]

H. Abolghasem, Jacobi stability of circular orbits in a central force, Journal of Dynamical Systems and Geometric Theories, 10 (2012), 197-214.  doi: 10.1080/1726037X.2012.10698621.

[2]

H. Abolghasem, Liapunov stability versus Jacobi stability, Journal of Dynamical Systems and Geometric Theories, 10 (2012), 13-32.  doi: 10.1080/1726037X.2012.10698604.

[3]

H. Abolghasem, Jacobi stability of Hamiltonian system, International Journal of Pure and Applied Mathematics, 87 (2013), 181-194.  doi: 10.12732/ijpam.v87i1.11.

[4]

P. L. AntonelliS. F. Rutz and V. S. Sabau, A transient-state analysis of Tyson's model for the cell division cycle by means of KCC-theory, Open Systems and Information Dynamics, 9 (2002), 223-238.  doi: 10.1023/A:1019752327311.

[5]

P. L. Antonelli and I. Bucataru, Volterra-Hamilton production models with discounting: General theory and worked examples, Nonlinear Anal. RWA, 2 (2001), 337-356.  doi: 10.1016/S0362-546X(00)00101-2.

[6]

P. L. AntonelliS. F. Rutz and C. E. Hirakawa, The mathematical theory of endosymbiosis I, Nonlinear Anal. RWA, 12 (2011), 3238-3251.  doi: 10.1016/j.nonrwa.2011.05.023.

[7]

B. C. BaoH. BaoN. WangM. Chen and Q. Xu, Hidden extreme multistability in memristive hyperchaotic system, Chaos Solitons & Fractals, 94 (2017), 102-111.  doi: 10.1016/j.chaos.2016.11.016.

[8]

C. G. BoehmerT. Harko and S. V. Sabau, Jacobi stability analysis of dynamical systems-applications in gravitation and cosmology, Advances in Theoretical and Mathematical Physics, 16 (2012), 1145-1196.  doi: 10.4310/ATMP.2012.v16.n4.a2.

[9]

C. G. Bohmer and T. Harko, Nonlinear stability analysis of the Emden-Fowler equation, Journal of Nonlinear Mathematical Physics, 17 (2010), 503-516.  doi: 10.1142/S1402925110001100.

[10]

E. Cartan and D. D. Kosambi, Observations sur le mémoire précédent, Mathematische Zeitschrift, 37 (1933), 619-622.  doi: 10.1007/BF01474603.

[11]

G. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurc. Chaos, 9 (1999), 1465-1466.  doi: 10.1142/S0218127499001024.

[12]

Y. Chen and Z. B. Yin, The Jacobi stability of a Lorenz-type multistable hyperchaotic system with a curve of equilibria, Int. J. Bifurc. Chaos, 29 (2019), 1950062, 10 pp. doi: 10.1142/S0218127419500627.

[13]

S.-S. Chern, Sur la geometrie d'un systeme d'equations differentielles du second ordre, Bulletin des Sciences Mathematiques, 63 (1939), 206-212. 

[14]

A. Cima and J. Llibre, Bounded polynomial vector fields, Trans. Am. Math. Soc., 318 (1990), 557-579.  doi: 10.1090/S0002-9947-1990-0998352-5.

[15]

B. Danila, T. Harko, M. K. Mak, P. Pantaragphong and S. V. Sabau, Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background, Advances in High Energy Physics, (2016), Article ID 7521464 26 pp.

[16]

C. Feng, Q. Huang and Y. Liu, Jacobi analysis for an unusual 3D autonomous system, International Journal of Geometric Methods in Modern Physics, 17 (2020), 2050062, 20 pp. doi: 10.1142/S0219887820500620.

[17]

M. K. Gupta and C. K. Yadav, KCC theory and its application in a tumor growth model, Mathematical Methods in the Applied Sciences., 40 (2017), 7470-7487.  doi: 10.1002/mma.4542.

[18]

M. K. Gupta and C. K. Yadav, Jacobi stability analysis of Rössler system, Int. J. Bifurc. Chaos, 27 (2017), 63-76.  doi: 10.1142/S0218127417500560.

[19]

M. K. Gupta and C. K. Yadav, Jacobi stability analysis of modified Chua circuit system, International Journal of Geometric Methods in Modern Physics, 14 (2017), 121-142.  doi: 10.1142/S021988781750089X.

[20]

M. K. Gupta and C. K. Yadav, Jacobi stability analysis of Rikitake system, International Journal of Geometric Methods in Modern Physics, 13 (2016), 1650098. doi: 10.1142/S0219887816500985.

