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Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes Jiri Petrzela Department of Radio Electronics, FEEC, Brno University of Technology, 601 90 Brno, Czech Republic; [email protected] Received: 20 August 2018; Accepted: 9 September 2018; Published: 12 September 2018







Abstract: This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of MVMS represents autonomous deterministic dynamical system with three degrees of freedom and smooth vector field. Based on the very recent results achieved for piecewise-linear MVMS numerical values of the parameters are calculated such that funnel and double spiral chaotic attractor is observed. Existence of such types of strange attractors is proved both numerically by using concept of the largest Lyapunov exponents (LLE) and experimentally by computer-aided simulation of designed lumped circuit using only commercially available active elements. Keywords: chaos; Lyapunov exponents; multiple-valued; static memory; strange attractors

1. Introduction A general property of chaos is long-time unpredictability; i.e., random-like evolution of dynamical system even if the describing mathematical model does not contain stochastic functions or parameters. Because of its nature, chaotic behavior was often misinterpreted as noise. The first mention of this kind of the complex solution was in [1] where Lorenz noticed the extreme sensitivity of autonomous deterministic dynamics to tiny changes of the initial conditions. After this very milestone, chaos started to be reported in many distinct scientific fields as well as daily life situations. Chaotic motion has been observed in chemical reactions [2], classical mechanics [3], hydrodynamics [4], brain activity [5], models of biological populations [6], economy [7] and, of course, in many lumped circuits. Two basic vector field mechanisms are required for evolution of chaos: stretching and folding. The first mechanism is responsible for exponential divergence of two neighboring state trajectories and second one bounds strange attractor within a finite state space volume. Pioneering work showing the presence of robust chaotic oscillation within dynamics of simple electronic circuit is [8]. So far, the so-called Chua´s oscillator was subject of laboratory demonstrations, deep numerical investigations and many research studies [9–11]. Several interesting strange attractors associated with different vector field local geometries have been localized within the dynamics of three-segment piecewise-linear Chua systems [12–14]. However, the inventors of Chua´s oscillator built it intentionally to construct a vector field capable of generating chaotic waveforms. Progress in computational power together with development of the parallel processing allows chaos localization in standard functional blocks of radiofrequency subsystems such as in harmonic oscillators [15,16], frequency filters [17,18], phase-locked loops [19], power [20] and dc-dc [21] converters, etc. From a practical point of view, chaos represents an unwanted operational regime that needs to be avoided. It can be recognized among

Entropy 2018, 20, 697; doi:10.3390/e20090697

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regular behavior because of the specific features in the frequency domain: continuous and broadband It can be recognized among regular behavior because of the specific features in the frequency domain: frequency spectrum. However, an approach that is more sophisticated is to derive a set of describing continuous and broadband frequency spectrum. However, an approach that is more sophisticated is differential equations and utilize the concept of LLE to find regions of chaotic solutions [22]. to derive a set of describing differential equations and utilize the concept of LLE to find regions of Searching for chaos in a mathematical model that describes simplified real electronic memory chaotic solutions [22]. block is also topic of this paper. Three programs were utilized for the numerical analysis of MVMS: Searching for chaos in a mathematical model that describes simplified real electronic memory Matlab 2015 for the search-through-optimization algorithm including CUDA-based parallelization, block is also topic of this paper. Three programs were utilized for the numerical analysis of MVMS: Mathcad 15 for graphical visualization of results and Orcad Pspice 16 for circuit verification. The Matlab 2015 for the search-through-optimization algorithm including CUDA-based parallelization, content of this paper is divided into four sections with the logical sequence: model description, Mathcad 15 for graphical visualization of results and Orcad Pspice 16 for circuit verification. The numerical analysis, circuit realization and verification; both through simulation and measurement. content of this paper is divided into four sections with the logical sequence: model description, numerical analysis, circuit realization and verification; both through simulation and measurement. 2. Dynamical Model of Fundamental MVMS Basic mathematical model of MVMS [23,24] is given in Figure 1 and can be described by three 2. Dynamical Model of Fundamental MVMS first-order ordinary differential equations in the following form: Basic mathematical model of MVMS [23,24] is given in Figure 1 and can be described by three dvdifferential equations dv2in the following form: di first-order ordinary C1 1 = i − f 1 (v1 ) C2 = i − f 2 (v2 ) L = vbias − v1 − v2 − R·i (1) dt dt dt 𝑑𝑣1 𝑑𝑣2 𝑑𝑖 (1) 𝐶1 = 𝑖 − 𝑓1 (𝑣1 ) 𝐶2 = 𝑖 − 𝑓 (𝑣2 ) 𝐿 = 𝑉𝑏𝑖𝑎𝑠 − 𝑣1 − 𝑣2 − 𝑅 ∙ 𝑖 𝑑𝑡 𝑑𝑡1 and C2 2is parasitic 𝑑𝑡 where the state vector is x = (v1 , v2 , i)T , C capacitance of first and second RTD, L and Rthe is state summarized lead inductance resistance, respectively. where vector is(RTDs x = (v1are , v2,connected i)T, C1 andin C2series) is parasitic capacitanceand of first and second RTD, L Details modeling(RTDs of high RTD including valuesand of parasitic elements can be and R isabout summarized arefrequency connected in series) leadtypical inductance resistance, respectively. found in [25]. We can express both nonlinear functions (for k = 1, 2) as: Details about modeling of high frequency RTD including typical values of parasitic elements can be found in [25]. We can express both nonlinear functions (for k = 1, 2) as: f k ( x ) = a k ( x − d k ) 3 + bk ( x − d k ) + c k 𝑓𝑘 (𝑥) = 𝑎𝑘 (𝑥 − 𝑑𝑘 )3 + 𝑏𝑘 (𝑥 − 𝑑𝑘 ) + 𝑐𝑘 .

(2) (2)

Figure Figure 1. 1. Basic Basic network network configurations configurations of of MVMS: MVMS: original original (left (left schematic), schematic), dual dual (right (right schematic). schematic).

Thus, AVC AVC of each RTD is a cubic polynomial that should form an N-type curve with a negative segment. dx/dt ==0.0.For segment. Fixed Fixed points points xee are are all all solutions solutions of of the the problem problem dx/dt Forfurther furthersimplicity, simplicity, let’s let’s assume assume that R == 00 Ω. We can determine global conditions and position for its existence within state space Ω. We determine global conditions and position for its existence within state space as as each namely: each solution solution of of system system of of the the nonlinear nonlinear algebraic algebraic equations, equations, namely: 𝑎1 (𝑥𝑒 − 𝑑1 )3 + 𝑏1 (𝑥𝑒 − 𝑑1 ) + 𝑐1 = 𝑎2 (𝑉𝑏𝑖𝑎𝑠 − 𝑥𝑒 − 𝑑2 )3 + 𝑏2 (𝑉𝑏𝑖𝑎𝑠 − 𝑥𝑒 − 𝑑2 ) + 𝑐2 a1 ( xe − d1 )3 + b1 ( xe − d1 ) + c1 = a2 (Vbias − xe − d2 )3 + b2 (Vbias − xe − d2 ) + c2 ye𝑦𝑒==V𝑉 −−xe𝑥𝑒 𝑏𝑖𝑎𝑠 bias 3 3 b1 ( x(𝑥 ze𝑧 ==a1𝑎( x(𝑥 d1𝑑) )+ d1𝑑) + c1𝑐 e −− )+ + 𝑏 e −− 𝑒

