Computer assisted proof of the existence of the Lorenz attractor in the Shimizu–Morioka system

Abstract

We prove that the Shimizu-Morioka system has a Lorenz attractor for an open set of parameter values. For the proof we employ a criterion proposed by Shilnikov, which allows to conclude the existence of the attractor by examination of the behaviour of only one orbit. The needed properties of the orbit are established by using computer assisted numerics. Our result is also applied to the study of local bifurcations of triply degenerate periodic points of three-dimensional maps. It provides a formal proof of the birth of discrete Lorenz attractors at various global bifurcations.

Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors

“…In [43,44], it was checked with the use of rigorous numerics that the classical Lorenz system satisfies the conditions of [2,3]. The same is true for an open set of parameter values in the Morioka-Shimizu model [8] and the extended Lorenz model [32].…”

“…Note that the Lorenz system (4) does not satisfy condition (C4 ) at classical parameter values, while the Morioka-Shimizu system (5) fulfils this condition for the set of parameter values for which a proof of the existence of Lorenz attractor is obtained in [8]. Therefore, Theorem 4 is applicable to time-periodic perturbations of the Lorenz attractor in the Lorenz system, and the stronger Theorem 3 is applicable to the periodic perturbation of the Lorenz attractor in the Morioka-Shimizu system.…”

“…For that, we show that T n 0 (W u ) cannot intersects v 1 and v 2 . Indeed, formula (8) for the local map implies that x and z in (19) are of order of λ k 0 . Hence, the main contribution to the x-coordinate in (19) is given by the term b(y − y + ), which is of order δ y (recall that we let Π 1 = {(x, y, z) | |x| < δ, |y − y − | < δ y , z < δ}).…”

Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions

“…Here, we assume β as the adjustable control parameter. The integer order system of Equation (1) was first proposed in [1] and has attracted the interest of researchers to study the stability and various types of bifurcation such as in [2][3][4][5]. However, the fractional order or arbitrary order of the system as in Equation (1) have received less attention; also see [6,7].…”

“…However, it is still yet to be applied for other fractional systems such as the fractional Shimizu-Morioka system.On the other hand, the Shimizu-Morioka system can be considered as a simplified system for investigating the dynamic bahaviour of the well-known Lorenz system. However, due to its rich dynamic behaviour, and especially chaotic behaviour of its solutions, the Shimizu-Morioka system has its self-interest [2][3][4][5]. Unfortunately, there are not many published results regarding the stability and bifurcation analysis for the Shimizu-Morioka system in fractional orders.…”