Evidence of strange attractors in class C amplifier with single bipolar transistor: polynomial and piecewise-linear case

entropy Article Evidence of Strange Attractors in Class C Amplifier with Single Bipolar Transistor: Polynomial and Piecewise-Linear Case Jiri Petrzela   Citation: Petrzela, J. Evidence of Strange Attractors in Class C Amplifier with Single Bipolar Transistor: Polynomial and Piecewise-Linear Case. Entropy 2021, 23, 175. https://doi.org/10.3390/ e23020175 Received: 20 December 2020 Accepted: 26 January 2021 Published: 30 January 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Radio Electronics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 12, 616 00 Brno, Czech Republic; [email protected].cz; Tel.: +420-54114-6561 Abstract: This paper presents and briefly discusses recent observations of dynamics associated with isolated generalized bipolar transistor cells. A mathematical model of this simple system is considered on the highest level of abstraction such that it comprises many different network topologies. The key property of the analyzed structure is its bias point since the transistor is modeled via two-port admittance parameters. A necessary but not sufficient condition for the evolution of autonomous complex behavior is the nonlinear bilateral nature of the transistor with arbitrary reason that causes this effect. It is proved both by numerical analysis and experimental measurement that chaotic motion is miscellaneous, robust, and it is neither numerical artifact nor long transient motion. Keywords: admittance parameters; bipolar transistor; entropy; chaos; Lypunov exponent; non-unilateral two-port; strange attractors 1. Introduction Chaos can be considered as long-time unpredictable behavior of a dynamical system that is both nonlinear and, in the autonomous case, has at least three degrees of freedom. Chaotic systems are very sensitive to the changes of initial conditions; this sensitivity is caused by exponential divergency of neighborhood orbits but, at the same time, the generated strange attractor is bounded into finite state space volume. The boundedness of the strange attractor is due to the suitable nonlinearity of the vector field. Despite mature observations, recent studies reveal that fixed points are not crucial for the evolution of chaos. There are several mathematical models with equilibrium degenerated into higherdimensional geometric structures or chaotic systems without equilibrium. Additionally, a wide variety of chaotic dynamical systems that exhibit the so-called hidden attractors are available via internet search. After its analytical, numerical, and experimental confirmation within a very simple fully analog circuit, famous Chua’s oscillator [ 1 ], chaos started to receive considerable attention, especially among circuit design engineers. Many interesting theories and practical findings coupled with nonlinear dynamics in lumped circuits were discovered and published; several examples can be found in papers [ 2 – 8 ]. By following subsequent discoveries in chaos theory and by increasing the knowledge of the strange attractor ´ s evolution, this kind of complex motion was detected in many electronic systems. Let us mention a few cases of naturally non-chaotic building blocks of more complex systems dedicated for analog signal processing. Robust chaos was reported in switched capacitor circuits [ 9 ], switched regulators [ 10 ], power converters [ 11 ], different topologies of dc-dc converters [ 12 ], and in power electronics in general. From the viewpoint of subsystems of radio-frequency path, structurally stable chaotic oscillations were discovered in phaselocked loops [ 13 ], multi-state memory cells [ 14 , 15 ], and standard structures of harmonic oscillators such as Colpitts [ 16 ], Hartley [ 17 ], Wien-bridge [ 18 ] or other topology having resistor-capacitor feedback [19] topology. Recent papers [ 20 , 21 ] have revealed the existence of strange attractors in a fundamental stage of class C amplifier with a single bipolar transistor. However, the bipolar transistor Entropy 2021,23, 175. https://doi.org/10.3390/e23020175 https://www.mdpi.com/journal/entropy Entropy 2021,23, 175 2 of 24 in this paper is assumed to have linear forward trans-conductance y 21 (v 1 ), while work [ 20 ] assumes a cubic polynomial for both functions y 12 (v 2 ), y 21 (v 1 ). Therefore, all seven distinct chaotic cases revealed in this manuscript are algebraically simpler than the single example proposed in paper [ 20 ]. In contrast, paper [ 21 ] deals with smooth nonlinear function y 21 (v 1 ) and presents strange attractors discovered for only two shapes of forward transconductance. On the other hand, in the upcoming analysis, smooth polynomial nonlinearity up to the fifth order that describes backward trans-conductance could be found within the set of the ordinary differential equations. Of course, the different mathematical model considered here leads to completely different numerical results as well as much simpler circuitry implementation. From the practical perspective, the third-order deterministic chaotic systems provided in this paper generate waveforms with different features. Readers can pick and use our dynamical system that fits a specific application. In general, the upcoming sections represent more comprehensive analysis of the class C amplifier than case study [ 20 ]. Simple circuits with one or two transistors are analyzed in paper [ 22 ], again from the viewpoint of the evolution of chaotic behavior. In these networks, the parasitic properties of transistors are not considered for the numerical investigations. Driven lumped electronic systems are subjects of chaotic dynamics as well. Moreover, degrees of freedom can be lowered to two. The chaotic operational regime of a KHN (Kerwin–Huelsman–Newcomb) filter (or state variable filters in general) is analyzed in paper [ 23 ]. It is demonstrated that chaos occurs and disappears according to the frequency and amplitude of input useful harmonic signal. Based on the observations presented in this paper, a two-terminal electronic device marked as “chaotic admittance” can be developed. While applied input voltage acts as a driving force, the chaotic waveform measured at any independent internal node controls input current. The practical application of such two-terminal can be discovered in noise generators, the testing of frequency responses, analog and digital modulations, the masking of useful analog signals using chaotic waveform [24], etc. 2. Single Transistor Stage Assume the lumped electronic equivalent of a single transistor that is characterized by arbitrary bias point and connected as indicated in Figure 1a. For a useful signal, a simplified small-signal calculation model provided in Figure 1b can be derived. Describing a set of ordinary differential equations can be expressed in matrix form as: d dtx=A·x↔d dt   v1 v2 i` =   −y11 c1−y12 c10 −y21 c2−y22 c2−1 c2 01 `0   ·  v1 v2 i` , (1) where y jk are admittance parameters of a bipolar transistor considered as two-port in a common emitter configuration and the state vector is x=(v1,v2,i`)T . Symbol c 1 represents a parasitic base-emitter capacitance and c 2 is a sum of parasitic collector-emitter