Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

Abstract

We present two dual oscillating circuits having a wide spectrum of dynamical properties but relatively simple topologies. Each circuit has five bifurcating parameters, one nonlinear element of cubic current–voltage characteristics, one controlled element, LCR components and a constant biasing source. The circuits can be considered as two coupled oscillators (linear and nonlinear) that form dual jerk circuits. Bifurcation diagrams of the circuits show a rather surprising result that the bifurcation patterns are of the Farey sequence structure and the circuits’ dynamics is of a fractal type. The circuits’ fractal dimensions of the box counting (capacity) algorithm, Kaplan–Yorke (Lyapunov) type and its modified (improved) version are all estimated to be between 2.26 and 2.52. Our analysis is based on numerical calculations which confirm a close relationship of the circuits’ bifurcation patterns with those of the Ford circles and Stern–Brocot trees.

1 Introduction

In the earlier paper [16] two dual circuits similar to those shown in Fig. 1 (without the coupling conductance G and resistance R) were investigated from the point of view of their mixed-mode oscillations (MMOs), canard solutions and folded points of the corresponding system of differential-algebraic equations (DAEs).

Fig. 1

(A) The LCL circuit with L ₁=ϵ≪1, V=(1+b)x ₂, $V_{\mathrm{nonl}}=\alpha x_{1}^{2}+\beta x_{1}^{3}$. (B) The CLC circuit with C ₁=ϵ≪1, I=(1+b)x ₂, $I_{\mathrm{nonl}}=\alpha x_{1}^{2}+\beta x_{1}^{3}$. For both circuits: a>0, b>0, β<0 and α>0

Mathematical model of such circuits comprises two linear and one nonlinear ordinary differential equations (ODEs). Such a model was also investigated in [9] and several conclusions about the occurrence of the L ^s patterns (large and small amplitude oscillations, or LAOs and SAOs, respectively) were drawn.

Each of the two circuits in Fig. 1 can be considered as a coupling of two oscillators (nonlinear and linear ones). The nonlinear oscillator comprises a nonlinear element with a cubic characteristics V _nonl (or I _nonl), an inductor (Fig. 1(A)) or a capacitor (Fig. 1(B)), while the linear oscillator includes an inductor, a capacitor, a voltage controlled voltage source (or a current controlled current source) and a biasing constant voltage (or current) source. The inductance L and capacitance C in the circuits in Fig. 1 are split between the two oscillators as shown in Fig. 2 with C _a +C _b =C and L _a +L _b =L.

Fig. 2

Coupling of oscillators through conductance G and resistance R. The coupling is of full strength if G=0 for circuit A (equivalently R=0 for circuit B) while the oscillators are completely decoupled if G→∞ (equivalently R→∞)

If we assume that 0<L ₁≡ϵ≪1 and $C, L_{2} \sim\mathcal {O}(1)$ for the circuit in Fig. 1A (0<C ₁≡ϵ≪1 and $L, C_{2} \sim\mathcal{O}(1)$ for the circuit in Fig. 1B), then both circuits can be described by the singularly perturbed model

$$ \begin{array}{@{}rcl}\epsilon x_1' & = & -x_2 +\alpha x_1^2+\beta x_1^3\equiv g(x_1,x_2,x_3)\\[1mm]x_2' & = & x_1-x_3-Kx_2\equiv f_1(x_1,x_2,x_3)\\[1mm]x_3' & = &a-bx_2\equiv f_2(x_1,x_2,x_3)\end{array} $$

(1)

where K≡G for the circuit in Fig. 1A and K≡R for the circuit in Fig. 1B.

As reported in [9, 16], the MMOs phenomenon (a combination of LAOs and SAOs) may include oscillations with quite complex patterns. The SAOs trace the weak canard trajectory of (1) around the origin (0,0,0), while the LAOs are due to the cubic nonlinearity and return mechanism. Canard trajectories (weak and strong) are closely related to certain folded nodes of DAEs that one obtains from (1) by assuming ϵ=0. Furthermore, a desingularization process of DAEs results in a system of two ODEs with a folded nodal point. See [1, 2, 6, 10, 13–15, 19, 21, 23] for more details of these properties.

