Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

With new three-segment piecewise-linearity in the classic Chua’s system, two new types of 2-scroll and 3-scroll Chua’s attractors are found in this paper. By changing the outer segment slope of the three-segment piecewise-linearity as positive, the new 2-scroll Chua’s attractor has emerged from one zero index-1 saddle-focus and two symmetric stable nonzero node-foci. In particular, by newly introducing a piecewise-linear control function, an improved Chua’s system only with one zero index-2 saddle-focus and two stable nonzero node-foci is constructed, fromwhich a 3-scroll Chua’s attractor is converged. Some remarks for Chua’s nonlinearities and the generating chaotic attractors are discussed, and the stabilities at the three equilibrium points are then analyzed, upon which the emerging mechanisms of the novel 2-scroll and 3-scroll Chua’s attractors are explored in depth. Furthermore, an analog electronic circuit built with operational amplifier and analog multiplier is designed and hardware circuit experiments are measured to verify the numerical simulations.These novel 2-scroll and 3-scroll Chua’s attractors reported in this paper are completely different from the classic Chua’s attractors, which will enrich the dynamics of the classic Chua’s system.


Introduction
As it has been shown, Chua's circuit is a relatively simple circuit which has rapidly become a paradigm for chaos [1].In the past three decades, numerous works have been reported on this circuit, including realization schemes, experimental measurements, numerical observations, and theoretical proofs [2][3][4][5][6][7].The classic Chua's system with a simple algebraic structure is a dimensionless form of Chua's equations, and its nonlinearity formed by Chua's diode, called Chua's nonlinearity in this paper, is three-segment piecewise-linear.Consequently, the classic Chua's system has three unstable equilibrium points, one zero index-1 saddle-focus and two symmetric nonzero index-2 saddle-foci, resulting in the generation of a self-excited 2-scroll chaotic attractor [1].
By modifying Chua's nonlinearity with multiple-segment piecewise-linear or continuous nonlinear functions, Chua's system can be generalized to a system exhibiting more complex attractors [8][9][10], that is, self-excited multiscroll Chua's chaotic attractors.Generally, self-excited multiscroll or multiwing chaotic attractors are generated by disposing unstable index-2 saddle-foci in terms of added breakpoints in the model system [11,12], which shows great theoretical and practical significance due to the applications to encrypted communication, chaos synchronization, and some other fields of Chua's systems with multiscroll chaotic attractors [13].
Chaotic dynamics has been investigated in numerous fields of science and engineering including mathematics, physics, chemistry, electronic circuit, information, and mechanical engineering [14][15][16][17][18][19][20][21].In the past few years, a new type of attractor, defined as hidden attractor, has been found in large numbers of nonlinear dynamical systems [19][20][21][22][23][24][25].Different from self-excited attractor, hidden attractor, whose attraction basin does not intersect with small neighborhoods of the equilibria of the system [19][20][21], is sensitive to the initial conditions and special analytical-numerical procedure should be adopted to locate its attractive basin [20].Recently, by developing this procedure, hidden chaotic spiral attractor has been numerically observed in the classic Chua's system with one stable zero equilibrium point [20,26].Furthermore, two coexisting hidden attractors have been numerically revealed and a dynamical route from one hidden limit cycle to hidden twin chaotic attractors by period-doubling bifurcations has been reported in the classic Chua's system [26,27].
The mechanism to have different types of attractors in the classic Chua's system is due to different Chua's nonlinearities [1,[26][27][28].Different Chua's nonlinearities formulate the three equilibrium points with different characteristics, leading to the generations of different types of Chua's attractors.This paper finds 2-scroll and 3-scroll Chua's attractors in the classic Chua's system by setting the outer segment slope of threesegment piecewise-linearity to be positive and introducing a new piecewise-linear control function [10,28].A distinct characteristic is that the scroll number of the new Chua's attractor does not associate with the number of index-2 saddle-foci of the classic Chua's system.To the best of our knowledge, there are no such reports in public literature.These findings will further enrich the dynamics of the classic Chua's system [1].
This rest of the paper is organized as follows.Section 2 discusses two kinds of Chua's nonlinearities and the correspondingly generating Chua's attractors and introduces a new three-segment piecewise-linearity.Section 3 studies a 2-scroll Chua's attractor with one zero index-1 saddle-focus and two symmetric stable nonzero node-foci.Section 4 investigates a 3-scroll Chua's attractor only with one zero index-2 saddlefocus and two symmetric stable nonzero node-foci by newly introducing a piecewise-linear control function.Section 5 conducts hardware circuit experiments to confirm the theoretical analyses.The conclusions are summarized in the last section.