[21]

M. K. Gupta and C. K. Yadav, Rabinovich-Fabrikant system in view point of KCC theory in Finsler geometry, Journal of Interdisciplinary Mathematics, 22 (2019), 219-241.  doi: 10.1080/09720502.2019.1614249.

[22]

T. Harko and V. S. Sabau, Jacobi stability of the vacuum in the static spherically symmetric brane world models, Physical Review D, 77 (2008), 104009. doi: 10.1103/PhysRevD.77.104009.

[23]

T. Harko, C. Y. Ho, C. S. Leung and S. Yip, Jacobi stability analysis of the Lorenz system, International Journal of Geometric Methods in Modern Physics, 12 (2015), 1550081. doi: 10.1142/S0219887815500814.

[24]

Q. Huang, A. Liu and Y. Liu, Jacobi stability analysis of the Chen system, Int. J. Bifurc. Chaos, 29 (2019), 1950139. doi: 10.1142/S0218127419501396.

[25]

D. D. Kosambi, Parallelism and path-space, Mathematische Zeitschrift, 37 (1933), 608-618.  doi: 10.1007/BF01474602.

[26]

M. Kumar, T. N. Mishra and B. Tiwari, Stability analysis of Navier-Stokes system, International Journal of Geometric Methods in Modern Physics, 16 (2019), 1950157. doi: 10.1142/S0219887819501573.

[27]

G. A. Leonov, Lyapunov functions in the global analysis of chaotic systems, Ukrainian Mathematical Journal, 70 (2018), 42-66.  doi: 10.1007/s11253-018-1487-y.

[28]

C.-L. Li and Y.-B. Zhao, A unified Lorenz-like system and its tracking control, Communications in Theoretical Physics, 63 (2015), 317-324.  doi: 10.1088/0253-6102/63/3/317.

[29]

X. LiaoG. ZhouQ. YangY. Fu and G. Chen, Constructive proof of Lagrange stability and sufficient-Necessary conditions of Lyapunov stability for Yang-Chen chaotic system, Appl. Math. Comput., 309 (2017), 205-221.  doi: 10.1016/j.amc.2017.03.033.

[30]

Y. Liu and Q. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal. Real World Appl., 11 (2010), 2563-2572.  doi: 10.1016/j.nonrwa.2009.09.001.

[31]

Y. LiuS. Pang and D. Chen, An unusual chaotic system and its control, Mathematical and Computer Modelling, 57 (2013), 2473-2493.  doi: 10.1016/j.mcm.2012.12.006.

[32]

Y. Liu and Q. Yang, Dynamics of the Lü system on the invariant algebraic surface and at infinity, Int. J. Bifurc. Chaos, 21 (2011), 2559-2582.  doi: 10.1142/S0218127411029938.

[33]

Y. Liu, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the conjugate Lorenz-type system, Nonlinear Anal. Real World Appl., 13 (2012), 2466-2475.  doi: 10.1016/j.nonrwa.2012.02.011.

[34]

Y. Liu, Analysis of global dynamics in an unusual 3D chaotic system, Nonlinear Dyn., 70 (2012), 2203-2212.  doi: 10.1007/s11071-012-0610-0.

[35]

J. Llibre and M. Messias, Global dynamics of the Rikitake system, Physica D, 238 (2009), 241-252.  doi: 10.1016/j.physd.2008.10.011.

[36]

J. Llibre, M. Messias and P. R. da Silva, On the global dynamics of the Rabinovich system, J. Phys. A, Math. Theor., 41 (2008), 275210. doi: 10.1088/1751-8113/41/27/275210.

[37]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[38]

J. Lü and G. Chen, A new chaotic attractor coined, Int. J. Bifurc. Chaos, 12 (2002), 659-661.  doi: 10.1142/S0218127402004620.

[39]

Q. LuoX. Liao and Z. Zeng, Sufficient and necessary conditions for Lyapunov stability of Lorenz system and their application, Science China Information Sciences, 53 (2010), 1574-1583.  doi: 10.1007/s11432-010-4032-7.

[40]

M. Messias, Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system, J. Phys. A, Math. Theor., 42 (2009), 115101. doi: 10.1088/1751-8113/42/11/115101.

[41]

M. Messias, Dynamics at infinity of a cubic Chua's system, Int. J. Bifurc. Chaos, 21 (2011), 333-340.  doi: 10.1142/S0218127411028453.

[42]

S. Oiwa and T. Yajima, Jacobi stability analysis and chaotic behavior of nonlinear double pendulum, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750176. doi: 10.1142/S0219887817501766.