capacitance and working capacitance of the parallel resonant tank. Resistor R 2 given in the schematic could contain both the output admittance of transistor y 22 and inductance resistive losses. In practice, entries of 2 × 2 transistor ´ s admittance matrix could be complex numbers, especially if high-frequency applications are addressed. It is necessary to realize that the word “high” is relative—it could be tens of MHz depending on the type of bipolar transistor. Coefficients of the transistor ´ s admittance matrix could also be nonlinear functions, specifically in the case of power amplifiers or if signals with high amplitudes are processed. Of course, numerical values of admittance parameters depend significantly on biasing circuitry. These are not specified in the analyzed schematic. However, to maintain the maximum universality of final remarks, the input and output admittance of a bipolar transistor is supposed to be linear, i.e., constant real number independent of the amplitude of a processed signal. The fundamental amplification capability of a bipolar transistor characterized by trans-admittance y 21 will be scalar constant as well, which is a linear Entropy 2021,23, 175 3 of 24 function of input voltage v 1 without offset. Therefore, a characteristic polynomial associated with the isolated system (1) and evaluated at the fixed point xe=v0 1,v0 2,i0 `Tis: det(γ·E−A)=γ3+c2·y11 +c1·y22 c1·c2 ·γ2+1 c1·c2·` c1+`·y11·y22 −`·y21·∂y12 ∂v2v0 2!·γ+y11 c1·c2·`=0 , (2) where E is the unity matrix. A partial derivative of backward trans-conductance of a bipolar transistor is evaluated at equilibrium structure d x /dt = 0 , where 0 is a vector of zeroes. Note that, at this moment, accumulation elements are normalized with respect to time and impedance. This fact is emphasized by utilization of the small letters c1 , c2 , ` throughout this manuscript. Entropy 2021, 23, x FOR PEER REVIEW 3 of 24 Of course, numerical values of admittance parameters depend significantly on biasing circuitry. These are not specified in the analyzed schematic. However, to maintain the maximum universality of final remarks, the input and output admittance of a bipolar transistor is supposed to be linear, i.e., constant real number independent of the amplitude of a processed signal. The fundamental amplification capability of a bipolar transistor characterized by trans-admittance y21 will be scalar constant as well, which is a linear function of input voltage v1 without offset. Therefore, a characteristic polynomial associated with the isolated system (1) and evaluated at the fixed point 𝐱=󰇛𝑣,𝑣,𝑖ℓ󰇜 is: det󰇛𝛾∙𝐄−𝐀󰇜=𝛾+∙∙ ∙∙𝛾+ ∙∙ℓ󰇧𝑐+ℓ∙𝑦∙𝑦−ℓ∙𝑦∙ 󰇻󰇨∙𝛾+ ∙∙ℓ=0 , (2) where E is the unity matrix. A partial derivative of backward trans-conductance of a bipolar transistor is evaluated at equilibrium structure dx/dt = 0, where 0 is a vector of zeroes. Note that, at this moment, accumulation elements are normalized with respect to time and impedance. This fact is emphasized by utilization of the small letters 𝑐,𝑐,ℓ throughout this manuscript. Figure 1. General circuit concepts analyzed in this paper: (a) fundamental cell of class C amplifier, (b) equivalent schematic of class C amplifier for useful small-amplitude AC (Alternating Current) signals. 2.1. Local Polynomial Backward Trans-Conductance Let us rewrite the matrix system of differential Equation (1) into the more general form that considers the possible fractional-order nature of the individual accumulation elements and a polynomial backward trans-conductance of a bipolar transistor. New ordinary differential equations will be:  𝑣=− 𝑣−󰇛𝑎∙𝑣+𝑏∙𝑣+𝑐∙𝑣+𝑑∙𝑣+𝑒∙𝑣󰇜,  𝑣=− 𝑣− 𝑣−𝑖ℓ,  𝑖ℓ = ℓ𝑣 , (3) where orders α, β, and γ are real numbers between zero and one. In this case, system (1) has one fixed point located at the origin. Firstly, let us consider cases where coefficient a = 0. Eigenvalues, i.e., solutions of cubic polynomial (2) associated with 𝐱=󰇛0,0,0󰇜, imply that the origin is a non-repelling fixed point, at least for reasonable values of transistor cell components, i.e., nonzero positive c1, c2, ℓ and positive system dissipation, i.e., y11 > 0 and y22 > 0. 𝛾=−  ,𝛾,=− ∙∙𝑦∓  𝑦 −4 ℓ , (4) Figure 1. General circuit concepts analyzed in this paper: ( a ) fundamental cell of class C amplifier, ( b ) equivalent schematic of class C amplifier for useful small-amplitude AC (Alternating Current) signals. 2.1. Local Polynomial Backward Trans-Conductance Let us rewrite the matrix system of differential Equation (1) into the more general form that considers the possible fractional-order nature of the individual accumulation elements and a polynomial backward trans-conductance of a bipolar transistor. New ordinary differential equations will be: dα dtαv1=−y11 c1v1−1 c1a·v2+b·v2 2+c·v3 2+d·v4 2+e·v5 2, dβ dtβv2=−y21 c2v1−y22 c2v2−1 c2i`,dγ dtγi`=1 `v1, (3) where orders α , β , and γ are real numbers between zero and one. In this case, system (1) has one fixed point located at the origin. Firstly, let us consider cases where coefficient a= 0 . Eigenvalues, i.e., solutions of cubic polynomial (2) associated with xe=(0, 0, 0)T , imply that the origin is a non-repelling fixed point, at least for reasonable values of transistor cell components, i.e., nonzero positive c 1 ,c 2 , ` and positive system dissipation, i.e., y 11 > 0 and y22 > 0. γ1=−y11 c1 ,γ2,3 =−1 2·c2 ·y22 ∓ry2 22 −4c2 `, (4) Therefore, all nontrivial solutions including sought strange attractors belong to the so-called hidden attractors [25]. The local vector field near to the origin is spanned by the eigenvector → ε1 and the eigenplane defined by −→ ε2,3 written in the following symbolic form: → ε1=    −c2 1−`·c1·y11·y22+c2·`·y2 11 c2 1·y21 −`·y11 c1 1     ,−→ ε2,3 =    0 −`·y22∓qy2 22−4c2 ` 2·c2 1     . (5) Entropy 2021,23, 175 4 of 24 As it will be clarified later, case a= 0 covers four out of six discovered sets of parameters that lead to robust chaotic behavior. Now assume an arbitrary value of coefficient a. By following Cardan´s rule, one can obtain the following symbolic eigenvalues: γ=−1 3·β"1 c2·`−a·y21 c1·c2 +y11·y22 c1·c2 −1 3y11 c1 +y22 c22#+β−1 3y11 c1 +y22 c2, (6) where the first auxiliary parameter: β= 3 v u u u u t−ϑ 2±v u u u tϑ2 4+1 c2·`−a·y21 c1·c2+y11·y22 c1·c2−1 3y11 c1+y22 c223 27 , (7) and the second auxiliary parameter can be calculated as: ϑ=y11 `·c1·c2 +1 27"2y11 c1 +y22 c23 −9y11 c1 +y22 c2 1 c2·`−a·y21 c1·c2 +y11·y22 c1·c2#. (8) For conservative dynamics, i.e., if y 11 =y 22 = 0, formulas for the eigenvalues (6), (7) and (8) significantly simplify into the following relation: γ1,2 =±s`·a·y21 −c1 `·c1·c2 ,γ3=0. (9) A closer insight into these eigenvalues is not necessary since the optimization routine (see below) operates with numerical values of network elements and, consequently, with numerical eigenvalues. Although case a 6= 0 was considered during the searching-for-chaos procedure, its linearized analysis is not provided in this paper. Corresponding symbolic formulas are too complicated to be displayed using a reasonable format. 