Karatetskaia Efrosiniia, Shykhmamedov Aikan, Kazakov Alexey
National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia.
Chaos. 2021 Jan;31(1):011102. doi: 10.1063/5.0036405.
A Shilnikov homoclinic attractor of a three-dimensional diffeomorphism contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold and homoclinic orbits to this saddle-focus. The orientation-reversing property of the diffeomorphism implies a symmetry between two branches of the one-dimensional stable manifold. This symmetry leads to a significant difference between Shilnikov attractors in the orientation-reversing and orientation-preserving cases. We consider the three-dimensional Mirá map x¯=y,y¯=z, and z¯=Bx+Cy+Az-y with the negative Jacobian (B<0) as a basic model demonstrating various types of Shilnikov attractors. We show that depending on values of parameters A,B, and C, such attractors can be of three possible types: hyperchaotic (with two positive and one negative Lyapunov exponent), flow-like (with one positive, one very close to zero, and one negative Lyapunov exponent), and strongly dissipative (with one positive and two negative Lyapunov exponents). We study scenarios of the formation of such attractors in one-parameter families.
三维微分同胚的希利尼科夫同宿吸引子包含一个鞍 - 焦点不动点，其具有二维不稳定不变流形以及到该鞍 - 焦点的同宿轨道。微分同胚的反向定向性质意味着一维稳定流形的两个分支之间存在对称性。这种对称性导致了在反向定向和正向定向情况下希利尼科夫吸引子之间的显著差异。我们将具有负雅可比行列式（(B\lt0)）的三维米拉映射(\bar{x}=y)，(\bar{y}=z)，以及(\bar{z}=Bx + Cy + Az - y)作为展示各种类型希利尼科夫吸引子的基本模型。我们表明，根据参数(A)、(B)和(C)的值，此类吸引子可能有三种类型：超混沌（具有两个正的和一个负的李雅普诺夫指数）、流状（具有一个正的、一个非常接近零的和一个负的李雅普诺夫指数）以及强耗散（具有一个正的和两个负的李雅普诺夫指数）。我们研究了单参数族中此类吸引子的形成情况。