1

𝑒

1

1

𝑒

1

(3) (3)

1

Vector field fieldgeometry geometrydepends dependson onthe theeigenvalues; eigenvalues; i.e., roots a characteristic polynomial. It i.e., roots of aofcharacteristic polynomial. It can can be calculated as det(s· = 0 where is the unity matrix J is Jacobi the Jacobi matrix: be calculated as det(s ·E–J)E–J) = 0 where E is Ethe unity matrix andand J is the matrix:

 −𝑏1 − 3𝑎1 (𝑥𝑒 − 𝑑12)2 0 1 − b − 3a ( x − d ) 0 1 2 e 1 1 1 𝐉 =( ). 0 −𝑏2 − 3𝑎2 (𝑦𝑒 − 𝑑22) 1 J= 0−1 −b2 − 3a2 −1 (ye − d2 ) 1 0. −1 −1 0 Characteristic polynomial in symbolical form becomes: 2 𝑠 3 + {3𝑎2 𝑉𝑏𝑖𝑎𝑠 − 6𝑎2 𝑉𝑏𝑖𝑎𝑠 (𝑥𝑒 + 𝑑2 ) + 𝑏1 + 𝑏2 + 3𝑎1 (𝑥𝑒2 + 𝑑12 ) + 3𝑎2 (𝑥𝑒2 + 𝑑22 ) − 2 6𝑥𝑒 (𝑎1 𝑑1 − 𝑎2 𝑑2 )}𝑠 2 + {[𝑎1 𝑎2 (9𝑑12 − 18𝑑1 𝑥𝑒 + 9𝑥𝑒2 ) + 3𝑏1 𝑎2 ]𝑉𝑏𝑖𝑎𝑠 + [𝑎1 𝑎2 (36𝑑1 𝑥𝑒2 −

(4) (4)

(5)

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Characteristic polynomial in symbolical form becomes: Entropy 2018, 20, 697

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Characteristic polynomial in symbolical form becomes: 2 𝑠 3 + {3𝑎2 𝑉𝑏𝑖𝑎𝑠 − 6𝑎2 𝑉𝑏𝑖𝑎𝑠 (𝑥𝑒 + 𝑑2 ) + 𝑏1 + 𝑏2 + 3𝑎1 (𝑥𝑒2 + 𝑑12 ) + 3𝑎2 (𝑥𝑒2 + 𝑑22 ) − 2 6𝑥𝑒 (𝑎1 𝑑1 − 𝑎2 𝑑2 )}𝑠 2 + {[𝑎1 𝑎2 (9𝑑12 − 18𝑑1 𝑥𝑒 + 9𝑥𝑒2 ) + 3𝑏1 𝑎2 ]𝑉𝑏𝑖𝑎𝑠 + [𝑎1 𝑎2 (36𝑑1 𝑥𝑒2 − 18𝑑12 𝑑2 − 18𝑑12 𝑥𝑒 + 36𝑑1 𝑑2 𝑥𝑒 − 18𝑥𝑒3 − 18𝑑2 𝑥𝑒2 ) − 6𝑏1 𝑎2 𝑥𝑒 − 6𝑏1 𝑎2 𝑑2 ]𝑉𝑏𝑖𝑎𝑠 + 𝑏1 𝑏2 + 3𝑎1 𝑑12 𝑏2 + 3𝑏1 𝑎2 𝑑22 + 3𝑎1 𝑏2 𝑥𝑒2 + 3𝑏1 𝑎2 𝑥𝑒2 − 6𝑎1 𝑑1 𝑏2 𝑥𝑒 + 6𝑏1 𝑎2 𝑑2 𝑥𝑒 + 𝑎1 𝑎2 (9𝑥𝑒4 − 18𝑑1 𝑥𝑒3 − 18𝑑2 𝑥𝑒3 + 9𝑑12 𝑑22 + 9𝑑12 𝑥𝑒2 + 9𝑑22 𝑥𝑒2 − 36𝑑1 𝑑2 𝑥𝑒2 − 18𝑑1 𝑑22 𝑥𝑒 + 18𝑑12 𝑑2 𝑥𝑒 ) + 2 2}𝑠 + 𝑏1 + 𝑏2 + 3𝑉𝑏𝑖𝑎𝑠 𝑎2 + 3𝑎1 𝑑12 + 3𝑎2 𝑑22 + 3𝑥𝑒2 (𝑎1 + 𝑎2 ) − 6𝑉𝑏𝑖𝑎𝑠 𝑎2 (𝑑2 + 𝑎2 𝑥𝑒 ) − 6𝑥𝑒 (𝑎1 𝑑1 − 𝑎2 𝑑2 ) = 0,

(5)(5)