Remarks for Chua's Nonlinearities and the Generating Chaotic Attractors
The classic Chua's system is described by the following dimensionless equations [1]: where , , and  are constant parameters and is used for describing the three-segment piecewise-linear characteristic of system (1), with ,  being the slopes of the inner and outer segments of ().Note that the parameter  is usually ignored; that is,  = 0.
In [10], through replacing the characteristic function (2) by a new piecewise-linear function (in the  and  variables) with a controllable threshold value  in the -axis, a 3-scroll chaotic attractor with three index-2 saddle-foci is converged, as shown in Figure 1(b), where  = 10.0,  = 14.0,  = 0,  = −1.43, = −0.59,and  = 1.The mechanism for generating the 3-scroll chaotic attractor is that the zero index-1 saddlefocus of the classic Chua's system is changed as a zero index-2 saddle-focus by modifying the nonlinearity of (2), leading to the formation of an additional index-2 saddle-focus.
If the outer segment slope of the three-segment piecewise-linearity () is changed as positive, a new Chua's nonlinearity is presented [28], as shown in Figure 3(c), where the relations are built as  < 0,  > 0, and || > 1, and the characteristics of three equilibrium points are altered, which lead to the occurrence of a new type of Chua's chaotic attractor as discussed in the following sections.
It is concluded that Chua's system (1) has the same three equilibrium points, one zero equilibrium point  0 and two symmetric nonzero equilibrium points  ± , as shown in Figure 3.With different Chua's nonlinearities, these equilibrium points have different dynamical characteristics, from which different types of Chua's chaotic attractors have emerged.
Considering that the system parameters  = 5.8,  = 6.2,  = −1.1, and  = 0.02 and the initial conditions (0.001, 0, 0), the phase portraits of a 2-scroll Chua's chaotic attractor in two different planes are plotted in Figures 4(a) and 4(b), respectively, where the mark I stands for the zero equilibrium point, and the marks ◻ and * represent the two symmetric nonzero equilibrium points.Correspondingly, the eigenvalues at these three equilibrium points of (3) are calculated as It means that  0 is a saddle-focus having a positive real root and two complex conjugate roots with negative real parts, that is, index-1 saddle-focus, whereas  ± are two symmetric stable node-foci having two complex conjugate roots with negative real parts and a negative real root.Consequently, when the two parameters  and  of the three-segment piecewise-linearity in (2) satisfy the conditions the modified Chua's system (1) has two symmetric stable nonzero node-foci but can generate 2-scroll Chua's chaotic attractor.This type of attractor is completely different from the self-excited 2-scroll chaotic attractor reported in [1] and also different from the hidden chaotic spiral attractor reported in [20,26].

3-Scroll Chua's Attractor with One
Saddle-Focus and Two Stable Node-Foci Likewise, the nonlinearity ℎ() is also a three-segment piecewise-linearity.With (7a), the zero equilibrium point  0 of system ( 10) is an index-1 saddle-focus.In order to make  0 convert into an index-2 saddle-focus, a piecewise-linear control function with two variables  and  is introduced, which can be expressed by [10] where the parameter  is a controllable threshold value in the z-axis and satisfies 0 <  < 1.Thus, by replacing ℎ() with ℎ  (, ), an improved Chua's system is obtained as where  > 0 and  > 0.
Similarly, the improved Chua's system described by ( 12) has three equilibrium points given in (3).Considering the piecewise-linear control function ℎ  (, ) with the condition 0 <  < 1, the zero equilibrium point  0 is situated in the inner region of the x-z plane and the two symmetric nonzero equilibrium points P ± are located in the outer regions of the x-z plane.Thus, the stability of  0 is forcedly altered and the stabilities of  ± are still maintained unchanged.
For example, the system parameters of the improved Chua's system are chosen as ( The third condition of ( 15) cannot be satisfied, implying that  0 is unstable.In contrast, the Jacobian matrix evaluated at the two symmetric nonzero equilibria  ± is obtained as which is identical with the Jacobian matrix (4) for the case of  ± .Hence,  ± are always stable.
For the system parameters given in (16), the eigenvalues at three equilibrium points are calculated as It means that  0 is an index-2 saddle-focus having two complex conjugate roots with positive real parts and a negative real root, whereas  ± are two stable node-foci having two complex conjugate roots with negative real parts and a negative real root.
For the system parameters given in ( 16) and the initial conditions (0.001, 0, 0), the phase portraits of the chaotic attractor in three different planes are plotted in Figures 5(a and two stable nonzero node-foci, whereas the 3-scroll attractor reported in [10] has three index-2 saddle-foci.Therefore, the new 3-scroll Chua's chaotic attractor is different from the self-excited or hidden Chua's chaotic attractors [19][20][21]. Additionally, the system parameters given in ( 16) and other initial conditions are considered.When the initial conditions (1, 0, 0) are assigned, the improved Chua's system exhibits a chaotic spiral attractor, as shown in Figure 6(a), while when the initial conditions (−1, 0, 1) and (1, 0, −1) are selected, respectively, the improved Chua's system displays two stable point attractors settling down the two symmetric nonzero equilibrium points P ± [29], as shown in Figure 6(b), where the trajectories from the initial states of (−1, 0, 1) are colored in blue and those from (1, 0, −1) are colored in red.The above results illustrate that just like the dynamical systems reported in [29][30][31][32][33][34][35], the improved Chua's system can present multistability.

