SPROTT ATTRACTOR

Two different pulse control methods are proposed to generate multi-butterfly attractors based on the Sprott C system. By introducing a bipolar multilevel pulse signal to the boostable variable z of the Sprott C system, translational multi-butterfly attractors with constant Lyapunov exponents can be obtained. Another method for generating nested multi-butterfly attractors is to replace the DC voltage of the Sprott C system with the unipolar multilevel pulse signal, and the dynamical behavior involves the superposition of multiple attractors corresponding to different pulse amplitudes. By defining time as an additional state variable, the proposed non-autonomous systems can be transformed into autonomous systems for analysis. It can be seen that systems have no equilibria, so they belong to hidden attractors. A normalized circuit implementation is given, translational and nested hidden multi-butterfly attractors can be obtained by setting proper pulse signals

and a system of three ordinary differential equations describing the structure as-

$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=& y + a x y +x z \\ \frac{dy}{dt}&=& 1 - b x^2 +yz \\ \frac{dz}{dt}&=& x-x^2-y^2 \end{eqnarray*}\right. $

When a = 2.07, b = 1.79, x₀ = 0.63, y₀ = 0.47 and z₀ = -0.54 and , a solution curve of this system has the shape of sphere like structure and the position of particles will follow a similar path.

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