Chaos, Solitons and Fractals 95 (2017) 14–20 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Analysis of spatial chaos appearance in cascade connected nonlinear electrical circuits B. Samardzic a , B.M. Zlatkovic b,∗ a University of Nis, The Faculty of Science and Mathematics in Nis, Visegradska 33, 18000, Serbia b University of Nis, The Faculty of the Occupational Safety of Nis, Carnojevica 10a, 18000, Serbia a r t i c l e i n f o Article history: Received 29 July 2016 Revised 26 September 2016 Accepted 9 December 2016 Keywords: Chaos Cascade-connected nonlinear electrical system Bifurcation diagram Escape-time diagram a b s t r a c t The system consisting of several cascade connected electrical circuits is presented in this paper. Consid- ering the system structure and the fact that the tunnel diodes have nonlinear characteristics, one of the properties of this system is the possibility of the chaos appearance. Necessary conditions and sufficient condition for the chaos appearance in the nonlinear cascade connected systems are given and analyzed, too. The results are confirmed by bifurcation and escape-time diagrams simulation. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction It was noticed, in the last years that in some practical real- ization of cascade connected nonlinear systems, (for example, at systems consisting of several cascade connected transporters for transportation of plastic or rubber strip materials, [1,2]) very com- plex oscillations appear. Under large amplifications these oscilla- tions become complex motions which cannot be described in the classic manner, [3–6]. It was shown that these motions present a type of deterministic chaos and that bifurcations can appear when the control parameter of cascade system varies. Particularly, in this case chaos appearance is not the result of signal iteration through the time, but the signal running through the space, [1,2,7–10]. Mathematical model of these cascade connected nonlinear sys- tems is described as follows: x k+1 = f (x k , r ) (1) where x k is the input, x k +1 is the output, r is the amplification of the kth cascade and f is the nonlinear function which is the same for all cascades. In the Fig. 1 the block scheme of nonlinear cascade system, given by Eq. (1), is shown. With x 1 the input of the first cascade is marked, f = f (x i , r ), i = 1, k is the two argument nonlinear func- tion, x i is the output of the previous cascade and r is the amplifi- ∗ Corresponding author. Fax: 381358228034. E-mail addresses: biljana@pmf.ni.ac.rs (B. Samardzic), bojana.zlatkovic@mts.rs (B.M. Zlatkovic). cation which is the same for all cascades. Therefore, the output of the ith cascade is at the same time the input of the next (i + 1)th cascade. In order to have a chaos appearance in electrical circuits, there must be a nonlinear element in the circuit, i.e., the element with nonlinear current-voltage characteristic, for example nonlinear re- sistor, diode, etc. The simple examples of electric circuits where chaos appears are Van der Pol oscillator, and Chua’s circuit, [6]. Both circuits have one nonlinear element, i.e. nonlinear resistor. Professor L. Chua was the first who proved mathematically the existence of deterministic chaos in the system of differential equa- tions. He, also, proved obtained result using the computer and, at the end, by experiment on the real system. This real system called Chua’s circuit is described by the initial system of differential equa- tions. For the different shapes of the current-voltage characteristic of nonlinear resistor and different values of C 1 , C 2 , R and L, Chua’s circuit has very rich dynamic behavior in parametric plane (α, β ) where α = C 2 /C 1 , β = C 2 R 2 /L. For β = const. and for different val- ues of parameter α, Hopf bifurcation, Rossler attractor etc., appear in circuit. (Figs. 2 and 3) In order to have a chaos appearance in cascade connected non- linear electrical circuit it is necessary that two conditions are sat- isfied: 1 each cascade has one nonlinear element, 2 excitation is in the first cascade and the output of one cascade is the input of the next cascade. http://dx.doi.org/10.1016/j.chaos.2016.12.003 0960-0779/© 2016 Elsevier Ltd. All rights reserved.