Physics Letters A 377 (2013) 699–702 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla Elementary quadratic chaotic ﬂows with no equilibria Sajad Jafari a,∗ , J.C. Sprott b , S. Mohammad Reza Hashemi Golpayegani a a Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran b Department of Physics, University of Wisconsin, Madison, WI 53706, USA article info abstract Article history: Received 23 October 2012 Received in revised form 5 January 2013 Accepted 7 January 2013 Available online 16 January 2013 Communicated by C.R. Doering Keywords: Chaotic ﬂows No equilibrium Eigenvalue Lyapunov exponent Three methods are used to produce a catalog of seventeen elementary three-dimensional chaotic ﬂows with quadratic nonlinearities that have the unusual feature of lacking any equilibrium points. It is likely that most if not all the elementary examples of such systems have now been identiﬁed.  2013 Elsevier B.V. All rights reserved. 1. Introduction Recently there has been interest in ﬁnding and studying rare examples of simple chaotic ﬂows such as those in which there are no equilibria or in which any existing equilibria are stable [1–4]. There is little knowledge about the characteristics of such systems. Here we consider chaotic ﬂows with no equilibria. Such systems can have neither homoclinic nor heteroclinic orbits [5], and thus the Shilnikov method [6,7] cannot be used to verify the chaos. We are aware of only three such examples that have been pre- viously reported, although there may be additional cases for which the lack of an equilibrium was not speciﬁcally noted. The oldest and best-known example is the conservative Sprott A system [8] listed as NE 1 in Table 1. This is an important system since it is a special case of the Nose–Hoover oscillator [9] which describes many natural phenomena [10], and thus it suggests that such systems may have practical as well as theoretical importance. Re- cently, two other dissipative examples have been reported, which we call the Wei system [1] listed as NE 2 in Table 1 and the Wang– Chen system [2], a simpliﬁed version of which with one fewer term than previously published is listed as NE 3 in Table 1. 2. Main results We performed a systematic search to ﬁnd additional three- dimensional chaotic systems with quadratic nonlinearities and no equilibria. Our search was based on the methods proposed in [11] * Corresponding author. Tel.: +98 9357874169; fax: +98 2164542370. E-mail address: sajadjafari@aut.ac.ir (S. Jafari). and used our own custom software. Our objective was to ﬁnd the algebraically simplest cases which cannot be further reduced by the removal of terms without destroying the chaos. The search was inspired by the observation that each of the known examples con- tains a constant term, and that if the constant is set to zero, the resulting system is nonhyperbolic (the equilibria have eigenvalues with a real part equal to zero). Two of them (Wei and Wang–Chen) have a pair of imaginary eigenvalues. It is a general requirement that chaotic systems of this type include such a constant term since there would otherwise be at least one equilibrium point at the origin (0, 0, 0). Thus we proceeded to search for additional ex- amples using three basic methods: (1) We added a constant term a to other nonhyperbolic sys- tems. For example, the system ˙ x = y ˙ y =−x + z ˙ z = k 1 x 2 + k 2 z 2 + k 3 y 2 + a (1) with a = 0 has an equilibrium at (0, 0, 0) whose eigenvalues are zero and pure imaginary. Adjusting and simplifying the parameters k 1 , k 2 , k 3 , and a gives the chaotic system listed as NE 7 system in Table 1. (2) We looked at cases where we could show algebraically that the equilibrium points are imaginary. For example, any chaotic so- lution of a parametric system such as ˙ x = y ˙ y = z 0375-9601/$ – see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.01.009