Obviously, symbolical expressions for the individual eigenvalues are very complicated and cannot further contribute to the better understanding of a vector field configuration and chaos evolution; check well-known Cardan rules. In situation, where xe and ye coordinate of equilibrium point is close to offset voltages represented by d1 and d2 , characteristic polynomial simplifies into the relation: s3 + (b1 + b2 )s2 + (b1 b2 + 2)s + b1 + b2 = 0, (6) Obviously, symbolical expressions for the individual eigenvalues are very complicated and cannot further contribute to the better understanding of a vector field configuration and chaos and the eigenvalues depend only on linear part of polynomial approximation of AVCs of RTDs. evolution; check well-known Cardan rules. In situation, where xe and ye coordinate of equilibrium Until very recently, analysis of MVMS was focused only on a high-frequency modeling of RTDs, point is close to offset voltages represented by d1 and d2, characteristic polynomial simplifies into the influence of a pulse driving force on overall stability [26] and global dynamics [27] and specification of relation: the boundary planes [28]. However, existence of chaos has not been uncovered and examined. 𝑠 3 + (𝑏1 + 𝑏2 )𝑠 2 + (𝑏1 𝑏2 + 2)𝑠 + 𝑏1 + 𝑏2 = 0, (6) 3. Numerical Results and Discussion and the eigenvalues depend only on linear part of polynomial approximation of AVCs of RTDs. Numerical values analysis of MVMSofparameters be obtained optimizationmodeling techniqueofdescribed Until very recently, MVMS wascan focused only onbya the high-frequency RTDs, in [29]. In this case, the mathematical model of MVMS was considered piecewise-linear. Such kind influence of a pulse driving force on overall stability [26] and global dynamics [27] and specification vector field allows better understanding of chaos chaoshas evolution, partial solution ofof thea boundary planes [28]. However, existence of not beenallows uncovered andanalytic examined. and makes linear analysis generally more powerful. However, situation when AVCs of both RTDs approximated the polynomial functions is closer to reality. Thus, our problem stands as 3. are Numerical Resultsby and Discussion follows: couple of three-segment piecewise-linear functions needs to be transformed into the cubic values of MVMS polynomial parameters can be obtained by the optimization technique described (or Numerical higher-order if necessary) functions without losing robust chaotic solution having innumerically [29]. In this case, the mathematical model of MVMS was considered piecewise-linear. Such kind of close metric fractal dimension (Kaplan-Yorke is preferred over capacity because of rapid a and vector field allows better understanding of chaos evolution, allows partial analytic solution and precise calculation). For more details, readers should consult [30] where the inverse problem has makes linear analysis generally more powerful. situationRossler when attractor AVCs of[31] both RTDs are been successfully solved. Finally, chaotic attractorHowever, like the so-called was localized. approximated bywithin the polynomial functions is closer reality. Thus, our problem stands as follows: Searching smooth vector field (1) and bytoconsidering default normalized values C1 = 11 F, couple of three-segment piecewise-linear functions needs to be transformed into the cubic (or higherC2 = 37 F, L = 100 mH and R = 0 Ω leads to the following optimal values of the cubic polynomials (2): order if necessary) polynomial functions without losing robust chaotic solution having numerically close fractal preferred over capacity because a1 metric = 648.5, b1 =dimension −23.1, c1(Kaplan-Yorke = −0.13, d1 =is0.3, a2 = 58.1, b2 = −21.9, c2 =of−rapid 0.65, and d2 =precise 0.5. (7) calculation). For more details, readers should consult [30] where the inverse problem has been successfully solved. Finally, attractor like the so-callednumerical Rossler attractor [31] process was localized. Adopting these valueschaotic and fourth-order Runge-Kutta integration we get the Searching within smooth vector (1)2.and by considering default(3)normalized values C1points: = 11 reference chaotic orbits provided in field Figure Accordingly to Equation we have three fixed T and characterized F,first C2 =located 37 F, L =in100 mH and = 0 Ω 0.704, leads to the following optimal values of the cubic λ polynomials position xe =R(0.046, −4.768) by eigenvalues = − 101.241, 1 (2): λ2 = 0.062, λ3 = 13.915, second equilibria in position xe = (0.292, 0.458, 0.043)T having set of the eigenvalues λ1 = 0.09, λ2 = 22.97, λ3 = 21.72 and xe = (0.55, 0.2, 4.299)T with local behavior given by 𝑎1 = 648.5, 𝑏1 = −23.1, 𝑐1 = −0.13, 𝑑1 = 0.3, 𝑎2 = 58.1, 𝑏2 = −21.9, 𝑐2 = −0.65, 𝑑2 = 0.5. (7) eigenvalues λ1 = −99, λ2 = 0.13, λ3 = 7.146. This is an interesting geometric configuration of the vector Adopting these values fourth-order Runge-Kutta numerical integration process weindex get the field: a funnel-type strangeand attractor generated by saddle-node equilibria with the stability 1, 0 reference chaotic orbits provided in Figure 2. Accordingly to optimization Equation (3) we have three points:to and 0; saddle-focuses are completely missing. Developed algorithm canfixed be utilized first located unpredictability in position xe = (0.046, 0.704, −4.768) and increase characterized byentropy. eigenvalues λ1 =with −101.241, maximize of a dynamical flowT and system Starting valuesλ2(7) T = 0.062, λ3 = 13.915, second equilibria in position xe = (0.292, 0.458, 0.043) having set of the eigenvalues the following set of numerical values was obtained: λ1 = 0.09, λ2 = 22.97, λ3 = 21.72 and xe = (0.55, 0.2, 4.299)T with local behavior given by eigenvalues λ1 = 680, = −23, 0.13, d1 =geometric 0.3, a2 = configuration 55, b2 = −25,ofcthe −0.65,field: d2 =a0.5, = −99, λ2a=1 0.13, λ3 =b17.146. Thisc1is=an−interesting funnel-(8) 2 = vector

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leading to a double-hook [12] like chaotic attractor visualized by means of Figure 3. The position of the first fixed point changes slightly to xe = (0.045, 0.705, −5.48)T as well as the corresponding eigenvalues λ1 = −109.037, λ2 = 0.049, λ3 = 13.915. The second equilibrium moves into position xe = (0.29, 0.46, 0.089)T and possesses the set of eigenvalues λ1 = 0.084, λ2 = 22.764, λ3 = 24.817. Finally, the last fixed point moves towards position xe = (0.554, 0.196, 5.31)T where the local behavior will be given by the eigenvalues λ1 = −109.953, λ2 = 0.085, λ3 = 10.596. The mentioned equilibria together with important state space sections are provided in Figure 4. Note that local geometries of a4 vector field Entropy 2018, 20, x of 23 are not affected is 23 given by Entropy 2018,since, 20, x in a closed neighborhood of the fixed points, dynamical movement 4 of are not affected a closed of the fixed points, dynamical movement is givenofbythe flow three eigenvectors as since, in theincase of aneighborhood funnel chaotic attractor. Also note that one direction are not affected since, inthe a closed neighborhood of the fixed points, dynamical is given by three eigenvectors as in casewith of a funnel attractor. Also note onemovement direction of the flow (along the eigenvector associated λ1 ) is chaotic strongly attracting; thisthat nature is evident from Figure 5. three eigenvectors as inassociated the case ofwith a funnel attractor. Also note that is one direction ofFigure the flow (along the eigenvector λ1) ischaotic strongly attracting; this nature evident from 5. Interesting fragments of the bifurcation diagrams are depicted in Figure 6. (along the eigenvector associated with λ1) is strongly attracting; this nature is evident from Figure 5. Interesting fragments of the bifurcation diagrams are depicted in Figure 6. Interesting fragments of the bifurcation diagrams are depicted in Figure 6.

Figure 2. Three-dimensional perspective viewson on aa typical attractor generated by polynomial Figure 2. Three-dimensional perspective views typicalchaotic chaotic attractor generated by polynomial T; Poincaré Figure 2. perspective a typical sections chaotic attractor polynomial MVMS forThree-dimensional the initial conditions x0 = (0.1,views 0.3, 0)on in planesgenerated shifted bybythe offsets of T MVMS for the initial conditions x0 = (0.1, 0.3, 0)T ; Poincaré sections in planes shifted by the offsets MVMS for the initial conditions 0.3, 0) fixed ; Poincaré in (blue planesdots), shifted byspace the offsets of polynomial functions: x = c1 (red),xy0 == c(0.1, 2 (yellow); point sections of the flow state rotation of polynomial functions: xx==system). (red), c2 (yellow); fixed of(blue the flow (blue dots), state space polynomial functions: cc11(red), yNumerical =yc2=(yellow); fixed point of point the flow dots), state space rotation (no deformations of axis integration with final time 10 4 and time step 0.01. 4 and time step 0.01. rotation(no (nodeformations deformations of axis system). Numerical integration time of axis system). Numerical integration with finalwith time final 10 4 and time10step 0.01.

Figure 3. Three-dimensional perspective views on chaotic attractor with increased entropy generated Figure 3. Three-dimensional views xon with increased generated by polynomial MVMS forperspective theperspective initial conditions 0 =chaotic (0.1, 0.3,attractor 0)T; Poincaré sections inentropy planes shifted by Figure 3. Three-dimensional views on chaotic attractor with increased entropy generated T; Poincaré sections in planes shifted by by polynomial MVMSfunctions: for the initial (0.1, 0.3, 0) offsets of polynomial x = conditions c1 (red), y =x0c2= (yellow); fixed point of the flow (blue dots), state T by polynomial MVMS for the initial conditions x0 = (0.1, 0.3, 0) ; Poincaré sections in planes shifted by offsetsrotation of polynomial functions: of x =axis c1 (red), y = Calculation c2 (yellow); fixed pointtime of the (bluestep dots), state space (no deformations system). with final 10 4flow and time 0.01. offsets of polynomial functions: x = c (red), y = c (yellow); fixed point of the flow (blue dots), state 4 2 space rotation (no deformations of 1axis system). Calculation with final time 10 and time step 0.01.

space rotation (no deformations of axis system). Calculation with final time 104 and time step 0.01.

