Dynamics at infinity and other global dynamical aspects of Shimizu–Morioka equations

The Hopf bifurcation in the Shimizu–Morioka system

“…The statement (a) and (b) of the next proposition are not new, in fact they are well know in the literature see for instance [13,5].…”

“…Later the system gained self-interest and several articles have appeared in the literature, dealing mainly with the chaotic behavior of the solutions and the emergence of strange attractor , see for instance [6,17,18,20,21,22]. It was shown in [17] among other properties that system (1) presents Lorenz-like strange attractors, for example taking α = 0.45 and λ = 0.75 (see Figure 1 of [13]). …”

Hidden Attractors and Dynamical Behaviors in an Extended Rikitake System

“…In this section, we use the Poincaré compactification method [Dumortier et al, 2006;Llibre et al, 2008;Messias & Gouveia, 2012] to make an analysis of the flow of the system (2) at infinity. Let S 3 = {r = (r 1 , r 2 , r 3 , r 4 ) ∈ R 4 | r = 1} be a Poincaré unit sphere.…”

Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system

“…Furthermore, some quadratic systems similar to system (2) also appeared, arising from physical models or proposed from an abstract point of view, which are shown to be chaotic [4,[10][11][12][14][15][16][17]19,20]. More recently, a series of papers have been published which present global dynamical analysis of Lorenz and Lorenz-like systems, giving a contribution to the understanding of these complex kinds of differential systems in R 3 ; see, for instance [5][6][7][8]12,13].…”

“…Beyond to complement the results presented in [4], the analysis of system (1) presented here makes part of a program aiming to describe global properties of quadratic three-dimensional differential systems defined in R 3 , which is being developed by several authors; see, for instance [5][6][7][8]12,13]. It is important to observe that these types of systems appear in the literature as mathematical models for several natural problems, coming from Physics, Biology and Engineering mathematical modeling; see, for instance, [3,5,6,9,13,21] and references therein. Also, one has recently found that the chaotic dynamics of differential systems is a very useful tool which has great potentials for applications in many branches of science and technology, such as information and computer science, power system protection, biomedical system analysis, encryption and communication, electronic circuits and so on (see [4] and its references).…”