It is known that (1) is a prototypical model of mixed-mode oscillations with three time scales. Since a and b in (1) are usually of order ϵ, therefore one can consider x ₁, x ₂ and x ₃ as the fastest, medium and slowest variables, respectively. Two time scales are also possible to achieve in (1) by an appropriate choice of parameters. Many practical oscillating systems in chemistry, physics and neuroscience have multiple time scales, exhibit MMOs and chaos (e.g. the Belousov–Zhabotinski reaction, Bonhoeffer–van der Pol oscillator, neuron oscillations in human brain and others [6]).

In this paper we further analyze the two circuits in Fig. 1. The MMOs are not the only possible dynamical features of the circuits. In fact, all the parameters α, β, a, b and K in (1) may be considered as the bifurcation parameters. We shall analyze the local maximum values of LAOs and SAOs and show that, by slowly changing any of the above parameters, we can obtain the local maximum values of x _i (t), i=1,2,3, that can be described by the Farey arithmetic [17] and related to the Stern–Brocot tree [8, 24]. In our analysis we use the letters L and s in L ^s to describe the number of local maximum values of the steady-state LAOs and SAOs, respectively, and not the number of LAOs and SAOs. Thus, our L ^s is different from the L ^s used, for example, in [9, 16]. Through a series of numerical experiments we shall show that the L ^s oscillations for the circuits in Fig. 1 occur in the order of Farey sequence of coprime integers. Also, the circuits can be considered as ones having a chaotic strange attractor, and as such can be characterized by a special number called the fractal dimension. Various fractal dimensions of the circuits in Fig. 1 are discussed in Sects. 7 and 4.

The paper is written for the purpose of showing how relatively simple circuits with one nonlinear element exhibit an amazing spectrum of dynamical properties ranging from MMOs, through chaotic responses, various bifurcation diagrams related to the Farey sequence of coprime integers, to fractal properties. We also propose a new bifurcation tree illustrating a hierarchical structure in which the circuits oscillate (see Figs. 12 and 13 to follow). We also show a relationship between the dynamical properties of (1) and Newton’s second law (in Sect. 4.3), allowing for a “mechanical” interpretation of x ₁, $x_{1}'$ and $x_{1}''$ as “position”, “velocity” and “acceleration,” respectively. It is our belief that the circuits analyzed in this paper could become benchmark examples of nonlinear circuits’ studies.

The circuits shown in Fig. 1 also fall into the category of chaotic jerk circuits [22], that is, the circuits described by a nonlinear third order equation in the form x‴=J(x″,x′,x). The nonlinear function J is called a jerk since, in mechanical systems, it would correspond to the first derivative of acceleration of x(t). The jerk structure of (1) follows from the fact that variable x ₁(t) in (1) satisfies the following equation:

$$ x_1''+\bigl(K-\epsilon^{-1}\bigl(2\alpha x_1+3\beta x_1^2\bigr)\bigr)x_1' + \epsilon^{-1}\bigl(x_1-K\bigl(\alpha x_1^2+\beta x_1^3\bigr)\bigr)=\epsilon^{-1}x_3(t)$$

(2)

which is coupled with the first-order equation

$$ x_3'-b\epsilon x_1'=a-b\bigl(\alpha x_1^2+\beta x_1^3\bigr).$$

(3)

From (2) and (3) we obtain the jerk equation in variable x ₁

(4)

where $N(x_{1})\equiv\alpha x_{1}^{2}+\beta x_{1}^{3}$.

Alternatively, one can consider the second-order equation $x_{3}''+Kx_{3}'-bx_{3}=Ka-x_{1}$ coupled with (3). This gives another jerk equation in variable x ₃, as follows:

(5)

This equation is structurally different from the jerk equation (4). We comment about this difference and further properties of both jerk equations in Sect. 4.

The paper is organized as follows. In Sect. 2 we present numerically obtained representative bifurcation diagrams when parameters α, β, a and b vary. In Sect. 7 we compare the hierarchical oscillations of the circuits to the Farey sequences, Stern–Brocot trees and Ford circles. After analyzing several bifurcation diagrams, we also estimate the circuits’ counting box fractal dimension in Sect. 7. Then, we construct our own tree with hierarchical windows of a fractal nature. In the infinite sequence of such windows we focus on those windows denoted by 2^k, k being a positive odd integer, as they are formed in the shape of the Stern–Brocot trees. We discuss and compare the circuits’ fractal counting box, Kaplan–Yorke and modified (improved) Kaplan–Yorke dimensions in Sect. 4. The jerky Newtonian nature of (4) and its relationship to Newton’s second law F=mx″ is also discussed in Sect. 4. Finally, we draw our conclusions in Sect. 5. Throughout the paper the symbol ⊕ stands for the following operation on four integers p ₁, q ₁, p ₂ and q ₂: p ₁/q ₁⊕p ₂/q ₂=(p ₁+p ₂)/(q ₁+q ₂).

2 Bifurcation Diagrams

The three basic modes of operation of the circuits are illustrated in Fig. 3 [16] which shows the time series and 2D solutions of (1) with ϵ=0.01, α=1.5, β=−1, b=0.005, K=0 and three values of a. In the SAOs only case, the small amplitude oscillations around the origin (0,0,0) are due to the Hopf bifurcation for a=0. We used non-zero initial conditions only for the SAOs case in the top panel in Fig. 3. In all other calculations in this paper the initial conditions were all zero.

Fig. 3

(Color online) Left column: x ₁ versus x ₂; right column: time responses (172≤t≤194); both columns: SAOs only (top: a=0), LAOs only (middle: a=0.00105) & MMOs (bottom: a=0.00055) with x ₁ (blue), x ₂ (red), x ₃ (green)

In the LAOs only case, a trajectory bypasses the region of small amplitude oscillations around the origin. The MMOs case is in some sense a combination of the previous two cases. The mechanism in which SAOs and LAOs occur is quite complex and has been the topics of recent papers [2, 6, 9, 10, 16, 19, 23].

The occurrence of MMOs is not the only interesting feature of the circuits in Fig. 1. When parameters α, β, a or b change slowly, system (1) may bifurcate and show complex chaotic behavior with MMOs transitions of various types. Figures 4, 5, 6, 7 show typical bifurcation diagrams with the vertical axis representing the maximum values of x ₁(t) identified in the interval 300≤t≤500. All calculations were done for 0≤t≤500 (with zero initial conditions), but the interval 0≤t<300 was not used in the process of identifying the maximum values of x ₁(t). Throughout the whole analysis we kept the value of ϵ unchanged at 0.01.

Fig. 4

Bifurcation diagram for α=1, β=−1, a=0.0005, ϵ=0.01, K=0.3 and 0.005<b<0.035 with intervals of MMOs sequence 1⁰→1¹→1²→⋯→1⁹→0¹

Fig. 5

Bifurcation diagram for α=1, β=−1, b=0.005, ϵ=0.01, K=0.3 and 0.00015<a<0.0007 with intervals of MMOs sequence 0¹→1⁸→1⁷→⋯→1¹→1⁰

Fig. 6

Bifurcation diagram for α=1, a=0.0005, b=0.005, ϵ=0.01, K=0.3 and −1.15<β<−0.70 with intervals of MMOs sequence 1⁰→6¹→5¹→⋯→1¹→1²→1³

Fig. 7

Bifurcation diagram for β=−1, a=0.0005, b=0.005, ϵ=0.01, K=0.3 and 0.9<α<1.1 with intervals of MMOs sequence 1⁰→4¹→⋯→1¹→⋯→2³