2.2. Local Piecewise-Linear Backward Trans-Conductance The idea behind this sub-section is to assume the successful (in the sense of comparable values of positive largest Lyapunov exponent) approximation of backward transconductance y 12 (v 2 ) by twoor three-segment piecewise linear (PWL) curves. A set of three ordinary differential equations that describe the class C amplifier is: dα dtαv1=−y11 c1 v1−1 c1 y12(v2),dβ dtβv2=−y21 c2 v1−y22 c2 v2−1 c2 i`,dγ dtγi`=1 `v1(10) where α , β , and γ are real numbers between zero and one. The approximated vector field is symmetrical with respect to origin. Therefore, for odd-symmetrical PWL function, backward trans-conductance with four breakpoints { −ϕ2 , −ϕ1 , ϕ1 , ϕ2 } can be expressed as : y12(v2)=ρ2·x+ρ0−ρ1 2·(|x+ϕ1|−|x−ϕ1|)+ρ1−ρ2 2·(|x+ϕ2|−|x−ϕ2|), (11) where ρ0 is slope of segment around zero, ρ1 is slope of segment between breakpoint ϕ1 and ϕ2 ( ϕ1 < ϕ2 ), and ρ2 is slope of segment in the outer regions of the vector field, i.e., x>ϕ2 . For even-symmetry of the vector field, PWL function could possess three breakpoints {−ϕ, 0, ϕ} and be characterized by the following simple relation: y12(v2)=ρ0·|x|+ρ1−ρ0 2·(|x+ϕ|+|x−ϕ|)−ϕ·(ρ1−ρ0). (12) Obviously, such PWL function has slope ρ0 for 0 < x< ϕ , slope −ρ0 for −ϕ<x<0 , slope −ρ1 for x< −ϕ , and finally slope ρ1 for x> ϕ . Both PWL functions generate a single equilibrium point located at the origin and the entire vector field is separated into five and four affine segments for odd (11) and (12) even-symmetrical PWL function, respectively. Entropy 2021,23, 175 5 of 24 Note that the existence of other fixed points is not conditioned by the shape of PWL functions. In other words, for scalar PWL functions, there is one equilibrium point located at (−y12(0)/y11, 0, y21·y12(0)/y11)T . Assume constant term ρconst in PWL function (11) or (12). Then, ρconst can be used to move the equilibrium point to a new position in the state space along a line. 2.3. Alternative Mathematical Models of Class C Amplifier So far, a bipolar transistor substituted by two-port described by admittance matrix was considered to be a dynamical system dedicated for analysis. Firstly, note that a transistor cannot be directly substituted by the very popular Giacoletto ´ s model or a similar interconnection where backward trans-conductance is neglected. Secondly, an arbitrarily biased bipolar transistor can be modeled using other types of two-port equivalent parameters, such as impedance or hybrid matrix. However, this change does not bring benefits over the initial admittance matrix Y; neither from the viewpoint of linear analysis nor circuitry realization. For a bipolar transistor modeled by impedance matrix Z = Y−1 we can obtain the following algebraic relations: z11 =y22 y11·y22−y12·y21 =0 , z12 =−y12 y11·y22−y12·y21 =1 , z21 =−y21 y11·y22−y12·y21 =1 y12 ,z22 =y11 y11·y22−y12·y21 =−y11 y12 ,(13) where significant simplifications provided above are valid for a zero output admittance y 22 = 0 and a normalized forward trans-conductance y 21 = 0. Analogically, for a bipolar transistor described by the hybrid matrix, we can obtain: h11 =1 y11 ,h12 =−y12 y11 ,h21 =1 y11 ,h22 =y11·y22 −y12·y21 y11 =−y12 y11 , (14) where the combination of y 22 = 0 and y 21 = 1 provides simplification again. Obviously, a chaotic system based on a class C amplifier can be constructed using (13) and (14), or by introducing a linear transformation of coordinates applied on system (3) or (10). A single-transistor class C amplifier can be also modeled by a two-port equivalent circuit of a bipolar transistor where three two-terminal devices (initially admittances) are arranged into Π topology appended by one voltage controlled current source. This controlled source can be located at the input port, output port or between these ports. However, no such transformation reduces the complexity of the final circuit since much more complex polynomial nonlinearities need to be implemented as lumped electronic subcircuits. From the viewpoint of global dynamics, the Π -type configuration of an equivalent circuit can still generate robust chaotic waveforms. 2.4. Searching for Chaotic Case For the process of chaos localization using the largest Lyapunov exponent as the objective function [ 26 ], transfer characteristics of backward trans-conductance were approximated by a polynomial up to the sixth order. Finally, it turns out that the values of both capacitors and inductors can be kept constant (unity) during optimization without losing the chance to find a chaotic solution. Additionally, a bipolar transistor is supposed to work close to ideal current source with y 22→ 0 and forward trans-conductance y21 =1S . Normalized eigenvalues associated with the origin will be γ1 = − y 11 , γ2,3 = ± j, i.e., neighborhood trajectories are attracted to an eigenplane where limit cycle is evolved. This is a quite unusual situation in chaos theory. Therefore, the sixth-dimensional hyperspace of the internal parameters of a dynamical system (1) with the edges Ψ∈ {y 11 ,a,b,c,d,e} undergoes deep investigation. The last five parameters shape nonlinear feedback function (3). Since individual points in this hyperspace can be calculated independently (an arbitrary number of the fitness functions can be calculated simultaneously), Matlab and CUDA-based parallel processes represent a good choice for high precision and fast calculation. Objective function is a combination of three phenomena: a bounded state attractor (verified during numerical integration), a positive value of LLE (taken as a final value Entropy 2021,23, 175 6 of 24 after integration), and predefined local geometry near the fixed point. Since there is no closed-form analytic solution associated with chaotic dynamics, stochastic nature-inspired optimization (a combination of genetic algorithm and swarm intelligence) was utilized. So far, several configurations Ψ with a reasonable (from the viewpoint of potential practical applications involving experimental construction of the chaotic circuit) seven-sided volume have been found. More details can be found in Table 1for polynomial vector field and Table 2for PWL case. The provided cases represent differently shaped (in the geometric sense) strange attractors and this list is by no means complete. Table 1. Numerical values of internal parameters of system (3) with mathematical orders α=β=γ= 1 that result in robust chaotic motion. Case y11 a b c d e Ψ10.56 0 2.1 0 −1.1 0 Ψ20.50 0 0 3 0 −1.5 Ψ30.30 5 0 −2 0 0 Ψ40.40 0 2.7 0 −2 0 Ψ50.30 0 0 3 0 −2 Ψ60.50 2 0 0 0 −0.5 Ψ70.40 0 0 2 0 –1 Table 2. Numerical values of internal parameters of system (10) with mathematical orders α=β=γ= 1 and either (11) or (12) that result in structurally stable chaotic motion (NA means Not Available). Case y11 ϕ ϕ1ϕ2ρ0ρ1ρ2 Ψ80.56 1.1 NA NA 1 −4.3 NA Ψ90.5 NA 0.3 1.1 0.3 2 −7 Ψ10 0.3 NA 0.6 1.18 4.6 0.6 −9.9 Ψ11 0.3 NA 0.4 1 0.2 1.5 −9.5 Dynamical system (1) can be rewritten in the form of a jerk function, that is, as the third order differential equation, namely: d3 dt3iL+y11 c1 +y22 c2·d2 dt2iL+1 c2·L+y11·y22 c1·c2 −y12·y21 c1·c2·d dtiL+y11 c1·c2·L·iL=0 , (15) or a simplified form by considering all assumptions provided above: d3 dt3iL+y11·d2 dt2iL+(1−y12·y21)·d dtiL+y11·iL=0 . (16) In both (15) and (16), y 12 is a nonlinear scalar function of variable v 2 . A single higherorder differential equation has a simple circuit representation: cascade connection of integrators with two-port feedback branches and an input summation/differentiation stage. Third-order differential Equation (15) or (16) can be instantaneously compared with the so-called jerk functions discovered during the excessive search performed by several scientists in the most recent three decades. Prof. Sprott was especially active in this research field and discovered many algebraically simple dynamical systems with a chaotic solution [ 27 – 29 ]. From this perspective, mathematical model (1) with parameters Ψ1 up to Ψ7 represents a new chaotic system that cannot be transformed into some known third-order system via a linear change of coordinates. 