Inductorless Chua ’ s Circuit : Experimental Time Series Analysis

We have implemented an operational amplifier inductorless realization of the Chua’s circuit. We have registered time series from its dynamical variables with the resistor R as the control parameter and varying from 1300Ω to 2000Ω. Experimental time series at fixed R were used to reconstruct attractors by the delay vector technique. The flow attractors and their Poincaré maps considering parameters such as the Lyapunov spectrum, its subproduct the Kaplan-Yorke dimension, and the information dimension are also analyzed here. The results for a typical double scroll attractor indicate a chaotic behavior characterized by a positive Lyapunov exponent and with a Kaplan-Yorke dimension of 2.14. The occurrence of chaos was also investigated through numerical simulations of the Chua’s circuit set of differential equations.


Introduction
Chaotic electronic circuits [1] have been widely studied during the last few decades due to their easy implementation, robustness, reproducibility of results, and also as a test platform for synchronization [2][3][4], chaos control [4][5][6], signal encryption [7], and secure communications [8,9]. Also it is easy, through Kirchhoff 's laws, to obtain the circuit described by a set of differential equations and carry on simulations which in most times, present good agreement with experimental data. The Chua's circuit [1,10]  (2) it exhibits a number of distinct routes to chaos and multistructural chaotic attractors [11]; (3) attractors that occur in Chua's circuit arise from very complex homoclinic tangencies and loops of a saddle focus [11]; (4) many opened questions on the system's behavior and the lack of a possibility to fully describe Chua's circuit from its equations [11]. The Chua's circuit has been the object of study of hundreds of papers, where its topological, numerical, physical, and dynamical characterizations are deeply investigated. See [12][13][14][15] and references therein.
Point (4) suggests that numerical analysis such as that carried on this work could provide some contributions to understand Chua's circuit dynamical behavior. Chua's circuit dynamical equations are given by where R,C 1 ,C 2 , and L are passive linear elements, r L is the inductor's resistance, i d is the current through Chua's diode with m 0 ,m 1 and B p as parameters.
R. M. Rubinger et al. 3 Chua's diode and the active component "inductor" were implemented according to Tôrres and Aguirre [16]. This inductor implementation turns easy and compact to construct the Chua's circuit. It has another advantage since it can be designed as resistance free as have been carried on this work. This paper is organized as follows: Section 2 is devoted to a detailed description of the parameters used to build and analyze Chua's circuit. A brief study of the equilibrium points of Chua's differential equations and the existence of a homoclinic loop is presented in Section 3. This was carried in order to identify the possible dynamical behavior for the chosen parameters of the circuit and to support the analyses carried in Section 4. In Section 4, we present the time series analysis of some illustrative experimental time series obtained from the Chua's circuit implementation. This section is the core of our work. Our aim is to characterize attractors obtained from this particular implementation of Chua's circuit with respect to its sensitivity to initial conditions and its dimension on the state space. Finally, concluding remarks are presented in Section 5.

Experimental details
Chua's circuit was constructed in a single face circuit board with the same scheme of [16] but with all capacitors 1000 times lower. This way C 1 = 23.5nF, C 2 = 235nF, and L = 42.3mH. These values were obtained from the combination of passive components and measured with a digital multimeter with a 3% precision. We evaluate the oscillation main frequency as a rough approximation by 1/(2π(LC 2 ) 1/2 ) which gives about 1600 Hz. This oscillation frequency allowed us to store large time series for data analysis. Other parameters were experimentally determined. From Chua's diode i(V ) characteristics linear fittings as B p = 1.8 V, m 1 = −0.758 mS, and m 0 = −0.409 mS with the significant digits limited by the fitting accuracy. Here S stands for inverse resistance unity. The resistor R, used as the control parameter, was a precision multiturn potentiometer and kept in the range of 1300 Ω to 2000 Ω.
A data acquisition (DAQ) interface with 16 bit resolution, maximum sampling rate of 200 k samples/s, and adjustable voltage range of maximal peak voltage of 10 V was applied for data storage. The Chua's circuit oscillations were measured at the x point depicted in Figure 1.1 after passing through an active buffer. Also Labview r was used to develop data acquisition software and analysis [17,18]. A Keithley 237 voltage/current source in series with the Chua's diode was applied to obtain the i(V ) data. For each time series the potentiometer R was detached from the circuit for resistance measurements with a 3(1/2) digit multimeter.
Four representative attractors obtained with R as 1480 Ω, 1560 Ω, 1670 Ω, and 1792 Ω will be presented in Section 4 with the respective analyses. Particular attention will be given to the double scroll attractor.