Bifurcation diagram in Fig. 4 clearly shows a pattern of sequence 1⁰→0¹ (through a series of oscillations: 1⁰→1¹→1³→⋯→0¹), while the sequence in Fig. 5 is of type 0¹→1⁰. On the other hand, Figs. 6 and 7 show the sequence of local maximum values 1⁰→1¹ with a few additional values of L ^s to the right of 1¹. The intervals S _i will be used in the sequel to determine the fractal dimensions of the circuits, as a detailed analysis of the L ^s patterns clearly indicates that the patterns are directly linked to the Farey sequence of the pairs of coprime integers (see the L ^s values shown in Figs. 4, 5, 6, 7 and in Figs. 8, 9, 10, 11, which follow). As a consequence, we shall conjecture that the dynamics of the circuits in Fig. 1 is in fact of a fractal type. Next, we shall use the bifurcation diagrams with varying b to proceed with a more detailed analysis.

Fig. 8

Detailed bifurcation sequence of MMOs 1⁰→1¹ with several lower level hierarchical oscillations (interval S ₀ in Fig. 4)

Fig. 9

Detailed bifurcation sequence of MMOs 1¹→1² with several lower level hierarchical oscillations (interval S ₁ in Fig. 4)

Fig. 10

Detailed bifurcation sequence of MMOs 1²→1³ with several lower level hierarchical oscillations (interval S ₂ in Fig. 4)

Fig. 11

Detailed bifurcation sequence of MMOs 1³→1⁴ with several lower level hierarchical oscillations (interval S ₃ in Fig. 4)

3 Fractal Windows and Trees

First, when a resolution of computation is increased, one can obtain more detailed bifurcation diagrams for any of the ten intervals S _i in Fig. 4. Figures 8, 9, 10, 11 show four such diagrams. These graphs yield a graphical representation of the first several layers of the L ^s sequences as is shown in Fig. 12. This tree is a modification of the Farey and Stern–Brocot trees [8, 24] and also corresponds to the Ford circles [7]. Any pair of coprime integers L ^s is marked in Fig. 12 as a single × point with its horizontal coordinate L/(L+s), sometimes called the firing number. The vertical coordinate depends on the window to which the pair L ^s belongs. For fixed values of coprime integers L and s, the tree in Fig. 12, just like the Farey sequence, includes all possible coprimes between L ^s and s ^L. The pairs of coprimes between L ^s and s ^L (coprimes at the same vertical level in Fig. 12) belong to different consecutive windows. For instance, at the vertical level 8 we have all coprimes between 1⁸ and 8¹ in the following sequence: 1⁸, 2¹³, 3¹⁷, 3¹⁶, 4¹⁹, 4¹⁷, 5¹⁹, 5¹⁸, 5¹⁷, 5¹⁶, 6¹⁷,…, 8⁷, 8⁵, 8³, 8¹, with their increasing firing numbers 1/9, 2/15, 3/20, 3/19, 4/23, 4/21, 5/24, 5/23, 5/22, 5/21, 6/23,…, 8/15, 8/13, 8/11, 8/9. Also, the coprimes 8¹, 8³, 8⁵, 8⁷ are located inside the window 2¹ (because (8+8+8+8)/(1+3+5+7)=2/1, see the vertical level 8). The coprimes 8⁹, 8¹¹, 8¹³, 8¹⁵ are located inside the window 2³ (because (8+8+8+8)/(9+11+13+15)=2/3, see the vertical level 9), etc. The total number of coprime pairs 8^s within each window equals φ(8)=φ(2³)=4, where φ(n) is Euler’s totient function [7]. Similarly, the total number of coprime pairs 15^s within each window is φ(15)=φ(3)φ(5)=8. For example, in the window 2¹ we have the coprimes 15¹, 15², 15⁴, 15⁷, 15⁸, 15¹¹, 15¹³, 15¹⁴ which yield $15^{1}\oplus15^{2}\oplus15^{4}\oplus15^{7}\oplus15^{8}\oplus15^{11}\oplus 15^{13}\oplus15^{14}=\frac{8\times15}{16+17+19+22+23+26+28+29}=\frac {2}{3}$, which is the firing number for 2¹. This sequence of coprime pairs 15^s continues in window 2³ with the coprimes 15¹⁶, 15¹⁷, 15¹⁹, 15²², 15²³, 15²⁶, 15²⁸, 15²⁹ which result (with the ⊕ operation) in value 2/5, that is, the firing number of 2³. This pattern continues indefinitely for windows 2⁵, 2⁷,…. The hierarchical oscillating sequences of L ^s grouped by windows in Fig. 12 (also in Fig. 13) are interesting as each of the 2^k (k being odd positive) windows is in fact the Stern–Brocot tree [24]. We discuss this relationship later on.

