Hidden and self-excited attractors in Chua circuit: SPICE simulation and synchronization

TL;DR

This paper investigates the synchronization of two Chua circuits using SPICE simulations, highlighting challenges in control signal selection due to multistability and hidden attractors in chaotic systems.

Contribution

It demonstrates the complexities of synchronizing chaotic circuits with hidden and self-excited attractors, emphasizing the difficulties in control signal design.

Findings

Synchronization achieved in SPICE simulations

Control signal selection is complex with multistability

Hidden attractors pose challenges for synchronization

Abstract

Nowadays various chaotic secure communication systems based on synchronization of chaotic circuits are widely studied. To achieve synchronization, the control signal proportional to the difference between the circuits signals, adjust the state of one circuit. In this paper the synchronization of two Chua circuits is simulated in SPICE. It is shown that the choice of control signal is be not straightforward, especially in the case of multistability and hidden attractors.

Equations

$$\frac{d{v}_1}{dt}=\frac{1}{C_1}\left(\frac{1}{R}(v_2-v_1)-f(v_1)\right),$$

$$\frac{d{v}_2}{dt}=\frac{1}{C_2}\left(\frac{1}{R}(v_1-v_2)+i_L\right),$$

$$\frac{d{i}_L}{dt}=-\frac{1}{L}(v_2+R_0{i}_L),$$

$$\displaystyle f(v_{1})=\left\{\begin{array}[]{lll}G_{b}v_{1}+(G_{b}-G_{a})E_{1},&\text{if}&v_{1}\leq-E_{1},\\ G_{a}v_{1},&\text{if}&|v_{1}|<E_{1},\\ G_{b}v_{1}+(G_{a}-G_{b})E_{1},&\text{if}&v_{1}\geq E_{1}.\\ \end{array}\right.$$

$$x=v_1,y=v_2,z=R{i}_L,$$

$$\alpha =\frac{1}{R{C}_1},\beta =\frac{R}{L},\gamma =\frac{R_0}{L}$$

$$\psi (x)=Rf(x),R{G}_a=m_0,R{G}_b=m_1,$$

$$\frac{dx}{dt}=\alpha (y-x-\psi (x)),$$

$$\frac{dy}{dt}=x-y+z,$$

$$\frac{dz}{dt}=-(\beta y+\gamma z),$$

$$\psi (x)=m_{1}x+(m_0-m_1)\text{sat}(x),$$

$$\displaystyle\text{sat}(x)=\left\{\begin{array}[]{lll}-1,&\text{if}&x\leq-1,\\ x,&\text{if}&|x|<1,\\ 1,&\text{if}&x\geq 1.\\ \end{array}\right.$$

$$\frac{dx}{dt}=\alpha (y-x-\psi (x)+\delta (\tilde{x}-x)),$$

$$\frac{dy}{dt}=x-y+z,$$

$$\frac{dz}{dt}=-(\beta y+\gamma z),$$

$$\frac{d\tilde{x}}{dt}=\alpha (\tilde{y}-\tilde{x}-\psi (\tilde{x})+\delta (x-\tilde{x})),$$

$$\frac{d\tilde{y}}{dt}=\tilde{x}-\tilde{y}+\tilde{z},$$

$$\frac{d\tilde{z}}{dt}=-(\beta \tilde{y}+\gamma \tilde{z})б,$$

Full text

1 Introduction: hidden and self-excited attractors in Chua circuit

Since the first chaotic behavior in dynamical systems was revealed by numerical integration [1], the researchers started to be interested in circuit implementation of chaos (see, e.g. [2, 3] and others), which allows one to ensure that the pseudo-orbits can be traceable by actual orbits (see, e.g. the corresponding discussion on shadowing in [4, 5]). At the same time various engineering perspectives of chaotic circuits application have been found [6].

The Chua circuit, invented in 1983 by Leon Chua [7, 8], is the simplest electronic circuit exhibiting chaos. Consider one of classical Chua circuits shown in Fig. 1.

