A Family of 1D Chaotic Maps without Equilibria

Abstract

In this work, a family of piecewise chaotic maps is proposed. This family of maps is parameterized by the nonlinear functions used for each piece of the mapping, which can be either symmetric or non-symmetric. Applying a constraint on the shape of each piece, the generated maps have no equilibria and can showcase chaotic behavior. This family thus belongs to the category of systems with hidden attractors. Numerous examples of chaotic maps are provided, showcasing fractal-like, symmetrical patterns at the interchange between chaotic and non-chaotic behavior. Moreover, the application of the proposed maps to a pseudorandom bit generator is successfully performed.

Keywords

• chaos
• discrete map
• hidden attractor
• fixed point
• equilibrium
• chaotification