3. Numerical Results The core engine for all routines used for numerical analysis presented in this paper is a fourth order Runge–Kutta method with fixed step size. The numerical integration of both mathematical descriptions of the discovered chaotic system, i.e., matrix expression (1) Entropy 2021,23, 175 7 of 24 and its normal form, calculated in Mathcad is provided by means of Figure 2. For this fundamental analysis, final time was set to 10,000 s, time step was 0.01 s, initial conditions were chosen x0 = ( − 1, 0, 0) T for parameter set Ψ1,2,3,5,6 and x0 = (2, 0, 0) T for parameter set Ψ4. Individual cases of parameters Ψ1–6 are associated with Table 1. Entropy 2021, 23, x FOR PEER REVIEW 8 of 24 Figure 2. Plane projection v 1 vs. v 2 (black) and rainbow colored three-dimensional perspective views on the typical strange attractors generated by: (a) parameter set Ψ 1 substituted into the expression (1), (b) parameter set Ψ 1 substituted into system (6), (c) parameter set Ψ 2 substituted into differential Equation (1), (d) parameter set Ψ 2 substituted into jerk dynamics (6), (e) parameter set Ψ 3 numerically integrated using Equation (1), (f) integration of system (1) with parameter set Ψ 4 , (g) parameter set Ψ 5 substituted into Equation (1), and (h) parameter set Ψ 6 substituted into system (1) and integrated. Table 3 provides calculated values that can quantify the complexity of typical strange attractors generated by the subclasses of a class C amplifier cell with a single transistor Ψ 1 up to Ψ 7 . The first flow quantifier is the largest Lyapunov exponent (LLE) calculated using the mathematical model, see [30,31] for an overall description and algorithm explanation. Based on the spectrum of one-dimensional Lyapunov exponents (real numbers calculated with transient behavior omitted), the so-called Kaplan–Yorke dimension (KYD) of a generated strange attractor is established [32,33]. Capacity dimension (CD) of the state space attractor established by using the box counting method [34] is also provided. Mentioned flow quantifiers adapted for third-order dynamical systems can be calculated as follows: LLE=lim → lim |𝒁  |→|𝛿𝒁󰇛𝑡󰇜| |𝛿𝒁|,KYD=2+𝐿𝐸+𝐿𝐸 𝐿𝐸 ,CD=lim →ln𝑁󰇛𝜀󰇜 ln1/𝜀 , (17) where LLE is calculated using flow linearization and change in cube volume after small integration step δZ(t), LE 1 > LE 2 > LE 3 are one-dimensional Lyapunov exponents arranged in a decreasing order, and N(ε) is the number of cubes with edge ε required to fully cover the inspected state attractor. Figure 2. Plane projection v 1 vs. v 2 (black) and rainbow colored three-dimensional perspective views on the typical strange attractors generated by: ( a ) parameter set Ψ1 substituted into the expression (1), ( b ) parameter set Ψ1 substituted into system (6) , ( c ) parameter set Ψ2 substituted into differential Equation (1), ( d ) parameter set Ψ2 substituted into jerk dynamics (6 ), ( e ) parameter set Ψ3 numerically integrated using Equation (1), ( f ) integration of system (1) with parameter set Ψ4 , ( g ) parameter set Ψ5 substituted into Equation (1), and ( h ) parameter set Ψ6 substituted into system (1) and integrated. Figure 3shows that the chaotic system is extremely sensitive to the small variations of initial states, as required for the chaotic dynamics. This kind of analysis was performed for the Ψ1 case of the chaotified class C amplifier (see Table 1), but similar results can be obtained for the rest of the system cases, both polynomial and PWL. In these graphs, red dots represent 10 4 initial conditions with normal distribution, standard deviation 0.01 and nominal value x0 = (1, 0, 0) T . Other colors have the following meanings: final state is stored after 1 s (green points), ending state after 10 s (blue dots) and final state after 100 s (black dots). Note that neighborhood trajectories diverge slowly, and after 10 s fiducial points are still closely spaced. There is one exception: system case Ψ2 possesses the higher degree of long-time unpredictability. Figures 4–10 demonstrate the distribution of dynamic energy through the state space for individual cases Ψ1–7 (see Table 1). Here, red color denotes high kinetic energy, green marks average local energy and magenta indicates a very low local energy. Numerical values associated with these rainbow scaled plots normalized to unity time intervals are also provided. Initial conditions and time step were kept the same as for the analysis given in Figure 2; both with integration input parameters and a set of the initial conditions. Firstly, note that strange attractors occupy different sized volumes in the state space. Therefore, each plotted high-resolution plane has different axis ranges but uniform step size 0.01; concrete boundaries can be found within descriptors of the individual figures. For system cases Ψ1 and Ψ4 , it is obvious that average dynamic energy rises with the absolute value of state variable z. Using visualized plots, geometrical similarity between system cases Ψ1 and Ψ4 can be observed. Finally, strange attractors primarily do not evolve within areas with very high or low normalized energy. Along with kinetic energy distributions, Entropy 2021,23, 175 8 of 24 Poincaréreturn maps for the horizontal slices of the state space (z= const) are visualized. Obviously, geometrical shapes of generated strange attractors are distinct and attractors are dense in the state space, outside regions of a high local differential growth. If indicated in the plot, vector field symmetry causes the strange attractor to be mirrored with respect to the zero plane (z= 0) and Poincarésections could only be provided for upper or lower half state space (system cases Ψ2,3,6). Entropy 2021, 23, x FOR PEER REVIEW 9 of 24 Figure 3. Sensitivity to tiny changes of initial condition demonstrated for first case of chaotic system: starting situation (red points), short time evolution (green points), average time evolution (blue dots) and long time separation (black dots). Nominal starting position is chosen as follows: (a) x 0 = (1, 0, 0) T , (b) x 0 = (−1, 0, 0) T , (c) x 0 = (0, −1, 0) T and (d) x 0 = (0, 1, 0) T . Magnified areas showing states are demonstrated. Figure 4. Horizontal state space slices given by z = const. showing kinetic energy distribution of typical chaotic attractors of case Ψ 1 system, associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −0.9, z = −0.7, z = −0.5, z = −0.2, z = 0, z = 0.3, z = 1, z = 1.5, z = 2, z = 2.4, and z = 2.47. Figure 3. Sensitivity to tiny changes of initial condition demonstrated for first case of chaotic system: starting situation (red points), short time evolution (green points), average time evolution (blue dots) and long time separation (black dots). Nominal starting position is chosen as follows: ( a ) x0 = (1, 0, 0) T , ( b ) x0 = ( − 1, 0, 0) T , ( c ) x0 = (0, − 1, 0) T and (d)x0= (0, 1, 0)T . Magnified areas showing states are demonstrated. Entropy 2021,23, 175 9 of 24 Entropy 2021, 23, x FOR PEER REVIEW 9 of 24 Figure 3. Sensitivity to tiny changes of initial condition demonstrated for first case of chaotic system: starting situation (red points), short time evolution (green points), average time evolution (blue dots) and long time separation (black dots). Nominal starting position is chosen as follows: (a) x 0 = (1, 0, 0) T , (b) x 0 = (−1, 0, 0) T , (c) x 0 = (0, −1, 0) T and (d) x 0 = (0, 1, 0) T . Magnified areas showing states are demonstrated. Figure 4. Horizontal state space slices given by z = const. showing kinetic energy distribution of typical chaotic attractors of case Ψ 1 system, associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −0.9, z = −0.7, z = −0.5, z = −0.2, z = 0, z = 0.3, z = 1, z = 1.5, z = 2, z = 2.4, and z = 2.47. Figure 4. Horizontal state space slices given by z=const. showing kinetic energy distribution of typical chaotic attractors of case Ψ1 system, associated Poincarésections (black dots). Figures sorted from left to right and up to down: z= − 0.9, z=−0.7, z=−0.5, z=−0.2, z= 0, z= 0.3, z= 1, z= 1.5, z= 2, z= 2.4, and z= 2.47. Entropy 2021, 23, x FOR PEER REVIEW 10 of 24 Figure 5. Horizontal state space slices defined by the plane z = const. and providing dynamical energy distribution of typical chaotic attractors of case Ψ 2 system (white curve), associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −3.3, z = −3, z = −2.5, z = −2, z = −1.5, z = −1, z = −0.8, z = −0.6, z = −0.4, z = −0.2, and z = 0. Figure 6. Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case Ψ 3 system, associated Poincaré sections (black dots). Individual figures are sorted from left to right and up to down with respect to the planes: z = −9.4, z = −9, z = −8.5, z = −8, z = −7, z = −6.5, z = −6, z = −4, z = −2, z = −1, and z = 0. Figure 7. Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case Ψ 4 system (white trajectory), and associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −0.4, z = −0.2, z = 0, z = 0.2, z = 0.3, z = 0.5, z = 0.8, z = 1.2, z = 1.6, z = 2, and z = 2.5. Figure 5. Horizontal state space slices defined by the plane z = const. and providing dynamical energy distribution of typical chaotic attractors of case Ψ2 system (white curve), associated Poincarésections (black dots). Figures sorted from left to right and up to down: z = −3.3, z = −3, z = −2.5, z = −2, z = −1.5, z = −1, z = −0.8, z = −0.6, z = −0.4, z = −0.2, and z = 0. Entropy 2021, 23, x FOR PEER REVIEW 10 of 24 Figure 5. Horizontal state space slices defined by the plane z = const. and providing dynamical energy distribution of typical chaotic attractors of case Ψ 2 system (white curve), associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −3.3, z = −3, z = −2.5, z = −2, z = −1.5, z = −1, z = −0.8, z = −0.6, z = −0.4, z = −0.2, and z = 0. Figure 6. Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case Ψ 3 system, associated Poincaré sections (black dots). Individual figures are sorted from left to right and up to down with respect to the planes: z = −9.4, z = −9, z = −8.5, z = −8, z = −7, z = −6.5, z = −6, z = −4, z = −2, z = −1, and z = 0. Figure 7. Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case Ψ 4 system (white trajectory), and associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −0.4, z = −0.2, z = 0, z = 0.2, z = 0.3, z = 0.5, z = 0.8, z = 1.2, z = 1.6, z = 2, and z = 2.5. Figure 6. Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case Ψ3 system, associated Poincarésections (black dots). Individual figures are sorted from left to right and up to down with respect to the planes: z = − 9.4, z = − 9, z = − 8.5, z = − 8, z = − 7, z = − 6.5, z = − 6, z = − 4, z = −2, z = −1, and z = 0. Entropy 2021,23, 175 16 of 24 Entropy 2021, 23, x FOR PEER REVIEW 16 of 24 Figure 17. Colored basins of attraction, individual slices are the horizontal planes: (a) z0 = −3, (b) z0 = −2.5, (c) z0 = −2, (d) z0 = −1.5, (e) z0 = −1.25, (f) z0 = −1, (g) z0 = −0.75, (h) z0 = −0.5, (i) z0 = −0.3, (j) z0 = −0.1, (k) z0 = 0.0, (l) z0 = 0.1, (m) z0 = 0.2, (n) z0 = 0.3, (o) z0 = 0.4, (p) z0 = 0.5, (q) z0 = 0.75, (r) z0 = 1.0, (s) z0 = 1.5, and (t) z0 = 2. Figure 17. Colored basins of attraction, individual slices are the horizontal planes: ( a ) z0 = − 3, ( b ) z0 = − 2.5, ( c ) z0 = − 2, ( d ) z0 = −1.5 , ( e ) z0 = − 1.25, ( f ) z0 = − 1, ( g ) z0 = − 0.75, ( h ) z0 = − 0.5, ( i ) z0 = − 0.3, ( j ) z0 = − 0.1, ( k ) z0 = 0.0, ( l ) z0 = 0.1, (m) z0 = 0.2, (n) z0 = 0.3, (o) z0 = 0.4, (p) z0 = 0.5, (q) z0 = 0.75, (r) z0 = 1.0, (s) z0 = 1.5, and (t) z0 = 2. Entropy 2021,23, 175 17 of 24 4. Design of Flow-Equivalent Chaotic Oscillator Verification through practical experiment belongs to the common standard for the presentation of a new chaotic dynamical system. It is widely adopted that the observability of strange attractors represents satisfactory proof of the robustness of desired dynamics. Lumped circuit synthesis based on a prescribed mathematical model is a problem that can be easily solved using several different approaches. One of the most popular methods is based on an integrator block schematic, where basic mathematical operations are performed by three types of the two-port building blocks: inverting summing integrators, differential amplifiers and blocks having piecewise-linear or polynomial transfer curve. Each mentioned operation requires at least a single active element, usually a voltage feedback operational amplifier. The main drawback of this concept is evident: the necessity of using many active elements and a rather high power consumption. An integrator based kind of circuit realization is possible in three operational regimes: the most preferred is voltage-mode concept [ 38 ], current-mode is usually dedicated for the higher frequency bands [39] and mixed-mode circuits. It is worth nothing that the Orcad Pspice circuit simulator was used for the prevalidation of designed chaotic oscillators. Figure 18 shows maximally idealized case Ψ1 and Ψ3 systems with impedance norm 10 3 and frequency norm 10 6 . Of course, having ideal voltage-controlled current-sources (G) and ideal voltage multiplication blocks (MULT), both norms can be arbitrary; only simulation profile setup needs to be adjusted accordingly. In our case, final time was set to 10 ms (to visualize the robust strange attractor), maximum allowed time step was reduced to 1 µ s (to demonstrate the density of the strange attractor) and pseudo-components IC = − 1V (IC1) served to set nonzero initial conditions into the circuit. In real circuitry, the injection of certain initial conditions is a much more complicated task. Entropy 2021, 23, x FOR PEER REVIEW 17 of 24 4. Design of Flow-Equivalent Chaotic Oscillator Verification through practical experiment belongs to the common standard for the presentation of a new chaotic dynamical system. It is widely adopted that the observability of strange attractors represents satisfactory proof