Fig. 12

Bifurcation tree with L ^s, L≤20, s≤20: each window 1^k, k=1,2,…,20, comprises two disjoint windows 2ⁿ and 2^l such that $\frac{1}{k}=\frac{2}{n}\oplus\frac{2}{l}$ (e.g. the 1¹ window comprises windows 2¹ and 2³ with $\frac{1}{1}=\frac {2}{1}\oplus\frac{2}{3}=\frac{4}{4}$). Also, each 2^k window comprises two disjoint windows 3ⁿ and 3^l such that $\frac {2}{k}=\frac{3}{n}\oplus\frac{3}{l}$, etc. Within each window the L ^s elements at the same horizontal level have the same L value, their respective s values run through all possible integer values such that g.c.d.(L,s)=1 and ∑ _m ⊕L _m /s _m =L/s (e.g. the 2³ window includes the elements 8¹⁵, 8¹³, 8¹¹ and 8⁹ at the level marked 9 with $\frac{8}{15}\oplus\frac {8}{13}\oplus\frac{8}{11}\oplus\frac{8}{9}=\frac{32}{48}=\frac {2}{3}$). Other similarity patterns also exist for this tree, including vertical patterns within each window. See Fig. 13 with a partial illustration of such a vertical pattern for the 2¹ window. Also, each window 2^k has a pattern of elements as shown in Fig. 15. This tree is, if fact, the Stern–Brocot tree [24]

Fig. 13

Bifurcation tree as in Fig. 12 with a pattern of vertically overlapping windows inside 2¹. The same pattern exists for all windows 2^k, k≥1 and odd. There are no numbers L ^s in the areas of light-shaded tongues of 1^k. Each dark-shaded window can also be subsequently partition into light-shaded tongues and dark-shaded windows

We begin with the bifurcation diagram in Fig. 4. Using the values of S _i and S _T one can solve the following equation to compute the fractal dimension, D [17]:

$$ \sum_{i=0}^n(S_i/S_T)^D=1.$$

(6)

Equation (6) is known as the box counting (capacity) formula with D called the counting box (capacity) fractal dimension [4]. In our computations we have chosen i=0,…,9, that is, 10 intervals S _i between various L ^s sequences, as shown in Figs. 4, 8–11. Equation (6) for the diagram in Fig. 4 is

which yields D=0.8449. Due to a certain degree of uncertainty associated with a precise estimation of the boundaries of S _i and S _T in our bifurcation diagrams, it is impossible to compute the exact value of D, but we conjecture (based on further analysis) that the counting box (capacity) dimension for each of the three variables x _i (t), i=1,2,3 is about 0.84. This statement follows from the examination of the diagrams in Figs. 4, 8–11 and also those in Figs. 16 and 17 to be done later. Table 1 shows several estimates of the values of D based on (6) and the five bifurcation diagrams with varying parameter b. Further comparison of the fractal dimension D for the circuits and the fractal dimension of the sequences of Ford circles p/q (see Fig. 14) is given in Table 2. We only use the Ford circles with 0≤p/q≤1 since all the sequences L ^s (marked as points × in Fig. 12) have their first coordinate between 0 and 1. For example, the three L ^s sequences 2³, 2¹ and 3¹ have their representative points × with the first coordinates 2/5, 2/3 and 3/4, respectively (see Fig. 12).