Figure 3. Three-dimensional perspective views on chaotic attractor with increased entropy generated by polynomial MVMS for the initial conditions x0 = (0.1, 0.3, 0)T; Poincaré sections in planes shifted by Entropy 2018, 20, 697 offsets of polynomial functions: x = c1 (red), y = c2 (yellow); fixed point of the flow (blue dots), state space rotation (no deformations of axis system). Calculation with final time 10 4 and time step 0.01.

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Figure 4. Important geometric structures located located within within state state space: space: vv111–v222 plane plane projection projection of function f111(v 1 ) (orange), function f 2 (v 2 ) (green) and v 1 = V bias − v 2 (blue line), intersections of these planes are fixed 2 2 ) (green) and1 v1 = bias (v11 ) (orange), function2f 2 (v Vbias2 − v2 (blue line), intersections of these planes are points of dynamical flow. flow. Discovered chaotic attractor put into of vector field. field. fixed points of dynamical Discovered chaotic attractor putthe intocontext the context of vector

Figure Figure 5. 5. Graphical Graphical visualization visualization of of aa dynamical dynamical behavior behavior near near the the fixed fixed points points (red (red dots): dots): equilibrium equilibrium xeee ==0.046 0.046(left (lefttwo two images), images), for for fixed fixed point point located located at at xeee ==0.292 0.292(middle (middle two two plots) plots) and and fixed fixed point point with with xeee ==0.55 (right two state portraits). 0.55 (right two state portraits).

Figure 6. Gallery Figure 6. Gallery of of one-dimensional one-dimensional bifurcation bifurcation diagrams diagrams calculated calculated with with respect respect to to the the polynomial polynomial of RTDs; RTDs; individual individual plots plots from from left left to to right: right: horizontal horizontal axis represented by approximation approximation of AVCs of 1 ∈ parameter range rangea1a1∈ (600, calculated with∆step = 0.1; parameter range 11 ∈ (−30, 0) parameter (600, 700)700) calculated with step = 0.1;Δparameter range b1 ∈ (−30,b0) established 1 established with step Δ = 0.01; parameter range c 1 ∈ (−1.5, 0) together with step Δ = 0.001; parameter with step ∆ = 0.01; parameter range c1 ∈ (−1.5, 0) together with step ∆ = 0.001; parameter a2 ∈ (30, 100) a22 ∈ step (30, 100) withparameter step Δ = 0.1; parameter b22 ∈ (−25, −20) = 0.01 andc2parameter with ∆ = 0.1; b2 ∈ (−25, −20) together withtogether step ∆ =with 0.01step andΔparameter ∈ (−4, 0) c22 ∈ step (−4, 0) with with ∆= 0.01.step Δ = 0.01.

Here, transient motion has been removed and plane x = d11 has been used for individual slices. These are plotted against parameters akk, bkk and ckk of polynomial approximations of AVC of k-th RTD. Parameters a11 and a22 can burn or bury chaotic attractor since these affect eigenvalues in “outer” parts of the active vector field (given by size of the state attractor) while geometry around “middle” fixed point remains almost unchanged. Figure 7 shows numerical calculation of a gained energy with small time step with respect to the state space location; red regions mark large increment while dark blue stands for a small evolution. As stated before, chaotic solution is sensitive to the changes of parameters akk, bkk, ckk and dkk, where k = 1, 2. If we consider these values variable we create eight-dimensional hyperspace in which chaos

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Here, transient motion has been removed and plane x = d1 has been used for individual slices. These are plotted against parameters ak , bk and ck of polynomial approximations of AVC of k-th RTD. Parameters a1 and a2 can burn or bury chaotic attractor since these affect eigenvalues in “outer” parts of the active vector field (given by size of the state attractor) while geometry around “middle” fixed point remains almost unchanged. Figure 7 shows numerical calculation of a gained energy with small time step with respect to the state space location; red regions mark large increment while dark blue stands for a small evolution. Entropy 2018, 20, x 6 of 23

Figure Figure7.7.Rainbow-scaled Rainbow-scaledcontour contour plots plots of ofshort-time short-timeevolution evolution of ofMVMS MVMSenergy energyfor forthe thestate statespace space slices 5, zz00 = −4, −4,zz0 0==−3, −3, = −z02,=z−1, z0z=0 =0,1,z0z0==1,2,zz00== 2, slicesforming formingunity unitycube cube(left (lefttotoright): right):z0z0==−−5, z0 z=0−2, 0 1, = 0, 3, 0 = z− zz00 == 3, 4, z0 = 5. 4, z00 == 5.

As stated before, chaotic solution is sensitive to the of for parameters ak , bk , ck andKaplan–Yorke dk , where k = Maximal merit of LLE is 0.089 for a value set (7) changes and 0.103 (8) and associated 1,dimension 2. If we consider these values variable we create eight-dimensional hyperspace in which chaos alternates [33] is 2.016 and 2.021 respectively. Specification of sufficiently large “chaotic” area is with periodic orbits or practical fixed-point solution. Two-dimensional subspaces important also from viewpoint; as will be clarified later. hewed-out from such hyperspace are provided in Figure 8. Each graph is composed of 101 × 101 = 10,201 points, calculation routine deals with time interval t ∈ (100, 104 ), random initial conditions inside basin of attraction and Gram-Smith orthogonalization [32]. Here, dark blue represents a trivial solution, light blue a limit cycle, green color stands for weak chaos and yellow marks strong chaotic behavior. Note that LLE for set of values (7) can be found in each visualized plot. Maximal merit of LLE is 0.089 for a value set (7) and 0.103 for (8) and associated Kaplan–Yorke dimension [33] is 2.016 and 2.021 respectively. Specification of sufficiently large “chaotic” area is important also from practical viewpoint; as will be clarified later.

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Figure 8. Gallery of the rainbow-scaled surface-contour plots of LLE as functions of two parameters; vertical range dedicated for LLE is −0.004 to 0.103, counting of plots from left to right and up to down: a1 –b1 , a1 –c1 , a1 –a2 , a1 –b2 , a1 –c2 , b1 –c1 , b1 –a2 , b1 –b2 , b1 –c2 , c1 –a2 , c1 –b2 , c1 –c2 , a2 –b2 , a2 –c2 , b2 –c2 .

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Figure 8. Gallery of the rainbow-scaled surface-contour plots of LLE as functions of two parameters; Entropy 2018,range 20, 697dedicated for LLE is −0.004 to 0.103, counting of plots from left to right and up to down: 8 of 23 vertical

a1–b1, a1–c1, a1–a2, a1–b2, a1–c2, b1–c1, b1–a2, b1–b2, b1–c2, c1–a2, c1–b2, c1–c2, a2–b2, a2–c2, b2–c2.