The circuit consists of passive resistors ($R$ and $R_{0}$), capacitors ($C_{1}$ and $C_{2}$), conductor $L$, and one nonlinear element with characteristics $f(\cdot)$, called Chua diode. It is described by the following equations

[TABLE]

Here $v_{1}$ and $v_{2}$ are voltages across capacitors $C_{1}$ and $C_{2}$, respectively, $i_{L}$ is a current through conductor $L$, function $f(v_{1})$ is volt-ampere characteristics of Chua’s diode. In the following discussion we choose $C_{2}$ such that $RC_{2}=1$ and put $E_{1}=1$. By the introduction of new variables

[TABLE]

system (LABEL:chua-circuit-equations) is transformed to the following form

[TABLE]

Until recently there had been found Chua attractors, which are excited from unstable equilibria only and, thus, can be easily computed (see, e.g a gallery of Chua attractors in [9]). Note that L. Chua [8], analyzing various cases of attractors in Chua’s circuit, did not admit the existence of attractors of another type — so called hidden attractors, being discovered later in his circuits. An attractor is called a self-excited attractor if its basin of attraction intersects an arbitrarily small open neighborhood of equilibrium, otherwise it is called a hidden attractor [10, 11, 12, 13]. Hidden attractor has basin of attraction which does not overlap with an arbitrarily small vicinity of equilibria.

For example, hidden attractors are attractors in systems without equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). The hidden vs self-excited classification of attractors was introduced in connection with the discovery of the first hidden Chua attractor [14, 15, 10, 16, 17]. The Leonov-Kuznetsov’s classification of attractors as hidden or self-excited is captured much attention of scientists from around the world and hidden Chua attractors have become intensively studied (see, e.g. [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In Fig. 2 is shown an example of self-excited and hidden Chua attractors visualized by numerical integration of system (3) in MATLAB.

2 Visualization of hidden Chua attractor in SPICE

Nowadays various Simulation Programs with Integrated Circuit Emphasis (SPICE) are widely used to analyze and design analog circuits [33]. Consider simulation of hidden Chua attractor in SIMetrix SPICE111 https://www.simetrix.co.uk/

for the following parameters: $C_{1}=118.2u,\ C_{2}=1m,\ L=82.8281,\ R0=430.7m,\ R=1k,\ G_{b}=-0.0001768,\ G_{a}=-0.0011468$ ($\alpha=8.4562$, $\beta=12.0732,\gamma=0.0052,m_{0}=-0.1768,m_{1}=-1.1468$).

⬇

1*#SIMETRIX

2R1 L_P 0 430m

3X$psi psi_OUT psi_OUT $$arbsourcepsi pinnames: N1 OUT

4.subckt $$arbsourcepsi N1 OUT

5B1 OUT 0 I=-0.0011468*V(N1) + (-0.00017680+0.0011468)*LIMIT(V(N1),-1,1)

6.ends

7C1 psi_OUT 0 118.2u IC=2 BRANCH={IF(ANALYSIS=2,1,0)}

8C2 C2_P 0 1m IC=1 BRANCH={IF(ANALYSIS=2,1,0)}

9L L_P C2_P 82.8281 IC=-4m BRANCH={IF(ANALYSIS=2,0,1)}

10R psi_OUT C2_P 1k

11.graph ”XY(C2_P, psi_OUT)” initxlims=false

12.TRAN 0 50 0 10m UIC

In Fig. 3 is shown corresponding SPICE realization of Chua circuit (see Fig. 1). Here $L,R0,R,C1,C2$ correspond to the elements of Chua circuit, and element $XYProbe$ is used to measure the voltage on capacitors (and plot projection of trajectories on $(v_{1},v_{2})$-plane). The Chua diode with characteristic $f(v_{1})$ is realized as “Arbitrary Source” (voltage-controlled current source) psi.

For the considered values of parameters there are three equilibria in the system: the zero equilibrium $F_{0}=(0,\,0,\,0)$ is a stable focus-node and two symmetric saddle equilibria $S_{\pm}$. To check that an attractor is hidden, we have to demonstrate that the trajectories from certain small vicinities of equilibria are not attracted by the attractor. Figs. 4 and 5 show SPICE simulation222 In our experiment “Max time step” is set to $10m$.

of trajectory in a vicinity of zero equilibrium and a trajectory with initial condition corresponding to the unstable manifold of the saddle (in both cases the considered trajectories are attracted to the zero equilibrium). Projection of twin symmetric chaotic attractors on the plane $(v_{1},v_{2})$ and attraction to the zero equilibrium are shown in Fig. 6.