Fig. 14

Several Ford circles with 0≤p/q≤1. More Ford circles with p/q>1 can be found in [24]

Table 1 Fractal dimensions D (6)

Table 2 Comparison of D: circuits and Ford circles

The fractal dimension values in Table 2 of such different fractal objects (circuits and Ford circles) are surprisingly in a very good agreement. It is also known that the total area of all Ford circles for 0<p/q<1 (therefore the fractal dimension of the Ford circles) can be computed as follows. Since the Ford circles associated with fractions 0<p/q<1, g.c.d.(p,q)=1, are centered at (p/q,1/(2q ²)) and have their radii equal 1/(2q ²) (see Fig. 14), therefore the total area of such circles is¹

(7)

Using the Euler’s totient and Riemann zeta functions, φ and ζ, respectively, we obtain

$$ A={\pi\over4}\sum_{q\ge1}{{\varphi(q)}\over{q^4}} ={{\pi\zeta(3)}\over{4\zeta(4)}} ={{45\zeta(3)}\over{2\pi^3}}\approx0.8723$$

(8)

where ζ(4)=π ⁴/90 was used.

Next, it is interesting to notice that each of the windows 2^k in Fig. 12 is in fact the Stern–Brocot tree. If we consider each such window as a polygon with the top point 2^k and the distance between the boundary points on the left and right sides of the window to be 1, then each window 2^k can be represented by the tree shown in Fig. 15. This is the Stern–Brocot tree. All the coprime pairs (L,s) represented by the × sign in Fig. 12 are now the red dots (color online) in Fig. 15. The tree in Fig. 15 has hierarchical symmetries corresponding to the symmetries of the internal (smaller) windows of each 2^k window. For example, by selecting the window 2¹ in Fig. 12 we see two subsequent internal windows 3² and 3¹. These two windows correspond to the left and right subtrees in Fig. 15. Each of the windows 3² and 3¹ has its own consecutive symmetry, etc. Infinite self-similarity fractal pattern is clearly seen in the splitting of the trees in Fig. 15.

Fig. 15

(Color online) General structure of all windows 2^k in Fig. 12 (Stern–Brocot tree [24]). The top red dot corresponds to the top 2^k point of each window followed by horizontal segments with 2,2,4,2,6,4,6,4,10,… points. This sequence of numbers is equivalent to the values of Euler’s totient function φ(n), n=2,3,…

4 More Estimates of Fractal Dimensions and Comparison with Other Chaotic Systems

4.1 Fractal Dimensions D _KY and D _Σ

By using the bifurcation diagram of x ₁(t) in Fig. 4 and formula (6) in Sect. 7 we estimated the counting box fractal dimension D to be 0.84. Bifurcation diagrams of the remaining variables x ₂(t) and x ₃(t) are shown in Figs. 16 and 17. If the formula (6) is applied to both of those diagrams, then one obtains the same values of the counting box fractal dimensions as we obtained for x ₁(t) (D=0.84). Thus, all three variables together yield the counting box fractal dimension, say D _c , of the circuits as D _c =2.52 (that is 3×0.84).

Fig. 16

Bifurcation sequence of MMOs 1⁰→1¹→⋯→1⁹→0¹ for x ₂(t) and the same parameters as in Fig. 4

Fig. 17

Bifurcation sequence of MMOs 1⁰→1¹→⋯→1⁹→0¹ for x ₃(t) and the same parameters as in Fig. 4