Circuitry Realization MVMS-Based Chaotic Oscillators 4. 4. Circuitry Realization ofof MVMS-Based Chaotic Oscillators Design analog equivalent circuit common way how prove existence structurally stable Design ofof analog equivalent circuit is is common way how to to prove existence of of structurally stable strange attractors within the dynamics of a prescribed set of ordinary differential equations. Realization strange attractors within the dynamics of a prescribed set of ordinary differential equations. of such the chaotic oscillators is a simple straightforward task thattask wethat can we solve Realization of so-called such the so-called chaotic oscillators is aand simple and straightforward canby using several approaches [34–38], both using discrete components and in integrated form. A favorite solve by using several approaches [34–38], both using discrete components and in integrated form. allows to utilize availableavailable active elements follows follows the concept of analog A method favorite that method thatusallows us tocommercially utilize commercially active elements the concept Thus, only three building blocks are required for circuit inverting summing ofcomputers. analog computers. Thus, only three building blocks are required forconstruction: circuit construction: inverting integrator, summing amplifier and, in the case of a polynomial nonlinearity, four-segment analog summing integrator, summing amplifier and, in the case of a polynomial nonlinearity, four-segment multiplier. Fully analog circuit implementation is shown in Figure 9. analog multiplier. Fully analog circuit implementation is shown in Figure 9.

Figure 9. 9. Chaotic oscillator designed to to audio band based onon integrator block schematic associated Figure Chaotic oscillator designed audio band based integrator block schematic associated with mathematical model of MVMS, numerical values of passive network components are included. with mathematical model of MVMS, numerical values of passive network components are included.

Note that this circuit synthesis requires many active devices: two TL084, two AD844 and a single Note that this circuit synthesis requires many active devices: two TL084, two AD844 and a single four channel four quadrant analog multiplier MLT04. VV but, forfor four channel four quadrant analog multiplier MLT04.Supply Supplyvoltage voltageisissymmetrical symmetrical±15 ±15 but, MLT04, this voltage is lowered to ±5 of theofanalog realizations of the of chaotic systemssystems with MLT04, this voltage is lowered toV. ±Majority 5 V. Majority the analog realizations the chaotic a polynomial nonlinearity utilize AD633, single channel It is possible in ouralso case in with a polynomial nonlinearity utilize i.e., AD633, i.e., singlemultiplier. channel multiplier. It isalso possible but with cost of the eight active devices. Dynamical range for correct operation MLT04 of is only ±2 is our casethe but with cost of eight active devices. Dynamical range for correct of operation MLT04 V.only However, prescribedprescribed strange attractor smaller in v1–v2 dimension. Advantage Advantage of this circuit is ±2 V. However, strangeisattractor is smaller in v1 –v2 dimension. of this that individual MVMS parameters can becan adjusted independently using potentiometers. circuit is that individual MVMS parameters be adjusted independently using potentiometers. Theoretically, using different decomposition of the polynomial functions, total number of the active

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elements can be lowered to four. Proposed chaotic oscillator is uniquely by the following Theoretically, using different decomposition of the polynomial functions,described total number of the active set of the differential elements canequations: be lowered to four. Proposed chaotic oscillator is uniquely described by the following set of the differential equations: 







   3   Rb Rd dv1 v3 Rz3 Rb 3 11 1 2 v − 𝑅 Rd V 1 1 𝑉− Vc1 𝑑𝑣 1 𝑣 𝑅 𝑅 𝑅 𝑅 = − − − v − V − K 1 3 𝑏 𝑑 𝑧3 𝑏 𝑑 𝑐1 cc cc 1 1 R C2− R dt 1 = −a (𝑣 −Rd + Rc 𝑉 ){R8 − Rz4 ( Ra) 𝐾 2 (𝑣1 − Rd + Rc𝑉𝑐𝑐 ) R}7−  ] R x   𝐶2 [− 𝑑𝑡 𝑅1 𝑅𝑎 1 𝑅𝑑 + 𝑅𝑐 𝑐𝑐 𝑅8 𝑅𝑧4  𝑅𝑎 3  𝑅𝑑 + 𝑅𝑐 𝑅7 𝑅𝑥  R Rh Rz1 R f Vc2 dv2 1 1 2 = − C13 − Rv32 − Ref v2 − RhR+hRg Vcc R110 −𝑅Rz2 𝑅 Re 3 K v2 − 𝑅 Rh + R g Vcc 1 R9 𝑉− Ry dt 𝑑𝑣 𝑅 1 𝑣 𝑅 ℎ 𝑐2  3 − 𝑓 (𝑣  − iℎ 𝑉𝑐𝑐 ) { − 𝑧1 ( 𝑓 ) 𝐾 2 (𝑣2 − =h− [− 𝑉 ) }− ] 𝑑𝑡 1 −𝐶R35 v𝑅12 + 𝑅v𝑒2 2+ V𝑅bias 𝑅10 𝑅𝑧2 𝑅𝑒 𝑅ℎ + 𝑅𝑔 𝑐𝑐 𝑅9 𝑅𝑦 ℎ + 𝑅𝑔 = R6 R R5 R C 2

dv3 dt

1

4

(9)

(9)

bias

𝑑𝑣3 1 𝑅5 𝑣1 𝑣2 𝑉𝑏𝑖𝑎𝑠 = [− ( + ) + ] 𝑑𝑡 𝐶dc 𝑅 𝑅 𝑅 𝑅 where VC1 , VC2 and Vbias are independent voltage sources and 1 6 4 5 𝑏𝑖𝑎𝑠 K = 0.4 is internally-trimmed scaling

constant of the cells MLT04. Fundamental time of this chaotic scaling oscillator is where VC1,analog VC2 andmultiplier Vbias are independent dc voltage sources and K =constant 0.4 is internally-trimmed chosenconstant to be τ of = the R·Cv = 103multiplier ·10−7 = 100 analog cellsµs. MLT04. Fundamental time constant of this chaotic oscillator is to be τ =also R·Cvpossible = 103·10−7to= 100 μs.both analog networks provided in Figure 1 directly. Instead Ofchosen course, it is build Of course, it is also possible to build both analog networks provided in Figure 1 directly. Insteaddesign of nonlinear two-ports we must construct a couple of resistors with polynomial AVC; systematic of nonlinear two-ports we must construct a couple of resistors with polynomial AVC; systematic towards these network elements can be found in [39,40]. Circuitry realization of original MVMS with design towards these network elements can be found in [39,40]. Circuitry realization of original state vector x = (v1 , v2 , i)T is demonstrated by means of Figure 10; i.e., the state variables are voltages MVMS with state vector x = (v1, v2, i)T is demonstrated by means of Figure 10; i.e., the state variables across are grounded capacitors and current flowing through the inductor. Note that both polynomial voltages across grounded capacitors and current flowing through the inductor. Note that both function (2) need to be rewritten into the form i = f (v) = ai·v=3f(v) + b=·va·2v3++ cb··vv2 ++c· d.v +Thus, a new polynomial function (2) need to be rewritten into the form d. Thus, a newset setof the dimensionless ordinary differential equations to be implemented as lumped analog electronic of the dimensionless ordinary differential equations to be implemented as lumped analog electroniccircuit circuit as follows: as follows: 𝑑𝑥 𝑑𝑦 dy 33 dx = −𝑧 − 648 ∙ 𝑥 33 − 583 ∙ 𝑥 22 − 152 ∙ 𝑥 − 10.7 ==𝑧 z−− 5858 ∙ 𝑦·y ++ 8787 ∙ 𝑦·y2 2−−2020∙ 𝑦 = −z − 648· x − 583· x − 152· x − 10.7 ·y−−3.3 3.3 dt 𝑑𝑡 𝑑𝑡dt dz (10) = x − y + 0.75 dt 𝑑𝑧 𝑑𝑡

= 𝑥 − 𝑦 + 0.75

Figure 10. Chaotic system obtained directly from fundamental MVMS with ideal multipliers and ideal

Figure 10. Chaotic system obtained directly from fundamental MVMS with ideal multipliers and ideal second-generation current-conveyors, numerical values of the circuit components are included. second-generation current-conveyors, numerical values of the circuit components are included.

