When considering the effects of fractional derivatives on systems with hidden attractors, a few fractional-order forms of systems with hidden attractors have been introduced. Authors have focused on local fractional diffusion equations in fractal heat transfer, fractal LC-electric circuit, and new rheological models. Moreover, it is worth noting that local fractional derivatives with special functions have received significant attention in different areas. In recent years, there has been an increasing interest in the stability of fractional systems. The fractional derivatives play important roles in the field of mathematical modeling of numerous models such as fractional model of regularized long-wave equation, Lienard’s equation, fractional model of convective radial fins, modified Kawahara equation, etc. ĭifferent definitions and main properties of fractional calculus have been reported in the literature. Hidden attractors have received considerable attention recently because of their roles in theoretical and practical problems. It is worth noting that systems with stable equilibria are systems with ‘hidden attractors’. In addition, the generalized Sprott C system with only two stable equilibria was introduced in. Interestingly, the six-term system with stable equilibria exhibited a double-scroll chaotic attractor. A six-term system with stable equilibria was presented in. By using the center manifold theory and normal form method, Wei investigated delayed feedback on such a chaotic system with two stable node-foci. found an unusual Lorenz-like chaotic system with two stable node-foci. In spite of the fact that the Yang-Chen system connected the original Lorenz system and the original Chen system, it was not diffeomorphic with the original Lorenz and Chen systems. Yang and Chen proposed a chaotic system with one saddle and two stable node-foci.

Several attempts have been made to investigate chaotic systems with stable equilibria. However, recent evidence suggests that chaos can be observed in 3D autonomous systems with stable equilibria. It has previously been observed that common 3D autonomous chaotic systems, such as Lorenz system, Chen system, Lü system, or Yang’s system, have one saddle and two unstable saddle-foci. Many three-dimensional (3D) autonomous chaotic systems have been found and reported in the literature. Applications of chaos have been witnessed in various areas ranging from path planning generator, secure communications, audio encryption scheme, image encryption, to truly random number generator. Finally, the designed systems are implemented by analog circuit and microcontroller to verify the multi-wing hidden attractors.There has been a dramatic increase in studying chaos and systems with chaotic behavior in the past decades. The local initial condition areas and complex boundaries for different coexisting attractors are addressed. By means of bifurcation diagram, finite-time Lyapunov exponents, phase portraits, and attraction basins, it is shown that the multi-wing chaotic attractors always coexist with stable point attractors. Furthermore, the rotation symmetry is applied to no-equilibrium system to generate multi-wing chaotic hidden attractors.

Two examples of 2-fold and 3-fold covers of the chaotic system are presented. Using rotation symmetry, we construct multi-fold cover of the quadratic system and obtain multiple symmetric stable equilibria. We first proposes a quadratic system with only two stable node-foci, which can generate double-wing chaotic hidden attractor. In this work, we aim at designing multi-wing 3D chaotic systems with hidden attractors. This category of chaotic attractors belong to hidden attractor, which is difficult to be located in the phase space. Some dissipative dynamical systems only have stable equilibria or have no equilibrium point but are able to generate chaos of non-Sil’nikov type.

Stable equilibrium 3d