It is worth comparing the above fractal dimension with some of the other fractal dimensions used in the analysis of fractals. It is well know that the Kaplan–Yorke (Lyapunov) fractal dimension is a widely used one when the dynamical equations are available. For a three-variable nonlinear dynamical system with a strange attractor the Kaplan–Yorke dimension is found by numerically calculating the Lyapunov exponents λ _i , i=1,2,3, and then using them in the formula [3]

$$ D_{KY}=1-\frac{\lambda_1}{\lambda_3}$$

(9)

with λ ₁>0, λ ₂=0, λ ₃<0 and λ ₁+λ ₃<0 (conditions for the existence of strange attractor). On the other hand, the modified (improved) Kaplan–Yorke dimension, denoted by D _Σ results from a quadratic interpolation rather than a linear one used in deriving (9). In the three-variable chaotic flow with a strange attractor the D _Σ is computed from [3]

$$ D_{\varSigma }=1.5+0.5\sqrt{1-8\frac{\lambda_1}{\lambda_3}}$$

(10)

with D _Σ ≥D _KY for −1≤λ ₁/λ ₃≤1. The maximum difference D _Σ −D _KY is 1/8 and it occurs at λ ₁/λ ₃=−3/8. Finally, it is known that for most chaotic flows we find that the Kaplan–Yorke and counting box (capacity) dimensions satisfy the following relation: D _KY ≤D _Σ .

To make our calculations comparable and consistent, we have used the same interval S _T (see Figs. 4, 16 and 17) to find D _KY and D _Σ for the circuits in Fig. 1. Figure 18 shows the D _c , D _KY and D _Σ (the two later dimensions vary with varying b). This figure certainly supports the generally accepted conjecture that D _KY ≤D _Σ ≤D _c .

Fig. 18

Fractal dimensions D _c =2.52 (const), D _Σ and D _KY with varying parameter b

4.2 Comparison of Fractal Dimensions

Fractal dimensions, in most cases, depend on the parameters and nonlinearities used. For example, in the case of Chua circuit, for most cases shown in Tables 1 and 2 in [20] we obtain the Kaplan–Yorke fractal dimensions, D _KY , in the interval 2.00–2.25, but there are also cases with D _KY equal 2.32, 2.40 or even 2.94. Other oscillators described by systems of three ODEs with only quadratic nonlinearities yield chaotic strange attractors with fractal dimensions D _KY that span practically the whole interval from 2 to 3. Those with values slightly greater than 2 come from simple flows with just one or two quadratic nonlinear terms in all three equations [18], while those systems with more quadratic terms yield higher values of D _KY , often between 2.5 and 3 [3]. The Rössler and Lorenz systems have their fractal dimensions of 2.01 and 2.06, respectively, since they contain only one (Rössler) and two (Lorenz) quadratic terms. On the other hand, the Newton–Leipnik system with three quadratic terms has D _KY between 2.24 and 2.30 (depending on parameters). The Rabinovich–Fabrikant system with five nonlinear terms has D _KY =2.30 (for parameters γ=0.87, α=1.1) [11]. The circuits studied in this paper have two nonlinear terms (quadratic, cubic) and their D _KY is between 2.26 and 2.36 (with varying parameter b, Fig. 18). This fits well with the general observation that simple three-variable ODEs with one or two nonlinear terms yield fractal dimensions D _KY closer to 2, while those with more nonlinear terms tend to have greater D _KY , around 2.5, 2.6, or closer to 3. Our system (1) falls in between: the system has slightly more complicated nonlinear terms than the Rössler and Lorenz systems, but still not too complicated to yield D _KY values above 2.5.

4.3 The jerky Newtonian Dynamics

The two jerk equations in the introductory section are quite different in nature with (4) being jerky Newtonian while (5) is not Newtonian. A jerk Newtonian equation originates from Newton’s law x″−(1/m)F=0 to become x‴−(1/m)(d/dt)F=0, where m is a mass and (d/dt) denotes the total derivative, yielding

$$ x'''-\frac{1}{m}\bigl[(\partial_{x'}F)x''+(\partial_xF)x'+\partial_tF \bigr]=0.$$

(11)