Circuit Realizations and Hardware Experiments
An analog electronic circuit built with operational amplifier and analog multiplier is designed to physically realize the improved Chua's system for generating the novel 3-scroll Chua's chaotic attractor, as shown in Figure 7 [9,30,31].The designed circuit is constructed by three parts: the first part is a basic three-dimensional Chua's system with three integration channels, as shown in Figure 7(a); the second part is the circuit for realizing the piecewise-linearity ℎ(), as shown in Figure 7(b); and the third part is the circuit for implementing the control function ℎ  (, ), as shown in Figure 7(c).In Figure 7, the system parameters given in ( 16) are used, and the time constant R 0 C 0 of the integrator is determined by  0 = 10 kΩ and  0 = 33 nF.The operational amplifiers OP07CP and analog multipliers AD633JNZ with ±15 V power supplies are utilized.Thus, the saturation voltage of the  operational amplifier output is  sat = 13.5 V.Note that in Figure 7(b) the following relations should be satisfied as The experimental prototype of the improved Chua's chaotic system is photographed in Figure 8.By Tektronix TDS 3034C digital oscilloscope, the phase portraits of novel 3-scroll Chua's chaotic attractor in three different planes are experimentally measured, as shown in Figures 9(a), 9(b), and 9(c), respectively.Furthermore, if the feedback resistance up the op-amp 7 in Figure 7(a) is changed as 62 kΩ and the realization circuit of ℎ() is directly linked to the input terminal of the op-amp 1 in Figure 7(a), the circuit realization of the modified Chua's system for generating 2-scroll Chua's chaotic attractor is established, from which the phase portraits of 2-scroll Chua's chaotic attractor in two different planes are experimentally captured, as plotted in Figures 10(a) and 10(b), respectively.Obviously, the experimental measurements of Figures 9 and 10 are consistent with the numerical simulations of Figures 5 and 4, respectively.

Conclusion
In this paper, by changing Chua's nonlinearity and introducing piecewise-linear control function, a new type of 2scroll and 3-scroll Chua's chaotic attractors generated from an improved Chua's system are studied through theoretical analyses, numerical simulations, and hardware circuit experiments.The traditional 2-scroll and 3-scroll Chua's chaotic  attractors are all excited from index-2 saddle-foci, whereas the proposed 2-scroll and 3-scroll Chua's chaotic attractors have emerged from two symmetric stable node-foci or one index-2 saddle-focus and two symmetric stable node-foci.Therefore, it is very interesting and particularly valuable to discover this 3-scroll chaotic attractor in the improved Chua's system only with one zero index-2 saddle-focus and two symmetric stable nonzero node-foci.Due to this, the proposed 3-scroll Chua's chaotic system may be regarded as a special paradigm for investigating special nonlinear phenomena different from self-excited and hidden Chua's chaotic attractors.

Figure 4
Figure 4: 2-scroll Chua's chaotic attractor with two symmetric stable nonzero node-foci.(a) Phase portrait in the x-y plane; (b) phase portrait in the x-z plane.

Figure 5 :
Figure 5: Novel 3-scroll Chua's chaotic attractor with one zero index-2 saddle-focus and two stable nonzero node-foci.(a) Phase portrait in the x-y plane; (b) phase portrait in the x-z plane; (c) phase portrait in the y-z plane; (d) Poincaré mapping in the x-z plane.

GNDFigure 8 :Figure 9 :
Figure 8: Photograph of the experimental prototype, the left part is basic circuit with three integration channels, whereas the right part is ℎ() piecewise-linearity circuit and ℎ  (, ) control function circuit.

Figure 10 :
Figure 10: Experimentally measured phase portraits of 2-scroll Chua's attractor.(a) Phase portrait in the x-y plane; (b) phase portrait in the x-z plane.