(10)

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Now assume that impedance and frequency norm are 105 and 104 , respectively. Such values lead to the nominal inductance 1H. This simplified concept of chaotic oscillators is given in Figure 10 and described by following set of the ordinary differential equations: 1 C1 dv dt + i L +

v31 R a1

+

v21 R a2

+

v1 R a3

+

VX R a4

=0

v32 v22 VY v2 2 C2 dv dt + Rb1 + Rb3 + Rb4 = i L + Rb2 L didtL + RS i L + v2 = v1 + Vbias

(11)

where a small value RS can still model lead to resistances of both RTDs that are parts of MVMS in Figure 1. Note that this kind of realization utilizes second generation current conveyors implemented by using ideal voltage-controlled voltage-source E and current-controlled current-source F. A positive variant of this active three-port element is commercially available as the AD844 while a negative variant is EL2082 (only one negative device is required). In practice, the inductor should be substituted by the synthetic equivalent; i.e., active floating gyrator (Antoniou’s sub-circuit) with a capacitive load, check Figure 11. In this case, the number of the active elements raises to eight: a single TL084, six AD844s and a single MLT04. Dynamical behavior is uniquely determined by the following mathematical model: dv1 dt dv2 dt

=

1 C1

=

1 C2



−i −

K2 v31 R a1 + Rin

−i −

K2 v32 Rb1 + Rin

 di dt

=

−

Kv21 R a2 + Rin

−

Kv22 Rb2 + Rin

R g2 R g1 R g3 R g4 Cg1 [ v1

−

v1 R a3

−

v2 Rb3

−

Vx R a4 + Rin

−

Vy Rb4 + Rin

 

(12)

− v2 + Vbias ]

where Rin represents input resistance of current input terminal of AD844. Its typical value is 50 Ω and, due to the high values of the polynomial coefficients and in the case of small impedance norm chosen, it cannot be generally neglected. On the other hand, we must avoid output current saturation of each AD844 and consider frequency limitations of each active device. Thus, choice of both normalization factors is always a compromise. Thanks to the symmetry inside floating Antoniou’s structure Rg5 = Rg3 , Rg7 = Rg1 , Rg6 = Rg2 and Cg2 = Cg1 . Behavior of this chaotic oscillator is extremely sensitive to the working resistors connected in the nonlinear two-terminal devices. Thus, calculated values were specified by Orcad Pspice optimizer where fitness functions (several should be defined to create tolerance channel) are absolute difference between polynomials in (11) and actual input resistance of designed circuit. Corresponding dc sweep analysis were estimated for input voltages from 0 V to 2 V with step 10 mV.

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Figure 11. 11. Chaotic Chaotic system directly from from fundamental fundamental MVMS MVMS network network with with floating floating synthetic synthetic Figure system obtained obtained directly inductor, numerical values of the passive circuit components are included, ready for verification. inductor, numerical values of the passive circuit components are included, ready for verification.

Last Last circuitry circuitry implementation implementation is provided provided in in Figure Figure 12; 12; namely namely dual dual network network to to the the original original MVMS. MVMS. Impedance Impedance norm norm is is chosen chosen to to be be 10 1033 and and frequency frequency normalization normalization factor factor is is 10 1055.. To get get the the reasonable values of the resistors further impedance rescaling is possible. Set of describing differential equations can be expressed as follows: (𝑟𝑇1 𝑖𝐿1 )3 𝑟𝑇1 𝑖𝐿1 𝑉1 (𝑟𝑇1 𝑖𝐿1 )2 𝑑𝑖𝐿1 1 = [−𝑅𝑠1 𝑖𝐿1 + 𝑟𝑇2 { + − } − 𝑟𝑇3 ] 𝑑𝑡 𝐿1 𝑅1 𝑅3 𝑅4 𝑅2 (𝑟𝑇4 𝑖𝐿2 )3 𝑟𝑇4 𝑖𝐿2 𝑉2 (𝑟𝑇4 𝑖𝐿2 )2 𝑑𝑖𝐿2 1 = [−𝑅𝑠2 𝑖𝐿2 + 𝑟𝑇5 { + − } − 𝑟𝑇6 ] 𝑑𝑡 𝐿2 𝑅5 𝑅7 𝑅8 𝑅6

(13)

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reasonable values of the resistors further impedance rescaling is possible. Set of describing differential equations can be expressed as follows: di L1 dt

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di L2 dt

= =

1 L1 1 L2





− Rs1 i L1 + r T2 



− Rs2 i L2 + r T5

(r T1 i L1 )3 R1 (r T4 i L2 )3 R5

+

r T1 i L1 R3

−

V1 R4

+

r T4 i L2 R7

−

V2 R8

 

2 − r T3 (rT1Ri2L1 )



2



− r T6 (rT4Ri6L2 )

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(13)

 𝑑𝑣 1  𝑣𝑍 dv𝑍Z = = C1X(−− RvZp −−𝑖𝐿1 i L1−−𝑖𝐿2 i L2++𝐼1I)1 dt 𝑑𝑡 𝐶𝑋 𝑅𝑝 where rTk is trans-resistance trans-resistance of k-h ideal current-controlled voltage-source and I11 is independent dc Tk is current source.

Figure 12. Hypothetic Hypothetic analog analog circuit circuit realization realization dual to original original MVMS network network with ideal controlled sources: trans-resistance of output current-to-voltage conversion is considered considered as as global global parameter. parameter.

Polynomial nonlinearity nonlinearity is is implemented implemented by by ideal ideal multiplication multiplication using using block block MULT. MULT. Current Polynomial respectively. However, However, flowing through both inductors can be sensed via small small resistor resistor RRS1 S1 and RS2 S2 respectively. differential equations; equations; similarly similarly these resistors represent error terms that are inserted into describing differential as a parasitic shunt resistor Rp . Such Such parasitic parasitic properties properties change change global global dynamics, dynamics, can boost system dimensionality (not in this situation) and desired chaotic behavior can eventually eventually disappear. disappear. active elements elementswhere whereoutput outputvoltage voltage is controlled input current are offnot Unfortunately, active is controlled by by input current are not off-the-shelf components. However, trans-resistance amplifiercan canbebeconstructed constructedusing usingsingle single standard standard the-shelf components. However, trans-resistance amplifier operational amplifier and feedback resistor. To do operational do this, this, input input is is fed fed directly directly to to − − terminal, node ++ is andOUT. OUT. Node Node OUT OUT also also represents represents output output of a designed connected to ground, resistor between − − and transimpedance amplifier. amplifier. Equivalently, Equivalently,the theAD844 AD844can canalso alsodo dothe thetrick. trick.Input Inputcan canbe beconnected connectedto to− − terminal, resistor resistor between between C C and and ground ground and and output output to to OUT. OUT. Considering the terminal, ground to ++ terminal, latter case, number of the active devices becomes seven: one MLT04 and six AD844. Colored plots provided in Figure 8 demonstrate that region of chaos around discovered values (7) is wide enough to provide a structurally stable strange attractor; both funnel and double-scroll type. The same analysis was performed for set (8); this strange attractor should be also observable. 5. Circuit Simulation, Experimental Verification and Comparison As mentioned before, AVC of both nonlinear resistors in Figure 11 should be as precise as possible to reach desired state space attractor. To fulfil this requirement, build-in Orcad Pspice