Thus, from (4) we obtain

$$ \begin{array}{@{}rcl}\partial_{x_1'}F & = & m\bigl[\epsilon^{-1}N'(x_1)-K\bigr] \\[1mm]\partial_{x_1}F & = & m\bigl[\epsilon^{-1}N''(x_1)x_1'-\epsilon^{-1}+\epsilon^{-1}KN'(x_1)+b\bigr] \\[1mm]\partial_tF & = & m\bigl[\epsilon^{-1}a - \epsilon^{-1}bN(x_1)\bigr]\end{array}$$

(12)

and (12) yields the “acceleration” $x_{1}''$ (=F/m) as follows:

(13)

where C is a constant.

It is easy to check from (12) that $\partial_{x_{1}x_{1}'}F=\partial_{x_{1}'x_{1}}F=m\epsilon^{-1}N''(x_{1})$. The integral term in (13) indicates that F depends not only on the instantaneous “position” x ₁(t) and “velocity” $x_{1}'(t)$, but also on the “memory” term $\epsilon^{-1}b\int_{0}^{t} N(x_{1}(\tau))d\tau$. This term depends on x ₁, but not on $x_{1}'$. The jerk equation (4) is of polynomial and Newtonian type.²

Because of the nonlinear function $N(x_{1})=\alpha x_{1}^{2}+\beta x_{1}^{3}$, the jerk equation (5) is not linear in $x_{3}''$. Thus, (5) is of a polynomial, but not Newtonian type. One cannot find the F/m expression in the way it was possible for variable x ₁. Also, it is not possible to find a single jerk equation for the remaining variable x ₂. This is, again, due to the polynomial nonlinearity $\alpha x_{1}^{2}+\beta x_{1}^{3}$ in (1). It remains to be seen what consequences of such a “mechanical” characterization of (1) are to the “electrical” variables x ₁, x ₂ and x ₃ (currents and voltages). An application of the electromechanical Appell function seems to be of interest here and should shed more light on the jerky dynamics of (1) [25].

To compare, the celebrated Rössler system x′=−y−ez, y′=x+ey and z′=1+(x−c)z is jerky polynomial and Newtonian in y, jerky rational but not Newtonian in both x and z. The Lorenz systems is non-Newtonian in all three variables [12].

5 Conclusions

We have analyzed two dual circuits with one nonlinear element described by a cubic nonlinearity (parameters α and β), a linearly controlled voltage (current) source defined by parameter b, a small constant voltage (current) source (parameter a) and a linear coupling resistance (conductance) K. Each of those parameters can be considered as a bifurcation parameter. For particular sets of the five parameters we obtain various sequences of local maximum values corresponding to the regular periodic MMOs, chaotic oscillations, simple relaxations, or small amplitude oscillations. Various bifurcation diagrams show the fractal nature of the local maximum sequences. Graphical representation of those sequences results in special two-dimensional windows trees (Figs. 12–13). Also, the hierarchical oscillatory sequences have their fractal dimensions very close to those of the Ford circles and possibly other similar structures (e.g. the Cantor set complementary to the devil’s staircase associated with the circle map [4]). We estimate the fractal dimensions of our circuits to be ∼2.52 (capacity dimension, formula (6)), 2.38–2.49 (modified Kaplan–Yorke dimension, formula (10)) and 2.26–2.36 (Kaplan–Yorke dimension, formula (9)).

The circuits considered in this paper, while simple in their structures, show a very rich spectrum of dynamical properties. This makes the circuits quite intriguing, not only from the point of view of the bifurcation oscillations following the Farey sequence of coprime integers, but also from the point of view of fractal properties, interaction of oscillators and singularly perturbed circuits. The jerk Newtonian variable x ₁ relates the circuits to Newton’s law $x_{1}''=F/m$ where $x_{1}''$ is the “acceleration” given by (13). It is expected that the circuits, considered as a coupling of oscillators (through the K element), may also show the mode entrainment phenomenon leading to the existence of the mode locking Arnold’s tongues.