of the robustness of desired dynamics. Lumped circuit synthesis based on a prescribed mathematical model is a problem that can be easily solved using several different approaches. One of the most popular methods is based on an integrator block schematic, where basic mathematical operations are performed by three types of the two-port building blocks: inverting summing integrators, differential amplifiers and blocks having piecewise-linear or polynomial transfer curve. Each mentioned operation requires at least a single active element, usually a voltage feedback operational amplifier. The main drawback of this concept is evident: the necessity of using many active elements and a rather high power consumption. An integrator based kind of circuit realization is possible in three operational regimes: the most preferred is voltage-mode concept [38], current-mode is usually dedicated for the higher frequency bands [39] and mixed-mode circuits. It is worth nothing that the Orcad Pspice circuit simulator was used for the pre-validation of designed chaotic oscillators. Figure 18 shows maximally idealized case Ψ 1 and Ψ 3 systems with impedance norm 10 3 and frequency norm 10 6 . Of course, having ideal voltage-controlled current-sources (G) and ideal voltage multiplication blocks (MULT), both norms can be arbitrary; only simulation profile setup needs to be adjusted accordingly. In our case, final time was set to 10 ms (to visualize the robust strange attractor), maximum allowed time step was reduced to 1 μs (to demonstrate the density of the strange attractor) and pseudo-components IC = −1V (IC1) served to set nonzero initial conditions into the circuit. In real circuitry, the injection of certain initial conditions is a much more complicated task. Figure 18. Idealized circuit realization of a chaotic system with the emulated bipolar transistor stage: (a) case Ψ 1 system with parameters taken from Table 1, (b) case Ψ 3 system with parameter set taken from Table 1, (c) Monge projections v y1 vs. v x1 (blue) and i L1 vs. v x1 (red), (d) frequency spectrum of generated signal v x1 (blue) and v y1 (red). Red areas represent polynomial feedback transfer functions. Voltage-mode realizations ready for simulation/construction are provided in Figure 19 and these circuits undergo deep experimental verification. Supply voltage is symmetrical ±15 V. The designed oscillator consumes two cheap voltage feedback operational Figure 18. Idealized circuit realization of a chaotic system with the emulated bipolar transistor stage: ( a ) case Ψ1 system with parameters taken from Table 1, ( b ) case Ψ3 system with parameter set taken from Table 1, ( c ) Monge projections v y1 vs. v x1 (blue) and i L1 vs. v x1 (red), ( d ) frequency spectrum of generated signal v x1 (blue) and v y1 (red). Red areas represent polynomial feedback transfer functions. Voltage-mode realizations ready for simulation/construction are provided in Figure 19 and these circuits undergo deep experimental verification. Supply voltage is symmetrical ± 15 V. The designed oscillator consumes two cheap voltage feedback operational Entropy 2021,23, 175 18 of 24 amplifiers TL082 (in a single package), one current-feedback operational amplifier with compensation node (denoted by letter C) AD844 and two four-quadrant analog multipliers AD633. Schematic and associated realization using a breadboard are provided by means of Figure 20 . The first designed chaotic oscillator (Figure 19a) is described by following ordinary differential equations: d dt v1=−v1 R1C−v2 R3·C+K2v3 2 R2C,d dt v2=−v1 R·C−v3 R·C,d dt v3=v2 R·C, (18) where state vector transforms into x = (v 1 ,v 2 ,v 3 ) T and K= 0.1 is the internally trimmed transfer constant of AD633. Note that this chaotic system models the behavior of function (3) with nonzero values aand c, whereby other terms are zero. Entropy 2021, 23, x FOR PEER REVIEW 19 of 24 Figure 19. Chaotic system with emulated bipolar transistor stage: a) circuit realization of differential Equations (1) and (3) and parameter set Ψ 3 taken from Table 1, b) circuit implementation of Equations (1) and (3) and parameter set Ψ 1 or Ψ 4 taken from Table 1. Different realization of the chaotic oscillator offers the principal schematic given in Figure 24b. This system is described by following a set of ordinary differential equations: 𝐶𝑑𝑑𝑡𝑣=−𝑣 𝑅−𝐾 𝑅𝑣+𝐾 𝑅𝑣 , 𝐶𝑑𝑑𝑡𝑣=−𝑖−𝑦∙𝑣 , 𝐿𝑑𝑑𝑡𝑖=𝑣 , (20) where the forward transconductance y 21 is realized by a single-input single-output operational trans-conductance amplifier. Nonlinear transfer function is implemented by couple (third-order polynomial for Ψ 3 , fourth-order polynomial for Ψ 1 and Ψ 4 ) or three (fifthorder polynomial to reach sets of parameters Ψ 2 , Ψ 5 and Ψ 6 ) AD633. Figure 20. Photos captured during experimental investigation: a) PCB showing two two-ports where transconductances y 12 and y 21 are polynomials up to the fourth-order, b) two views onto breadboard with designed chaotic oscillator based on generalized class C amplifier. Figure 19. Chaotic system with emulated bipolar transistor stage: ( a ) circuit realization of differential Equations (1) and (3) and parameter set Ψ3 taken from Table 1, ( b ) circuit implementation of Equations (1) and (3) and parameter set Ψ1 or Ψ4 taken from Table 1. Entropy 2021, 23, x FOR PEER REVIEW 19 of 24 Figure 19. Chaotic system with emulated bipolar transistor stage: a) circuit realization of differential Equations (1) and (3) and parameter set Ψ 3 taken from Table 1, b) circuit implementation of Equations (1) and (3) and parameter set Ψ 1 or Ψ 4 taken from Table 1. Different realization of the chaotic oscillator offers the principal schematic given in Figure 24b. This system is described by following a set of ordinary differential equations: 𝐶𝑑𝑑𝑡𝑣=−𝑣 𝑅−𝐾 𝑅𝑣+𝐾 𝑅𝑣 , 𝐶𝑑𝑑𝑡𝑣=−𝑖−𝑦∙𝑣 , 𝐿𝑑𝑑𝑡𝑖=𝑣 , (20) where the forward transconductance y 21 is realized by a single-input single-output operational trans-conductance amplifier. Nonlinear transfer function is implemented by couple (third-order polynomial for Ψ 3 , fourth-order polynomial for Ψ 1 and Ψ 4 ) or three (fifthorder polynomial to reach sets of parameters Ψ 2 , Ψ 5 and Ψ 6 ) AD633. Figure 20. Photos captured during experimental investigation: a) PCB showing two two-ports where transconductances y 12 and y 21 are polynomials up to the fourth-order, b) two views onto breadboard with designed chaotic oscillator based on generalized class C amplifier. Figure 20. Photos captured during experimental investigation: ( a ) PCB showing two two-ports where transconductances y 12 and y 21 are polynomials up to the fourth-order, ( b ) two views onto breadboard with designed chaotic oscillator based on generalized class C amplifier. Entropy 2021,23, 175 19 of 24 Similarly, the second dynamical system is able to model differential equations with the polynomial function (3) with nonzero values of coefficients band d. The set of differential equations is: d dt v1=−v1 R3C−K+R2 R1+R2v2 2 R4C+K+R2 R1+R23v4 2 R5C,d dt v2=−v1 R·C−v3 R·C,d dt v3=v2 R·C, (19) whereargumentinbracketscanbechosenadvantageouslysuchthatequalityK+R 2 /( R1+R2) = 1 holds . The fundamental time constant of this circuit is τ=R·C= 104·10−8= 100 µs , but the main frequency components can be shifted toward the GHz