Differential equation analysis
Considering a resistance free inductor, that is, r L = 0, we have determined the operating points which coincide with the equilibrium points of (1.1), that is, its solution foṙ Here the dot over the variables stands for time derivatives. The solutions correspond to the state space points (−Ri d (x),0,i d (x)), which coincide with the interception of the load line with the graph of i d in the plane y = 0. The load line is a straight line with slope −1/R determined by the Kirchhoff 's laws applied to the circuit composed by R and the Chua's diode. One of these equilibrium points will always be the origin (0,0,0). For (1.1) presents only the equilibrium point at the origin. The origin is a saddle focus point, since the Jacobian matrix of (1.1) at (0,0,0) has one negative real eigenvalue and two complex eigenvalues with positive real parts.
, the origin is a (1-2)-saddle point, that is, the Jacobian matrix J f (0,0,0) has three real eigenvalues, being one negative and two positives.
(1.1) presents three equilibrium points p 0 = (0,0,0), For 1319.26Ω < R < 1323.93Ω, p 0 is a (2-1)-saddle point, and for 1323.93Ω ≤ R ≤ 2000Ω, the equilibrium point p 0 is of saddle-focus type, since the Jacobian matrix J f (p 0 ) has one real positive eigenvalue λ 00 and two complex eigenvalues, λ 01 and λ 02 , with negative real parts. Therefore, p 0 has a 1-dimensional unstable manifold and a 2-dimensional stable manifold. The equilibrium points p 1 and p 2 are of saddle-focus type too, but their stable manifolds are 1-dimensional and their unstable manifolds are 2-dimensional, since the Jacobian matrix J f (p 1 ) = J f (p 2 ) has one real negative eigenvalue and two complex eigenvalues with positive real parts.
The presence of homoclinic loops connecting p 0 to itself, that is, p 0 possesses a 2dimensional stable manifold and a 1-dimensional unstable manifold which intersect nontransversely, for some value of the parameter R, plays a fundamental role in the existence of chaos in (1.1).

R. M. Rubinger et al. 5
The existence of a homoclinic loop at p 0 is now outlined, according to [19]. Equation (1.1) can be written in dimensionless form (3.6) and the dimensionless variables and parameters are given by λ 00 > −Re(λ 01 ) > 0 [11]. In this work, Shil'nikov saddle-focus condition is satisfied by In Figure 3.1 it is possible to identify the stable and unstable manifolds associated with it. The value of R for the homoclinic loops is near of the value found for the experimental measurements of the cycle-one attractor obtained with R = 1480Ω as will be presented in the next section. It should be pointed out that the nominal values of capacitors and resistors used in this implementation were selected by measurements with digital multimeters which are subjected to experimental errors between 1% and 3%. Thus the value of R for the occurrence of the homoclinic loops is compatible with our experimental results.

