Recurrent solutions of the Lorenz system of differential equations

Abstract

By introducing new variables in the systems of difference equations from the paper [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contrib. Sec. Math. Tech. Sci. 33 (2012) 75–96], new systems of difference equations are obtained. The solutions of these new systems of difference equations depend only on the initial values [Formula: see text] and the coefficients [Formula: see text]. The power series whose coefficients are these solutions present the recurrent solutions for the Lorenz system of differential equations. A comparison between the behavior of the Lorenz system of differential equations and the local behavior of its recurrent solutions for small time steps is examined in examples by computer simulations, as in the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contrib. Sec. Math. Tech. Sci. 33 (2012) 75–96; B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. Fifth Int. Scientific Conf. FMNS - 2013, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107 and B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61].

CMOS Design of Chaotic Systems Using Biquadratic OTA-C Filters

“…The high-level simulation of the chaotic Lorenz system can be performed by using Simulink, whose equations in the frequency domain are given in (2). The block description is sketched in Figure 1, where it can be appreciated that the equations are synthesized using addition, subtraction, multiplication, and gain blocks, while three 1/s integrator blocks are used to obtain each state variable, x, y, and z.…”

“…In this manner, despite its apparent disorder, chaos has at least three remarkable characteristics, i.e., high sensitivity to initial conditions, lack of periodicity, and deterministic behavior, which means that its time evolution is completely determined by its mathematical model. In the continuous-time domain, a chaotic system must have at least three ODEs, as for the case of the Lorenz system [2], which is one of the most famous examples of chaotic systems.…”