band easily by appropriate frequency rescaling. Considering the normalized numerical values provided in Table 1, impedance rescaling 10 4 and frequency norm 10 8 circuit components for (18) with parameter set Ψ3 are: C= 10 nF, R= 10 k Ω ,R 1 = 33 k Ω ,R 2 = 50 Ω , and R 3 =2 k Ω . Analogically, circuit components for (19) with parameter set Ψ2 ( Ψ4 ) are the following: C= 10 nF, R= 10 k Ω , R 1 = 1 k Ω ,R 2 = 9 k Ω ,R 3 = 18 k Ω (25 k Ω ), R 4 = 4.8 k Ω (3.7 k Ω ), and R 5 = 91 Ω (50 Ω ). Figure 20a shows the PCB (Printed Circuit Board) of two uncoupled two-ports modeled by the adjustable admittance parameters. While input and output admittance is linear and represented by a variable resistor, trans-admittance y 12 and y 21 are polynomials up to the fourth order. The unoccupied socket is dedicated for integrated circuit TL084 (four operational amplifiers in a single package), or its empty pins can be used to connect PCB with the breadboard. PCB is designed such that a user can use switches to change signs of all coefficients of the polynomial trans-conductance y 12 (v 2 ) and/or y 21 (v 1 ). Figure 20b demonstrates the simplicity of the designed chaotic system. 5. Experimental Verification Selected strange attractors observed during experimental verification are provided in Figures 21–23. In the first two cases, individual plane projections captured by an oscilloscope are compared with numerically integrated results and Equations (1) and (3) for parameter set Ψ3 and Ψ1 , respectively—values are given in Table 1. A uniform 100 mV grid is used for numerical integration results. For the latter case, numerical mirrors of the visualized strange attractors are not provided. During measurement, strong sensitivity of type of the steady state to the initial conditions imposed into the chaotic oscillator has been confirmed. Nevertheless, goodish correspondence between theory and practical experiment was achieved. The route-to-chaos scenario can be traced via the shaping of nonlinear y 12 (v 2 ) function, namely by variable resistors R 4 and R 5 in Figure 19b. To obtain classes of the chaotic system characterized by sets Ψ2 , Ψ5 , Ψ6 , and Ψ7 , additional AD633 is necessary. However, a major part of the proposed oscillator remains unchanged. Entropy 2021, 23, x FOR PEER REVIEW 20 of 24 Figure 21. Dynamical system (1) with (3) and values Ψ 3 from Table 1, Comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 2 vs. v 3 plane, e) f) v 1 vs. v 2 plane. Figure 22. Dynamical system (1) with (3) and values Ψ 1 from Table 1, comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 1 vs. v 2 plane, e) f) v 2 vs. v 3 plane. Figure 23. Different Monge projections of strange attractors not mutually connected with numerical analysis of generalized chaotic class C amplifier. Figure 24. Two alternative lumped circuitry implementations of class C potentially chaotic amplifier: a) principal schematic of dynamical system with passive approximated fractional-order inductor, b) realization based directly on the state model (20). Figure 21. Dynamical system (1) with (3) and values Ψ3from Table 1, Comparison between numerical integration process (blue) and laboratory experiment (green): (a,b)v1vs. v3plane, (c,d)v2vs. v3plane, (e,f)v1vs. v2plane. Entropy 2021, 23, x FOR PEER REVIEW 20 of 24 Figure 21. Dynamical system (1) with (3) and values Ψ 3 from Table 1, Comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 2 vs. v 3 plane, e) f) v 1 vs. v 2 plane. Figure 22. Dynamical system (1) with (3) and values Ψ 1 from Table 1, comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 1 vs. v 2 plane, e) f) v 2 vs. v 3 plane. Figure 23. Different Monge projections of strange attractors not mutually connected with numerical analysis of generalized chaotic class C amplifier. Figure 24. Two alternative lumped circuitry implementations of class C potentially chaotic amplifier: a) principal schematic of dynamical system with passive approximated fractional-order inductor, b) realization based directly on the state model (20). Figure 22. Dynamical system (1) with (3) and values Ψ1 from Table 1, comparison between numerical integration process (blue) and laboratory experiment (green): (a,b)v1vs. v3plane, (c,d)v1vs. v2plane, (e,f)v2vs. v3plane. Entropy 2021,23, 175 20 of 24 Entropy 2021, 23, x FOR PEER REVIEW 20 of 24 Figure 21. Dynamical system (1) with (3) and values Ψ 3 from Table 1, Comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 2 vs. v 3 plane, e) f) v 1 vs. v 2 plane. Figure 22. Dynamical system (1) with (3) and values Ψ 1 from Table 1, comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 1 vs. v 2 plane, e) f) v 2 vs. v 3 plane. Figure 23. Different Monge projections of strange attractors not mutually connected with numerical analysis of generalized chaotic class C amplifier. Figure 24. Two alternative lumped circuitry implementations of class C potentially chaotic amplifier: a) principal schematic of dynamical system with passive approximated fractional-order inductor, b) realization based directly on the state model (20). Figure 23. Different Monge projections of strange attractors not mutually connected with numerical analysis of generalized chaotic class C amplifier. Different realization of the chaotic oscillator offers the principal schematic given in Figure 24b. This system is described by following a set of ordinary differential equations: C1 d dt v1=−v1 R1 −K Ry v2 2+K2 Rx v4 2,C2 d dt v2=−iL−y21·v1,Ld dtiL=v2, (20) where the forward transconductance y 21 is realized by a single-input single-output operational trans-conductance amplifier. Nonlinear transfer function is implemented by couple (third-order polynomial for Ψ3 , fourth-order polynomial for Ψ1 and Ψ4 ) or three (fifth-order polynomial to reach sets of parameters Ψ2,Ψ5and Ψ6) AD633. Entropy 2021, 23, x FOR PEER REVIEW 20 of 24 Figure 21. Dynamical system (1) with (3) and values Ψ 3 from Table 1, Comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 2 vs. v 3 plane, e) f) v 1 vs. v 2 plane. Figure 22. Dynamical system (1) with (3) and values Ψ 1 from Table 1, comparison between numerical integration process (blue) and laboratory experiment (green): a) b) v 1 vs. v 3 plane, c) d) v 1 vs. v 2 plane, e) f) v 2 vs. v 3 plane. Figure 23. Different Monge projections of strange attractors not mutually connected with numerical analysis of generalized chaotic class C amplifier. Figure 24. Two alternative lumped circuitry implementations of class C potentially chaotic amplifier: a) principal schematic of dynamical system with passive approximated fractional-order inductor, b) realization based directly on the state model (20). Figure 24. Two alternative lumped circuitry implementations of class C potentially chaotic amplifier: ( a ) principal schematic of dynamical system with passive approximated fractional-order inductor, ( b ) realization based directly on the state model (20) . Speaking in terms of commercially available active devices, trans-conductance amplifiers are available as LM13700, LT1228, MAX435, or diamond transistors OPA660, etc. Note that a nonlinear two-port needs to work in trans-admittance regime, i.e., with input voltage and output current. If both impedance and frequency scaling factors could