[43]

O. E. Rössler, An equation for continuous chaos, Phys. Lett. A, 57 (1976), 397-398.  doi: 10.1016/0375-9601(76)90101-8.

[44]

V. S. Sabau, Systems biology and deviation curvature tensor, Nonlinear Anal. RWA, 6 (2005), 563-587.  doi: 10.1016/j.nonrwa.2004.12.012.

[45]

G. van der Schrier and L. R. Maas, The diffusionless Lorenz equations; Silnikov bifurcations and reduction to an explicit map, Physica D, 141 (2000), 19-36.  doi: 10.1016/S0167-2789(00)00033-6.

[46]

A. Vanevcek and S. Celikovský, Control Systems: From linear analysis to synthesis of chaos, Prentice Hall International (UK) Ltd., (1996), 238-257. 

[47]

T. Yajima and K. Yamasaki, Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system, International Journal of Geometric Methods in Modern Physics, 13 (2016), 1650045. doi: 10.1142/S0219887816500456.

[48]

K. Yamasaki and T. Yajima, Lotka–Volterra system and KCC theory: Differential geometric structure of competitions and predations, Nonlinear Analysis: Real World Applications, 14 (2013), 1845-1853.  doi: 10.1016/j.nonrwa.2012.11.015.

[49]

K. Yamasaki and T. Yajima, Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory, Journal of Dynamical Systems and Geomatric Theories, 14 (2016), 137-153.  doi: 10.1080/1726037X.2016.1250500.

[50]

K. Yamasaki and T. Yajima, KCC analysis of the normal form of typical bifurcations in one-dimensional dynamical systems: Geometrical invariants of saddle-node, transcritical, and pitchfork bifurcations, Int. J. Bifurc. Chaos, 27 (2017), 1750145. doi: 10.1142/S0218127417501450.

[51]

Q. YangG. Chen and T. Zhou, A unified Lorenz-type system and its canonical form, Int. J. Bifurc. Chaos, 16 (2006), 2855-2871.  doi: 10.1142/S0218127406016501.

[52]

Q. Yang and G. Chen, A chaotic system with one saddle and two stable node-foci, Int. J. Bifurc. Chaos, 18 (2008), 1393-1414.  doi: 10.1142/S0218127408021063.

[53]

Q. Yang and Y. Liu, A hyperchaotic system from a chaotic system with one saddle and two stable node-foci, Journal of Mathematical Analysis and Applications, 360 (2009), 293-306.  doi: 10.1016/j.jmaa.2009.06.051.

[54]

Z. Zhang, T. Ding, W. Huang et al., Qualitative Theory of Differential Equations, Science Press, Beijing, in Chinese (1985).

Figure 1.  Attractors of Yang-Chen system with (a) $ a = 10 $, $ b = 8/3 $, and $ c = 16 $; (b) $ a = 35 $, $ b = 3 $, and $ c = 35 $
Figure 2.  Dynamics of the Yang-Chen system near the sphere at infinity in the local charts $ U_1 $ for (blue) $ (a, b, c) = (0.5, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.03) $, (red) $ (a, b, c) = (1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.03) $, (black) $ (a, b, c) = (0.1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.01) $, respectively
Figure 3.  Dynamics of the Yang-Chen system near the sphere at infinity in the local charts $ V_1 $ (blue) $ (a, b, c) = (0.5, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.03) $, (red) $ (a, b, c) = (1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.03) $, (black) $ (a, b, c) = (0.1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.01) $, respectively
Figure 4.  Phase portrait of the system (10), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $ U_2 $
Figure 5.  Phase portrait of the system (12), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $ U_3 $
Figure 6.  Phase portrait of system (1) at infinity
Figure 7.  Time variation of the deviation vector and its curvature near $ E_{1} $, for $ a = 35 $, $ b = 3 $
Figure 8.  Time variation of instability exponent $ \delta(E_{1}) $ for $ a = 35 $, $ b = 3 $, and different values of $ c $
Figure 9.  Time variation of the deviation vector and its curvature near $ E_{2, 3} $ with $ a = 35, b = 3 $. Initial conditions used to integrate deviation equations are $ \xi_{1}(0) = \xi_{2}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = 10^{-6} $
Figure 10.  Time variation of curvature $ \kappa_{0} $ of deviation vector near equilibrium points $ E_{1} $ with $ a = 35, b = 3 $
Figure 11.  Time variation of curvature $ \kappa_{0} $ of deviation vector near equilibrium points $ E_{2, 3} $ with $ a = 35, b = 3 $
Figure 12.  A large version of Fig. 11 at time $ 0.25 $ to $ 0.55 $
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