Research Article
Year 2022, Volume 4, Issue 1, 26 - 36, 30.03.2022

### References

• Algaba, A., Dominguez-Moreno, M. C., Merino, M. and Rodríguez- Luis, A. J., 2018 A Review on Some Bifurcations in the Lorenz System. Nonlinear Systems, 1:3–36.
• Alligood, K. T., Sauer, T. and Yorke, J., 1996 Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York.
• Anastassiou, S., Pnevmatikos, S. and Bountis T, 2002 Quadratic Vector Fields Equivariant Under the D2 Symmetry Group. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23, 1350017.
• Argyris, J., Faust, G., Haase, M. and Friedrich, R., 2015 An Exploration of Dynamical Systems and Chaos. Springer-Verlag, Berlin.
• Cencini, M., Cecconi, F. and Vulpiani, A., 2010 Chaos: From Simple Models to Complex Systems. World Scientific, Singapore.
• Chen, G. and Ueta, T., 1999 Yet another chaotic attractor. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9:1465–1466.
• Dumortier, F., Llibre, J. and Artés, J.C., 2006 Qualitative Theory of Planar Differential Systems. Springer-Verlag, New York.
• Guckenheimer, J. and Holmes, P. [2002] “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields", (Appl. Math. Sci. 42, Springer-Verlag, New York).
• Heidel, J. and Zhang, F., 1999 Nonchaotic behavior in threedimensional quadratic systems II. The conservative case. Nonlinearity 12:617–633.
• Heidel, J. and Zhang, F., 2007 Nonchaotic and chaotic behaviour in three-dimensional quadratic systems: Five–one conservative cases. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17:2049–2072.
• Jafari, S., Sprott, J. C., Pham, V-T, Volos, C., and Li, C., 2016 Simple chaotic 3D flows with surfaces of equilibria. Nonlinear Dynamics 86:1349–1358.
• Li, C., Peng, Y., Tao, Z., Sprott, J. C. and Jafari, S., 2021 Coexisting Infinite Equilibria and Chaos. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 31(5), 2130014, 17p
• Llibre, J., 2004 Integrability of polynomial differential systems. Handbook of differential equations, Elsevier/North-Holland, Amsterdam.
• Llibre, J. and Messias, M., 2009 Global dynamics of the Rikitake system. Phys D: Nonlinear Phenomena 238:241–252.
• Llibre, J., Messias, M. and da Silva, P.R., 2008 On the global dynamics of the Rabinovich system. J. Phys. A: Math. Theor. 41, 275210, 21p.
• Llibre, J., Messias, M. and da Silva, P.R., 2010 Global dynamics of the Lorenz system with invariant algebraic surfaces. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20:3137–3155.
• Llibre, J., Messias, M. and da Silva, P.R., 2012 Global dynamics in the Poincaré ball of the Chen system having invariant algebraic surfaces. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, 1250154, 17p.
• Llibre, J. and Oliveira, R. D. S., 2015 Quadratic systems with invariant straight lines with total multiplicity two having Darboux invariants. Comm. in Contemporary Math. 17, 1450018.
• Llibre, J. and Zhang, X., 2012 On the Darboux integrability of the polynomial differential systems. Qualit. Th. Dyn. Sys. 11:129– 144.
• Lorenz, E.N., 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20:130–141.
• Malasoma, J. -M., 2002 A new class of minimal chaotic flows equation for continuous chaos. Phys. Lett. A. 305:52–58.
• Malasoma, J. -M., 2009 Non-chaotic behavior for a class of quadratic jerk equations. Chaos, Solitons Fractals 39:533–539.
• Messias, M. and Silva, R. P., 2018 Nonchaotic Behavior in Quadratic Three-Dimensional Differential Systems wiht a Symmetric Jacobian Matrix. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28, 1830006.
• Messias, M. and Silva, R. P., 2020 Determination of nonchaotic behavior for some classes of polynomial jerk equations. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 30, 2050117, 12p.
• Osinga, H. M. and Krauskopf, B., 2002 Visualizing the structure of chaos in the Lorenz system. Computers and Graphics 26(5):815– 823.
• Osinga, H. M. and Krauskopf, B., 2004 The Lorenz manifold as a collection of geodesic level sets. Nonlinearity 17 C1.
• Ott, E., 2002 Chaos in Dynamical Systems. Cambridge University Press, London.
• Rössler, O. E., 1976 An equation for continuous chaos. Phys. Lett. A 57:397–398.
• Sparrow, C., 1982 The Lorenz Equations. Bifurcations, Chaos, and Strange Attractors. Springer-Verlag, New York.
• Strogatz, S.H., 2001 Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry and engineering.Westview Press, New York.
• Zhang, F. and Heidel, J., 1997 Nonchaotic behaviour in threedimensional quadratic systems. Nonlinearity 10:1289–1303.
• Zhang, F. and Heidel, J., 2012 Chaotic and nonchaotic behaviour in three-dimensional quadratic systems: 5-1 dissipative cases. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, 1250010.
• Zhang, F., Heidel, J. and Le Borne, R., 2008 Determining nonchaotic parameter regions in some simple chaotic jerk functions. Chaos, Solitons and Fractals 11:1413–1418.
• Zhu C., Liu Y. and Guo Y., 2010 Theoretic and Numerical Study of a New Chaotic System. Intelligent Information Management 2:104-109.
• Yang, X. S., 2000 Nonchaotic behavior in nondissipative quadratic systems. Chaos, Solitons and Fractals 11:1799–1802.
• Yang, X. S., 2002 On non-chaotic behavior of a class of jerk systems. Far East J. Dyn. Syst. 4:27–38.
• Yang, X. S. and Chen, G., 2002 Non-chaotic behavior in a class of continuous dynamical systems. Far East J. Dyn. Syst. 4:87–95.
• Wiggins, S., 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Appl. Math. 2, Springer-Verlag, New York.

### Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces

Year 2022, Volume 4, Issue 1, 26 - 36, 30.03.2022

### Abstract

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

### References

• Algaba, A., Dominguez-Moreno, M. C., Merino, M. and Rodríguez- Luis, A. J., 2018 A Review on Some Bifurcations in the Lorenz System. Nonlinear Systems, 1:3–36.
• Alligood, K. T., Sauer, T. and Yorke, J., 1996 Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York.
• Anastassiou, S., Pnevmatikos, S. and Bountis T, 2002 Quadratic Vector Fields Equivariant Under the D2 Symmetry Group. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23, 1350017.
• Argyris, J., Faust, G., Haase, M. and Friedrich, R., 2015 An Exploration of Dynamical Systems and Chaos. Springer-Verlag, Berlin.
• Cencini, M., Cecconi, F. and Vulpiani, A., 2010 Chaos: From Simple Models to Complex Systems. World Scientific, Singapore.
• Chen, G. and Ueta, T., 1999 Yet another chaotic attractor. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9:1465–1466.
• Dumortier, F., Llibre, J. and Artés, J.C., 2006 Qualitative Theory of Planar Differential Systems. Springer-Verlag, New York.
• Guckenheimer, J. and Holmes, P. [2002] “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields", (Appl. Math. Sci. 42, Springer-Verlag, New York).
• Heidel, J. and Zhang, F., 1999 Nonchaotic behavior in threedimensional quadratic systems II. The conservative case. Nonlinearity 12:617–633.
• Heidel, J. and Zhang, F., 2007 Nonchaotic and chaotic behaviour in three-dimensional quadratic systems: Five–one conservative cases. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17:2049–2072.
• Jafari, S., Sprott, J. C., Pham, V-T, Volos, C., and Li, C., 2016 Simple chaotic 3D flows with surfaces of equilibria. Nonlinear Dynamics 86:1349–1358.
• Li, C., Peng, Y., Tao, Z., Sprott, J. C. and Jafari, S., 2021 Coexisting Infinite Equilibria and Chaos. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 31(5), 2130014, 17p
• Llibre, J., 2004 Integrability of polynomial differential systems. Handbook of differential equations, Elsevier/North-Holland, Amsterdam.
• Llibre, J. and Messias, M., 2009 Global dynamics of the Rikitake system. Phys D: Nonlinear Phenomena 238:241–252.
• Llibre, J., Messias, M. and da Silva, P.R., 2008 On the global dynamics of the Rabinovich system. J. Phys. A: Math. Theor. 41, 275210, 21p.
• Llibre, J., Messias, M. and da Silva, P.R., 2010 Global dynamics of the Lorenz system with invariant algebraic surfaces. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20:3137–3155.
• Llibre, J., Messias, M. and da Silva, P.R., 2012 Global dynamics in the Poincaré ball of the Chen system having invariant algebraic surfaces. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, 1250154, 17p.
• Llibre, J. and Oliveira, R. D. S., 2015 Quadratic systems with invariant straight lines with total multiplicity two having Darboux invariants. Comm. in Contemporary Math. 17, 1450018.
• Llibre, J. and Zhang, X., 2012 On the Darboux integrability of the polynomial differential systems. Qualit. Th. Dyn. Sys. 11:129– 144.
• Lorenz, E.N., 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20:130–141.