be arbitrary real numbers, we experience two degrees of freedom for the calculation of inductance and capacitance. Therefore, the parallel inductor-capacitor resonant tank could be arbitrary as well, i.e., associated with an audio amplifier, an active part of a sensor element, a model of piezo-element, a matching subcircuit, etc. Entropy 2021,23, 175 21 of 24 Fractional-Order Chaotified Class C Amplifier Now assume that the transistor is loaded by the non-integer order LC tank. Fractionality will be represented by the presence of a fractional-order (FO) inductor. This FO inductor will be approximated in the frequency domain by a more complicated network [ 40 ] with a phase frequency response of impedance rippled around ideal value 90 ◦γ . Mentioned approximation should be valid at least in a finite frequency range that corresponds to the desired chaotic signal generated by the FO chaotic class C amplifier. In our case, frequency range turns to be from 1 Hz up to 2 MHz (see also Figure 18) and the minimal complexity of the FO inductor is 7 (number of required resistor-inductor sections). Note that the impedance of the RL approximation circuit tends to R c for very low frequencies and reaches to infinity for high frequencies. A set of ordinary differential equations that describes circuitry given in Figure 24b with an FO inductor L x with seven sections is as follows: d dt v1=−v1 R1·C1−K Ry·C1v2 2+K2 Rx·C1v4 2,d dt v2=−v2 R2·C2−y21 C2·v1−1 C2·iLc, d dt iLc=v2 Lc−Rc Lc·iLc −1 Lc·∑7 k=1RckiLc−iLck ,d dt iLck =Rck Lck iLc−iLck , (21) where k= 1, . . . , 7. Speaking in terms of FO network analysis, this is the place where conventional circuitoriented simulation software such as Orcad Pspice can be utilized. A gallery of passive ladder FO capacitors calculated for important decimal orders between zero and one can be found in paper [ 41 ]. Structures proposed in this paper have been optimized from the viewpoint of low phase error (less than 1.5 ◦ ) and wide frequency range (from 3 Hz up to 3 MHz). By following the duality principle, these findings can be extended and FO inductors for orders 9/10 (Table 4), 8/9 (Table 5), 4/5 (Table 6), and 3/4 (Table 7) are presented. Numerical values provided in these tables lead to FO inductors having unity pseudo-inductance, i.e., the module of impedance measured at the specific frequency f0= 1/(2π) Hz is 1 s1−α/F, where αrepresents math order. Table 4. Numerical values of fully passive series-parallel circuit realization of fractional-order (FO) inductor with mathematical order 9/10, i.e., phase shift between voltage and current 81◦. RaR1R2R3R4R5R6R7 0.6 Ω3.3 Ω22.7 Ω153 Ω1031 Ω6944 Ω46.7 kΩ313 kΩ LaL1L2L3L4L5L6L7 144 mH 120 mH 98 mH 79 mH 64 mH 52 mH 42 mH 37 mH Table 5. Numerical values of fully passive series-parallel circuit realization of FO inductor with mathematical order 8/9, i.e., phase shift between voltage and current 80◦. RaR1R2R3R4R5R6R7 1Ω6.3 Ω44.4 Ω319 Ω2286 Ω16.4 kΩ118 kΩ833 kΩ LaL1L2L3L4L5L6L7 203 mH 230 mH 194 mH 152 mH 120 mH 93 mH 73 mH 62 mH Table 6. Numerical values of fully passive series-parallel circuit realization of FO inductor with mathematical order 4/5, i.e., phase shift between voltage and current 72◦. RaR1R2R3R4R5R6R7 1.1 Ω4.7 Ω25.8 Ω141 Ω769 Ω4184 Ω22.7 kΩ133 kΩ LaL1L2L3L4L5L6L7 35 mH 237 mH 155 mH 101 mH 66 mH 43 mH 28 mH 22 mH Entropy 2021,23, 175 22 of 24 Table 7. Numerical values of fully passive series-parallel circuit realization of FO inductor with mathematical order 3/4, i.e., phase shift between voltage and current 67.5◦. RaR1R2R3R4R5R6R7 1.2 Ω4.5 Ω22 Ω108 Ω526 Ω2591 Ω12.7 kΩ55.6 kΩ LaL1L2L3L4L5L6L7 13 mH 210 mH 132 mH 78 mH 46 mH 27 mH 16 mH 10 mH Author encouragement for interested readers: please do not hesitate to contact me (via email) if a specific mathematical order, frequency range, approximation network complexity or different phase accuracy of the FO capacitor and/or inductor is required. 6. Discussion This paper brings an example of an electronic circuit for which, under very specific circumstances, the circuit can switch from regular into chaotic behavior. Conditions leading to chaotic motion can be summarized as follows: 1. General mathematical models analyzed in this paper (3) and (10) contain normalized values of all accumulation elements. After optimization, to observe strange attractors, resulting parasitic capacitance as well as capacitance and inductance located within the LC resonant tank are of comparable orders. Therefore, parasitic accumulation elements turn into functional. This fact increases the intrinsic number of degrees of freedom and forces a naturally non-chaotic analogue building block to behave chaotically. Because of the internal structure of bipolar transistors commonly used in class C amplifiers, this kind of motion is possible only for assumed high-frequency operation. In practice, generated chaotic waveform can be easily misinterpreted as noise. 2. The second condition for chaos evolution is the presence of a specific local nonlinear feedback. In the mathematical model of the analyzed dynamical system, either polynomial or PWL scalar function is the only nonlinearity. 3. The third specific property of a bipolar transistor is linear backward trans-conductance. Its value is non-zero and relatively large. 7. Conclusions This manuscript admits the parasitic properties of a bipolar transistor to be the working accumulation elements; base-emitter capacitance is mandatory, while collector-emitter capacitance may not be present. There is one consequence resulting from this research: chaos belongs to the natural behavior of sub circuits that contain at least one bipolar transistor, although the expected working regime may seem rather hypothetical for a practically oriented design engineer. This statement agrees with the conclusion reached in paper [ 42 ], where the JFET element is the single locally active element while the coil is the passive one. In our case, fingerprints of a bipolar transistor can be found in both folding and stretching mechanisms of a vector field. Additionally, a set of ordinary differential equations together with the six different sets of internal parameters proposed in this paper can be considered as a new chaotic dynamical system. This system is a member of autonomous deterministic systems with a single center-type stable equilibrium point. Strange attractors observed in the frame of numerical investigations are in good accordance with those captured as oscilloscope screenshots using a flow-equivalent circuit. This paper leaves significant space for further research, for example, to find system parameters close to the common operation of a single transistor stage where: 1. Parasitic capacitors are working ones, 2. Nonlinearity is typical for a large signal model of a bipolar transistor, 3. An additional degree of freedom is presented because driving force (processed signal) changes the operational point of an analyzed circuit. The results presented in this work are strictly associated with a single-stage class C amplifier with a single bipolar transistor. It is well known that the probability of chaos Entropy 2021,23, 175 23 of 24 rises with the total order of a circuit. 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