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Since the German biochemist Otto Rössler proposed the first hyperchaotic model in the years 1970s and showed to the scientific community how important hyperchaos can be in describing real life phenomena, it has become necessary to develop and propose various techniques capable of generating hyperchaotic attractors with more complex dynamics applicable in both theory and practice. We propose in this paper, an innovative method with analytical and numerical aspects able to generate a class of hyperchaotic attractors with many wings and different shapes. We use a fractal operator to obtain an expression of the modified fractal-fractional Lü system, which is therefore solved numerically. After showing that the initial model is hyperchaotic, we perform some numerical simulations that prove that the hyperchaotic status of the system remain unchanged. The results show that the modified system can generate hyperchaotic attractors of types n -wings, n × m -wings, n × m × p -wings and n × m × p × r -wings (m , n , p , r ∈ ℕ), using both linear and rotational variations. It appears that the system is involved in fractal designs comprising a linear or rotational self-duplication process happening in different scales across the system and ending up with the triangular or square shape. • An innovative fractal system with quadratic function is proposed. • The system is solved numerically and simulations are done. • It generates multi-wing hyperchaotic attractors with many shapes. • These attractors are of types N, NxM, NxMxP and NxMxPxR-wings. • There is fractal design with linear, triangular or square shape. [ABSTRACT FROM AUTHOR]

In this paper, we present mathematical analysis and numerical simulation of a three-dimensional autonomous fractional system with coexistence of multi-scroll chaotic attractors. We replaced the classical derivatives of such system with the Caputo-Fabrizio fractional derivative. This derivative combines both the exponential laws and non-singular kernels in its formulation which makes it special and useful. A two-step Adams-Bashforth scheme is derived for the approximation of the fractional derivative with exponential law and non-singular kernel. We then presented both numerical results and graphical results by considering many values of the fractional-order parameter β ∈ (0, ]. We demonstrate that the observed chaotic behavior conduct perseveres as the fractional-order parameter approaches 1. [ABSTRACT FROM AUTHOR]