New types of 3-D systems of quadratic differential equations with chaotic dynamics based on Ricker discrete population model

Keywords: System of ordinary quadratic differential equations Linear transformations Boundedness Limit cycle Chaotic attractor Saddle focus Ricker discrete population model a b s t r a c t The wide class of 3-D autonomous systems of quadratic differential equations, in each of which either there is a couple of coexisting limit cycles or there is a couple of coexisting chaotic attractors, is found. In the second case the couple consists of either Lorentz-type attractor and another attractor of a new type or two Lorentz-type attractors. It is shown that the chaotic behavior of any system of the indicated class can be described by the Rick-er discrete population model: z i+1 The values of parameters, at which in the 3-D system appears either the couple of limit cycles or the couple of chaotic attractors, or only one limit cycle, or only one sphere-shaped chaotic attrac-tor, are indicated. Examples are given. In 1963 Lorenz discovered a simple 3-D autonomous chaotic system. This system has only two quadratic non-linearity terms. In future many of other chaotic systems have been found. One of principal directions of searches was the finding it is possible more simple chaotic systems with quadratic non-linearities. It should be note that most found chaotic systems had the kind: