The energy conservative splitting FDTD scheme and its energy identities for metamaterial electromagnetic Lorentz system

We expose the chaotic attractors of time-reversed nonlinear system, further implement its behavior on electronic circuit, and apply the pragmatical asymptotically stability theory to strictly prove that the adaptive synchronization of given master and slave systems with uncertain parameters can be achieved. In this paper, the variety chaotic motions of time-reversed Lorentz system are investigated through Lyapunov exponents, phase portraits, and bifurcation diagrams. For further applying the complex signal in secure communication and file encryption, we construct the circuit to show the similar chaotic signal of time-reversed Lorentz system. In addition, pragmatical asymptotically stability theorem and an assumption of equal probability for ergodic initial conditions (Ge et al., 1999, Ge and Yu, 2000, and Matsushima, 1972) are proposed to strictly prove that adaptive control can be accomplished successfully. The current scheme of adaptive control—by traditional Lyapunov stability theorem and Barbalat lemma, which are used to prove the error vector—approaches zero, as time approaches infinity. However, the core question—why the estimated or given parameters also approach to the uncertain parameters—remains without answer. By the new stability theory, those estimated parameters can be proved approaching the uncertain values strictly, and the simulation results are shown in this paper.

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On stochastic properties of dynamic chaos in systems of autonomous differential equations, such as the Lorentz system

Trudy MAI ◽

10.34759/trd-2020-112-1 ◽

2020 ◽

pp. 1-1

Author(s):

Olga Khatuntseva

Keyword(s):

Differential Equations ◽

Dynamic Chaos ◽

Lorentz System ◽

Autonomous Differential Equations ◽

Stochastic Properties

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A Pseudorandom Signal Generator Based on the Lorentz System Subjected to Quasi-Resonant Action

2020 Wave Electronics and its Application in Information and Telecommunication Systems (WECONF) ◽

10.1109/weconf48837.2020.9131530 ◽

2020 ◽

Author(s):

S. S. Loginov ◽

M. Yu. Zuev ◽

Ya. G. Agacheva

Keyword(s):

Signal Generator ◽

Lorentz System ◽

Pseudorandom Signal

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A modified Lorenz system: Definition and solution

Asian-European Journal of Mathematics ◽

10.1142/s1793557120501648 ◽

2020 ◽

Vol 13 (08) ◽

pp. 2050164

Author(s):

Biljana Zlatanovska ◽

Donc̆o Dimovski

Keyword(s):

Differential Equations ◽

Difference Equations ◽

Lorenz System ◽

Local Approximation ◽

System Of Differential Equations ◽

Lorentz System ◽

Constant Coefficients ◽

Linear Differential ◽

The Lorenz System ◽

Fifth Order

Based on the approximations of the Lorenz system of differential equations from the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contributions Sec. Math. Tech. Sci. Manu. XXXIII 1–2 (2012) 75–96, B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. 5th Int. Scientific Conf. FMNS, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107, B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61], we define a Modified Lorenz system, that is a local approximation of the Lorenz system. It is a system of three differential equations, the first two are the same as the first two of the Lorenz system, and the third one is a homogeneous linear differential equation of fifth order with constant coefficients. The solution of this system is based on the results from [D. Dimitrovski and M. Mijatovic, A New Approach to the Theory of Ordinary Differential Equations (Numerus, Skopje, 1995), pp. 23–33].

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