• Malasoma, J. -M., 2002 A new class of minimal chaotic flows equation for continuous chaos. Phys. Lett. A. 305:52–58.
• Malasoma, J. -M., 2009 Non-chaotic behavior for a class of quadratic jerk equations. Chaos, Solitons Fractals 39:533–539.
• Messias, M. and Silva, R. P., 2018 Nonchaotic Behavior in Quadratic Three-Dimensional Differential Systems wiht a Symmetric Jacobian Matrix. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28, 1830006.
• Messias, M. and Silva, R. P., 2020 Determination of nonchaotic behavior for some classes of polynomial jerk equations. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 30, 2050117, 12p.
• Osinga, H. M. and Krauskopf, B., 2002 Visualizing the structure of chaos in the Lorenz system. Computers and Graphics 26(5):815– 823.
• Osinga, H. M. and Krauskopf, B., 2004 The Lorenz manifold as a collection of geodesic level sets. Nonlinearity 17 C1.
• Ott, E., 2002 Chaos in Dynamical Systems. Cambridge University Press, London.
• Rössler, O. E., 1976 An equation for continuous chaos. Phys. Lett. A 57:397–398.
• Sparrow, C., 1982 The Lorenz Equations. Bifurcations, Chaos, and Strange Attractors. Springer-Verlag, New York.
• Strogatz, S.H., 2001 Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry and engineering.Westview Press, New York.
• Zhang, F. and Heidel, J., 1997 Nonchaotic behaviour in threedimensional quadratic systems. Nonlinearity 10:1289–1303.
• Zhang, F. and Heidel, J., 2012 Chaotic and nonchaotic behaviour in three-dimensional quadratic systems: 5-1 dissipative cases. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, 1250010.
• Zhang, F., Heidel, J. and Le Borne, R., 2008 Determining nonchaotic parameter regions in some simple chaotic jerk functions. Chaos, Solitons and Fractals 11:1413–1418.
• Zhu C., Liu Y. and Guo Y., 2010 Theoretic and Numerical Study of a New Chaotic System. Intelligent Information Management 2:104-109.
• Yang, X. S., 2000 Nonchaotic behavior in nondissipative quadratic systems. Chaos, Solitons and Fractals 11:1799–1802.
• Yang, X. S., 2002 On non-chaotic behavior of a class of jerk systems. Far East J. Dyn. Syst. 4:27–38.
• Yang, X. S. and Chen, G., 2002 Non-chaotic behavior in a class of continuous dynamical systems. Far East J. Dyn. Syst. 4:87–95.
• Wiggins, S., 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Appl. Math. 2, Springer-Verlag, New York.

### Details

Primary Language English Mathematics, Interdisciplinary Applications Research Articles Marcelo MESSİAS (Primary Author) São Paulo State University 0000-0003-2269-7091 Brazil Rafael Paulino SİLVA This is me 0000-0001-5366-2524 Brazil State of São Paulo Foundation (FAPESP) Grant Number 2019/10269-3 The first author was supported by State of São Paulo Foundation (FAPESP) grant number 2019/10269-3 and by CNPq/Brazil grant number 311355/2018-8. The second author was supported by CAPES/Brazil through a PhD fellowship. March 30, 2022 Year 2022, Volume 4, Issue 1

### Cite

 APA Messias, M. & Silva, R. P. (2022). Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces . Chaos Theory and Applications , 4 (1) , 26-36 . DOI: 10.51537/chaos.1022368
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Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science