Experimental results and discussion
For this work we have carried out time series measurements of the variable x(t) for some R values and proceeded as described in Section 2.  [20] and the embedding dimension [21] were determined. Figure 4.3 presents the mutual information for attractor 4. The first minimum corresponds to the optimal time delay for the delayed vectors. For the double scroll attractor it is of 7-time steps of 33 μs.
The false nearest neighbors algorithm was applied to verify if the time series is sensitive to noise [22]. Since Chua's system is a three-variable system, it turns out that false nearest neighbors should indicate the embedding dimensions as three. A higher than three embedding dimension for this system would mean significant noise contamination [17, 18].  Since our results indicate no false nearest neighbors for embedding dimensions above 3, we can neglect noise contribution for the geometric invariants that will be presented in the following. Figure 4.4 presents the false nearest neighbor plot for attractor 4. As can be seen, the proper embedding dimension is 3. In Figure 4.5, we present the Poincaré section for the    continuous curve crossing the y = x line. In Figure 4.6(b) we have a more complex pattern obtained for attractor 4. It is basically composed by two curves, one corresponding to each side of the "scroll" of the flow attractor.
Time series analyses were carried for all attractors. The estimated parameters were the Lyapunov spectrum [23] with its subproduct the Kaplan-Yorke dimension (D KY ) [24] and the information dimension (D 1 ) [25]. D 1 was measured for both flow and map representations. We will present detailed analysis for attractor 4 and summarize the information for all attractors in a table that will follow.   4.7 presents the Lyapunov spectrum for the attractor 4, obtained by using the method described in [23] and implemented in [17,18]. It is characterized by a positive, a null, and a negative Lyapunov exponent. This configuration is a characteristic of chaotic attractors. The Kaplan-Yorke dimension for this attractor is evaluated as D KY = 2.14.
R. M. Rubinger et al. 11 Dimension analysis gives complementary information since it is common to find strange attractors with fractal shape.
D KY is considered as equivalent to D 1 [26]. Considering this we present in Figure 4.8 the D 1 for attractor 4. In  of zero slope independent of the embedding dimensions above the proper one (i.e., 3 for Chua's circuit). In According to [26] the dimension of a map attractor is related to the dimension of its flow attractor by a difference of one unity. This occurs because the map is obtained by eliminating the flow direction which is related to the null Lyapunov exponent. Since the null Lyapunov exponent is associated to a dimension of one, the map information dimension (D 1M ) must be related to the flow information dimension (D 1F ) by D 1M = D 1F − 1.
Considering that D KY ∼ D 1F we can infer that our measurements of D 1F are underestimated and that D 1M + 1 is compatible with the corresponding values of D KY . The reason for the low value of D 1F is yet unknown but certainly it is related to the direction of the flow and thus to the null Lyapunov exponent. Table 4.1 summarizes the results for the four presented attractors. The first column, assigned as #, indicates the number of the attractor as defined in the text. 1, 2, 3, and 4 correspond to the attractors obtained with R in ohms defined in column 2. The third column is D 1M , measured for the Poincaré maps and the fourth column presents D 1F , measured for the flow attractor. The fifth column is the Kaplan-Yorke dimension. The sixth column is the minimal embedding dimension obtained from the false nearest neighbor algorithm. The last column lists the three Lyapunov exponents in decreasing order.
Both periodic attractors presented three negative Lyapunov exponents, but the first two can be considered as null when compared with the third value. Considering this,  Attractors 3 and 4 presented one positive, one null, and one negative Lyapunov exponent. The sum of the exponents is negative, which means that attractors contract volume in state space. The Kaplan-Yorke dimension for them is above 2.0, whilst the D 1 was determined as 1.8 for the flow representation of the attractors. As discussed above the latter value is underestimated. Two facts corroborate for this assumption. One is that D 1 measured for the Poincaré maps of the attractors does not differ by one unity from the measurement carried on the flow attractors, but they do differ by approximately one unity from the Kaplan-Yorke dimension.
The other fact is also related to the Poincaré map of the attractors. The visual inspection of the Poincaré maps presented in Figure 4.6 indicates that they are objects with dimension greater than 1. Thus, the flow attractor must be an object with a dimension greater than 2, since by adding 1 to a number between 1 and 2 the resulting number must be between 2 and 3.

Summary
We have implemented experimentally an operational amplifier inductorless realization of the Chua's circuit.
A homoclinic loop was found by numerical analysis of normalized Chua's differential equations at a parameter corresponding to R = 1490.46Ω. Indeed, bifurcations were observed experimentally in the vicinity of R for the homoclinic loop.
We selected four representative attractors obtained with R as 1480 Ω, 1560 Ω, 1670 Ω, and 1792 Ω to present in this work. They correspond to a cycle one, cycle two, chaotic-like in one region, and the double scroll, respectively.
Considering the double scroll, that is, for R = 1792Ω, the information dimension of a three-dimensional delay vector reconstruction of the attractor (D 1F ) and of its Poincaré map (D 1M ) are 1.8 and 1.2, respectively. Also the Lyapunov spectrum gives positive, null, and negative exponents with a Kaplan-Yorke dimension as 2.14 characterizing the attractor as chaotic. This indicates that the flow attractor dimension has been underestimated and that the Kaplan-Yorke dimension is better suited for this attractor.