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Colored plots provided in Figure 8 demonstrate that region of chaos around discovered values (7) is wide enough to provide a structurally stable strange attractor; both funnel and double-scroll type. Entropy 2018, 20, x 13 of 23 The same analysis was performed for set (8); this strange attractor should be also observable. Of course, the operational regime of this nonlinear resistor needs to be limited; in our case to

5. Circuit Experimental Verification Comparison input Simulation, voltages into range starting with −0.5 V andand ending with 2 V. Note that up to thirteen fitness functions were supposed to cover predefined operational range. Also note that optimization was As mentioned before, AVC of both nonlinear resistors in Figure 11 should be as precise as possible stopped while some objective functions still have nonzero error (expressed in terms of the differential to reach desired state space attractor. To fulfil this requirement, build-in Orcad Pspice optimizer can be percentages). Important is the matter of similarity between simulated and numerically integrated adopted; seeattractor. Figure 13 where optimal AVC of one nonlinear two-terminal device is reached. strange

Figure 13. Orcad Pspice optimizationtoolbox: toolbox: definitions objective functions and and requested Figure 13. Orcad Pspice optimization definitionsofofthe the objective functions requested maximal errors (upper right field), error graph (upper left), new values of resistors (middle table). maximal errors (upper right field), error graph (upper left), new values of resistors (middle table).

Implementation of the chaotic based on the integrator robust; Of course, the operational regimeoscillator of this nonlinear resistor needsblock to beschematic limited; is inmore our case to input i.e., the desired strange attractor is less vulnerable to the passive component matching: fabrication voltages into range starting with −0.5 V and ending with 2 V. Note that up to thirteen fitness functions series (E12) and component tolerances (1%) were considered for design. We are also experiencing were supposed to cover predefined operational range. Also note that optimization was stopped while superior parasitic properties of the voltage-mode active elements: both integrated circuits TL084 and someMLT04 objective have termsofofresistors the differential havefunctions very highstill input and nonzero very low error output(expressed resistances.in Values that form percentages). external Important is the mattertoofAD844 similarity simulated and numerically integrated strange network connected werebetween chosen such that parasitic input and output impedances canattractor. be Implementation of the chaotic oscillator based on the integrator block schematic is more robust; neglected. This is the main reason why this kind of MVMS realization was picked and forwarded intodesired practicalstrange experiments and undergoes laboratoryto measurement. i.e., the attractor is less vulnerable the passive component matching: fabrication Generally, parasitic properties of the active elements play important roleWe in aare design of series (E12) and component tolerances (1%) were considered for design. alsoprocess experiencing analog chaotic oscillators of course, should beactive minimized. Besides input and output impedances superior parasitic propertiesand, of the voltage-mode elements: both integrated circuits TL084 and alsohave roll-off effects transfer constants need to be analyzed. Several publications havethat beenform devoted MLT04 very highofinput and very low output resistances. Values of resistors external to reveal and study problems associated with the parasitic properties of the specific active devices network connected to AD844 were chosen such that parasitic input and output impedances can be and how these affect global dynamics; for example [41]. neglected. This is the main reason why this kind of MVMS realization was picked and forwarded into In simulation profile of transient response, final time was set to 1 s and maximal time step 10 μs. practical andeach undergoes laboratory This experiments setup is kept for simulation mentionedmeasurement. in this section. Circuit simulation associated with Generally, parasitic properties of the active the elements rolechaotic in a design process of Figure 9 is provided in Figure 14. To transform funnel play into aimportant double-scroll attractor we analog chaotic oscillators and, course, should Besides input and output should change value of theofresistors Rz3, Rz1 be andminimized. R10. Simulation results associated withimpedances direct realization of original MVMS provided intoFigure 10 is demonstrated by meanshave of Figure 15. also roll-off effects of transfer constants need be analyzed. Several publications been devoted Computer-aided analysis ofassociated chaotic oscillator with idealized controlledofsources is showed in devices Figure and to reveal and study problems with the parasitic properties the specific active how 16. these affect global dynamics; for example [41]. Existence of observable strange response, attractors and numerically expected scenarios In simulation profile of transient final time was set to 1 s routing-to-chaos and maximal time step 10 µs. within dynamics of the fundamental MVMS has been proved experimentally; by its construction on This setup is kept for each simulation mentioned in this section. Circuit simulation associated with the bread-board (see Figure 17) and consequent laboratory measurement using the analog

Figure 9 is provided in Figure 14. To transform the funnel into a double-scroll chaotic attractor

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oscilloscope. Due to its simpler realization and increased robustness, only circuitry illustrated by means of Figure 9 was decided for a real measurement. Selected chaotic waveforms in a time domain are provided by means 18. Independent source Vbias = 750 mV in series with Rbias = 1 we should change value of ofFigure the resistors Rz3 , Rvoltage z1 and R10 . Simulation results associated with kΩ can be replaced by the positive supply voltage +5 V and the fixed series resistance Rbias = 6.7 kΩ. direct realization of original MVMS provided in Figure 10 is demonstrated by means of Figure 15. Analogically, combinations Vc1 = −130 mV, Rx = 1 kΩ and Vc2 = −650 mV, Ry = 1 kΩ can be replaced by Computer-aided analysis of chaotic oscillator with idealized controlled sources is showed in Figure 16. the negative supply voltage −5 V, Rx = 38.5 kΩ and the same voltage −5 V, Ry = 7.7 kΩ respectively.

Figure 14. Orcad Pspice circuit simulationassociated associated with circuit given in Figure 9: selected Figure 14. Orcad Pspice circuit simulation withchaotic chaotic circuit given in Figure 9: selected plane projection v3–v2 (blue, upper graph), v2–v1 (red) and v3–v1 (blue) of the chaotic attractor, plane projection v3 –v2 (blue, upper graph), v2 –v1 (red) and v3 –v1 (blue) of the chaotic attractor, generated chaotic signal v1 (red) and v2 (blue) in the time domain, chaotic waveform v1 (red) and v2 generated chaotic signal v1 (red) and v2 (blue) in the time domain, chaotic waveform v1 (red) and (blue) in the frequency domain. Note that significant frequency components are concentrated in audio v2 (blue) in the frequency domain. Note that significant frequency components are concentrated in range. audio range.

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Figure Figure 15. 15. Orcad Orcad Pspice Pspice circuit circuit simulation simulation associated associated with with network network given given in in Figure Figure 10: 10: selected selected plane plane projection projection vv11–v –v22 (blue, (blue, upper upper plot), plot), vv11–i –iLL(red) (red)and andvv22–i–iLL(blue) (blue)ofofchaotic chaoticattractor, attractor, generated generated signal signal vv11 (red) (red) and and vv22 (blue) (blue) in in time time domain, domain, chaotic chaotic waveform waveform vv11 (red) (red)and andvv22(blue) (blue)in infrequency frequencydomain. domain.