In Fig. 7 is shown simulation self-excited Chua attractor in SPICE for the parameters $C_{1}=66.667u,\ C_{2}=1m,\ L=35.7143,\ R0=0,\ R=1k,\ G_{b}=-0.00071429,\ G_{a}=-0.0011468$.

In this case the trajectory with initial data in a vicinity of zero equilibrium is attracted by the self-excited attractor.

3 Synchronization of Chua circuits

Nowadays various chaotic secure communication systems based on Chua circuits and other electronic generators of chaotic oscillations are of interest [34, 35, 36, 37, 38, 39]. The operation of such systems is based on the synchronization chaotic signals of two chaotic identical generators (transmitter and receiver) for different initial data. The control signal proportional to the difference between the circuits signals, adjust the state of one receiver. The multistability and existence of hidden attractors may lead to improper workreceiver of such systems.

Consider now two $x$-coupled Chua systems

[TABLE]

where $\delta$ is a coupling factor ($\delta=\frac{R}{R_{\text{coupling}}}$), and the corresponding circuit (see Fig. 8).

SIMetrix SPICE realization of the coupled Chua circuits is shown in Fig. 9.

Here the passive elements of the first circuit: $L,R0,R,C1,C2$, are identical to the corresponding elements of the second circuit: $L1,R1,R2,C3,C4$. Elements $XYProbe$ are used to measure voltages on capacitors and plot on $(v_{1},v_{2})$-plane. To simulate nonlinear elements with characteristic $f(v_{1})$, we use SPICE elements “Arbitrary Source” (voltage-controlled current source) psi and psi1. Resistor $R3$ connects two circuits and characterizes the distance between transmitter and receiver. Below it is shown that critical coupling value (i.e. maximum value for which synchronization takes place) is different for different choice of initial data because of the multistability and hidden attractors. In our experiments the simulation time is $100$ seconds and minimal value of $R3$ is 1000.

Example 1. Consider initial data of the circuits on one of the symmetrical hidden attractors: Chua circuit 1 — $v_{1}(0)=-3.7727,v_{2}(0)=-1.3511,i_{L}(0)=0.0047$; Chua circuit 2 — $v_{1}(0)=-3.6,v_{2}(0)=-1.3511,i_{L}(0)=0.0047$.

Example 2. Consider initial data of the circuits on two symmetrical hidden attractors: Chua circuit 1: $v_{1}(0)=-3.7727,v_{2}(0)=-1.3511,i_{L}(0)=0.0047$; Chua circuit 2: $v_{1}(0)=3.7727,v_{2}(0)=1.3511,i_{L}(0)=-0.0047$.

Example 3 Consider initial data of the circuits on one of the symmetrical hidden attractors and stable zero equilibrium: Chua circuit 1 — $v_{1}(0)=-3.7727,v_{2}(0)=-1.3511,i_{L}(0)=0.0047$; Chua circuit 2 — $v_{1}(0)=0,v_{2}(0)=0,i_{L}(0)=0$.

Example 4 Consider the case of symmetrical self-excited attractors for the parameters $\alpha=15$, $\beta=28$, $\gamma=0$, $m_{0}=-5/7$, $m_{1}=-8/7$ (corresponding SPICE parameters: $C_{1}=6.667\cdot 10^{-5},\ C_{2}=0.001,\ G_{a}=-7.1429\cdot 10^{-4},\ G_{b}=-1.1429,\ L=35.7134,\ R_{0}=0$). Take the following initial data of the circuits on one of the symmetrical self-excited attractors and unstable zero equilibrium: Chua circuit 1 — $v_{1}(0)=-0.780,v_{2}(0)=-0.525,i_{L}(0)=3.5m$; Chua circuit 2 — $v_{1}(0)=0,v_{2}(0)=0,i_{L}(0)=0$.

Conclusion

In conclusion we note that the simulation of circuit behavior by software, as well as the numerical integration of its dynamical model, is subject to numerical errors due to time discretization step (see, e.g. the corresponding examples with phase-locked-loop based circuits in [40, 41, 42, 43]). Observation of hidden Chua attractors in physical experiments is discussed, e.g., in [23, 44, 45, 47, 46].

Acknowledgements.

This work is supported by Leading Scientific School of Russian Federation (8580.2016.1, s.2) and DST-RFBR project (16-51-45062, s.3).