Dynamics of the Lorenz system having an invariant algebraic surface

Abstract

In this paper, we characterize all dynamics of the Lorenz system ẋ=s(y−x), ẏ=rx−xz−y, ż=xy−bz having an invariant algebraic surface.

Invariant Algebraic Surfaces and Constrained Systems

“…This leads us to start our studies with less degenerate cases. For this, we will consider a system in R 3 given by (15) ẋ = α(x, y), ẏ = n j=0 β j (x, y)z j , ż = γ(x, y, z), which has invariant algebraic surface M = H −1 (0), H(x, y, x) = f (x, y)z − g(x, y). As in the proof of Theorem 7 we obtain the constrained system…”

“…In [15] the authors gave all the six invariant algebraic surfaces for the Lorenz System (29). In [3] the flows on such invariant surfaces were considered and in [11] the authors studied the global dynamics of (29). Three out of six invariant algebraic surfaces can be written in the form M = {H −1 (0)} with H i (x, y, z) = f i (x, y)z − g i (x, y), i = 1, 2, 3.…”

New Chaotic Regimes in the Lorenz and Chen Systems

“…It maps into the forward-time Lorenz system provided the parameters are chosen according to σ = −a/c, ρ = a/c − 1, β = −b/c, which is a seldom considered region of the Lorenz system. Some special cases of the Lorenz system with negative and zero parameters have been studied [Celikovský & Vanȇcek, 1994;Celikovský & Chen, 2002, 2005Cao & Zhang, 2007;Llibre et al, 2010;Sprott, 2010;Li & Sprott, 2014], but apparently not the one considered here.…”

The zero-Hopf bifurcations of a four-dimensional hyperchaotic system

“…This system plays an important role in other areas such as in the modeling of lasers 2 and dynamos. 3 As one of the simplest models presenting chaos, the Lorenz system exhibits a rich range of dynamical properties, and it has been researched from different points of view, such as positive invariant, 4 integrability, [5][6][7] global dynamics, [8][9][10] and bifurcation. 11,12 After the Lorenz system, mathematicians and physicists from a physical or purely abstract mathematical point of view proposed various polynomial differential systems in R 3 , whose trajectories exhibit chaotic dynamics of the Lorenz system type.…”