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Figure 16. Orcad OrcadPspice Pspice circuit simulation associated with network Figure plane Figure 16. circuit simulation associated with network given ingiven Figurein10: plane 10: projection projection v Z –i L2 (blue, upper plot), i L2 –i L1 (red) and v Z –i L1 (blue) of the observed strange attractor, vZ –iL2 (blue, upper plot), iL2 –iL1 (red) and vZ –iL1 (blue) of the observed strange attractor, generated generated chaotic signal iL2v(red) andinvthe Z (blue) the time domain, chaoticiL2waveform vZ chaotic signal iL2 (red) and time in domain, chaotic waveform (red) and ivL2Z (red) (blue)and in the Z (blue) (blue) in the frequency Note that generated chaos isbynot affected levels by saturation levels of the frequency domain. Notedomain. that generated chaos is not affected saturation of the active devices. active devices.

Existence of observable strange attractors and numerically expected routing-to-chaos scenarios within dynamics of the fundamental MVMS has been proved experimentally; by its construction on the bread-board (see Figure 17) and consequent laboratory measurement using the analog oscilloscope. Due to its simpler realization and increased robustness, only circuitry illustrated by means of Figure 9 was decided for a real measurement. Selected chaotic waveforms in a time domain are provided by

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means of Figure 18. Independent voltage source Vbias = 750 mV in series with Rbias = 1 kΩ can be replaced by the positive supply voltage +5 V and the fixed series resistance Rbias = 6.7 kΩ. Analogically, combinations Vc1 = −130 mV, Rx = 1 kΩ and Vc2 = −650 mV, Ry = 1 kΩ can be replaced by the negative supply voltage −5 V, Rx = 38.5 kΩ and the same voltage −5 V, Ry = 7.7 kΩ respectively. Entropy 2018, 20, x Entropy 2018, 20, x

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Figure 17. Practical realization of the integrator-based chaotic MVMS using bread-board. Figure 17.17. Practical integrator-based chaotic MVMS using bread-board. Figure Practicalrealization realization of of the the integrator-based chaotic MVMS using bread-board.

Figure 18. Selected chaotic waveforms in time domain generated by integrator-based MVMS.

Figure Selected chaoticwaveforms waveforms in generated by by integrator-based MVMS. Figure 18. 18. Selected chaotic in time timedomain domain generated integrator-based MVMS. Thus, there is no need to introduce three additional independent dc voltage sources into Thus, However, there is notoneed introduce three additional independent voltage sources into oscillator. trace to route-to-chaos scenarios, MVMS parametersdc c1 and c2 represented by oscillator. However, to trace route-to-chaos scenarios, MVMS parameters c 1 and c2 represented by voltages Vc1 and Vc2 have been considered as the variables. In practice, two voltage dividers based on voltages Vc1 and Vc2 have been considered as the variables. In practice, two voltage dividers based on

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Thus, there is no need to introduce three additional independent dc voltage sources into oscillator. However, to trace route-to-chaos scenarios, MVMS parameters c1 and c2 represented by voltages Vc1 and Vc2 have been considered as the variables. In practice, two voltage dividers based on potentiometers followed by two voltage buffers (remaining part of fifth integrated circuit TL084) did the trick. Since extreme values of Vc1 , Vc2 voltages can be managed observed Monge projections given in Figure 19 (plane v3 vs. v2 ) originate both inside and outside prescribed bifurcation scheme pictured in Figure 6 (third and sixth plot). Of course, very small steps of the hand-swept parameters cannot be captured by the oscilloscope. Different plane projections, namely v3 vs. v1 , are shown in Figure 20. The provided screenshots are centered on captured state attractors not the origin of the state coordinates. Of course, the mentioned plane projections are not in full one-to-one correspondence with the theoretical results given in Figure 14, simply because transfer functions of the nonlinear two-ports cannot be defined precisely using imperfect discrete resistors; it does not get better shape even if the working resistors are replaced by the standard hand potentiometers. Finally, the sequence of periodic and chaotic windows with continuous change of the individual coefficients of the polynomials has been confirmed; roughly proving the correctness of bifurcation diagrams in Figure 6. During laboratory experiments, unexpected and a very interestingly-shaped strange attractors were identified (see Figure 21). Unfortunately, numerical values of the mathematical model parameters were not found, and reference state trajectories cannot be created. Initial conditions need not to be imposed on the outputs of the inverting integrators (capacitors need not to be pre-charged) since neighborhood of zero belongs into the basin of attraction for both strange attractors.

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Figure 19. Selected v3–v2 plane projections measured of integrator-based MVMS realization, see text.

Figure 19. Selected v3 –v2 plane projections measured of integrator-based MVMS realization, see text.

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Figure 20. Few selected v3–vplane 1 plane projections measured within dynamics of the integrator-based Figure 20. Few selected v3 –v projections measured within dynamics of the integrator-based 1 Figure 20. Few selected 3–v1 plane projections measured within dynamics of the integrator-based implementation MVMS, vvoltage sources Vc1 and Vc2 are swept, i.e., system parameters c1 and c2 are implementation MVMS, voltage sources Vc1c1 and V areswept, swept,i.e., i.e., system parameters c1 and implementation MVMS, voltage sources andvoltages Vc2c2are system parameters c1 and c2 arec2 are considered as variable parameters, changeVof the starts within bifurcation diagrams provided considered as variable parameters, change of the voltagesstarts startswithin within bifurcation diagrams provided considered as variable parameters, change of the voltages bifurcation diagrams provided by means of Figure 6 but finally goes far beyond it. by means of Figure 6 but finally goes by means of Figure 6 but finally goesfar farbeyond beyond it. it.

Figure 21. Gallery of interesting (still robust and real-time observable) strange attractors generated by

Figure 21. Gallery of realization interesting (still robustdifferent and real-time strange generated by by Figure 21. Gallery of interesting (still robust and real-time observable) strange generated the integrator-based of MVMS, plane observable) projections, see textattractors forattractors details. the integrator-based realization of MVMS, different plane projections, see text for details. the integrator-based realization of MVMS, different plane projections, see text for details.

6. Conclusions Existence of robust chaotic attractors in the smooth vector field of MVMS have been demonstrated in this paper. This work represents a significant extension of the discoveries discussed in [29]. Both

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referred chaotic attractors are self-excited attractors; hidden attractors were not sought after and, in the case of analyzed MVMS dynamics, remain a mystery. The proposed mathematical model: (1) together with the nonlinear functions (2) and parameters (7) or (8) can be also considered as a new chaotic dynamical system. This is still an up-to-date problem for many engineers and the topic of many recent scientific papers. However, algebraically much simpler dynamical systems that exhibit chaotic attractors exist; reading [42–47] is recommended. Three different circuit realizations are presented. Each was verified by circuit simulation and the most robust implementation undergoes experimental measurement. Two designed oscillators utilize off-the-shelf active elements and can serve for various demonstrations. The fundamental motivation of this work is to show that structurally stable strange attractors can be observed in smooth dynamical system naturally considered as nonlinear but with non-chaotic limit sets. However, designed autonomous chaotic oscillators can serve as the core circuits in many practical applications such as cryptography [48], spread-spectrum modulation techniques [49], useful signal masking [50,51], random number generators [52,53], etc. This work also leaves several places for future research. For example, following interesting questions needs to be answered: Are there some hidden attractors? Are multi-scroll chaotic attractors observable if many RTD will be connected appropriately? Can MVMS generate chaos if values of the internal parameters are close to microelectronic memory cell fabricated in common technology? Funding: Research described in this paper was financed by the National Sustainability Program under Grant LO1401. For the research, infrastructure of the SIX Center was used. Conflicts of Interest: The author declares no conflict of interest.

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