A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design

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1 entrop Article A New Chaotic Sstem with Multiple Attractors: Dnamic Analsis, Circuit Realiation and S-Box Design Qiang Lai, ID, Akif Akgul, Chunbiao Li, Guanghui Xu and Ünal Çavuşoğlu School of Electrical and Automation Engineering, East China Jiaotong Universit, Nanchang, China Department of Electrical and Electronics Engineering, Facult of Technolog, Sakara Universit, Serdivan, Turke; aakgul@sakara.edu.tr School of Electronic and Information Engineering, Nanjing Universit of Information Science and Technolog, Nanjing, China; chunbiaolee@nuist.edu.cn School of Electrical and Electronic Engineering, Hubei Universit of Technolog, Wuhan, China; xgh@hbut.edu.cn Department of Computer Engineering, Facult of Computer and Information Sciences, Sakara Universit, Serdivan, Turke; unalc@sakara.edu.tr * Correspondence: chaos@ecjtu.jx.cn or laiqiang@.com; Tel.: +--- Received: November ; Accepted: December ; Published: December Abstract: This paper reports out a novel three-dimensional chaotic sstem with three nonlinearities. The sstem has one stle equilibrium, two stle equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel sstem is its multiple attractors caused b different initial values. With the change of parameters, the sstem experiences mono-stilit, bi-stilit, mono-periodicit, bi-periodicit, one strange attractor, and two coexisting strange attractors. The complex dnamic behaviors of the sstem are revealed b analing the corresponding equilibria and using the numerical simulation method. In addition, an electronic circuit is given for implementing the chaotic attractors of the sstem. Using the new chaotic sstem, an S-Box is developed for crptographic operations. Moreover, we test the performance of this produced S-Box and compare it to the existing S-Box studies. Kewords: new chaotic sstem; multiple attractors; electronic circuit realiation; S-Box algorithm. Introduction The discover of the well-known Loren attractor [] in opened the upsurge of chaos research. In the decades thereafter, a large number of meaningful achievements on chaos control, chaotification, snchroniation and chaos application have emerged continuousl. Great changes have also been made to the understanding of chaos. Scholars began to think more out a wa to produce chaos rather than blindl suppress chaos. The generation of chaotic attractors in three-dimensional autonomous ordinar differential sstems has been of particular interest. As we all know, a multitude of tpical sstems with chaotic attractors were found, including Rössler sstem, Chen sstem, Sprott sstem, Lü sstem, etc. [ ]. With the further research of chaos, scientists found that some nonlinear dnamic sstems not onl have a chaotic attractor but also coexist with multiple attractors for a set of fixed parameter values. The coexisting attractors ma be fixed points, limit ccles, strange attractors, etc. The number and tpe of attractors are usuall associated with parameters and initial conditions of the sstem. Each attractor has its own basin of attraction which is composed of the initial conditions leading to long-term behavior that settles onto the attractor. The phenomenon of multiple attractors can be seen in man biological sstems and phsical sstems [ ]. In recent ears, the low-dimensional autonomous chaotic sstems with multiple attractors have aroused scholars research enthusiasm. Entrop,, ; doi:./e

2 Entrop,, of Li and Sprott found multiple attractors in chaotic sstems b numerical analsis and introduced the offset boosting method and conditional smmetr method for producing multiple attractors in differential sstems [ ]. Kengne et al. analed the multiple attractors of simple chaotic circuits, which can be described b differential equations [,]. Bao and Xu put forward some memristor-based circuit sstems with multiple chaotic attractors [,]. Lai et al. proposed some three-dimensional and four-dimensional continuous chaotic sstems with multiple attractors [ ]. Wei et al. attempted to reveal the intrinsic mechanism of the multiple attractors b analing the bifurcation of the sstem []. The investigation of chaos and multiple coexisting attractors is indeed a ver interesting research issue in academia. It helps to recognie the dnamic evolution of the actual sstem and promote the stud of complexit science. Chaotic sstems have been found to be used in man areas. The most valule application is crptolog. Chaotic sstem, in view of its rich dnamic behaviors and initial sensitivit, provides the mixing and spreading properties, which are the general requirements of encrption [,]. The S-Box is known as the most basic unit with scrambling function in block encrption algorithms. A good S-Box can make the encrption algorithm have higher securit and better ilit to withstand attacks. Although there have been man works on chaotic S-Box design, it is still important to generate S-Box according to some unique chaotic sstems. Before appling the chaotic sstem to engineering fields, it is necessar to realie it through electronic circuits in order to prove its real existence. Based on circuit theor and simple circuit elements, chaotic signals can be generated in oscilloscopes. So far, the electronic circuit has become an important tool for the analsis of chaotic sstems [ ]. This present paper considers a special polnomial chaotic sstem with the following features: (i) it has three nonlinearities x,, x and the invariance of transformation (x,, ) ( x,, ); (ii) it performs a butterfl attractor; and (iii) it has a stle equilibrium, an unstle equilibrium and two stle equilibria, three unstle equilibria for different parameter conditions, and experiences mono-stilit, bi-stilit, mono-periodicit, bi-periodicit, one strange attractor and two coexisting strange attractors. After investigating the dnamic behavior of the sstem, an electronic circuit and an S-Box are designed according to the sstem. The paper is arranged as follows: Section describes the chaotic sstem and shows its butterfl attractor. Section anales the stilit of the equilibria. Section studies the dnamic behavior of the sstem. Section considers the electronic circuit realiation of the sstem. Section estlishes the S-Box according to the sstem, and Section summaries the conclusions of this paper.. The Description of a Chaotic Sstem The chaotic sstem proposed in this paper can be expressed as the following set of differential equations: ẋ = ax, ẏ = b + x, ż = c + x + k, with state vector (x,, ) R and parameter vector (a, b, c, k) R. A butterfl attractor can be observed b numerical simulation on Matl software (Matl., MathWorks, Natick, MA, USA). The phase portraits of sstem () under parameters (a, b, c, k) = (,,, ) and initial condition (,, ) are shown in Figure. It visuall demonstrates that sstem () displas an attractor as the sstem trajectories will eventuall move to a bounded region. The Lapunov exponents of the sstem are calculated as l =., l =., l =.. The Lapunov dimension is D l = l /l =., so it can be determined that the attractor is a chaotic attractor. The time series of generated from two ver close initial conditions (,, ) and (,,.) are plotted in Figure. At the beginning, the are almost the same, but their differences are increasing after a number of iterations. That is to sa, sstem () is sensitive dependence on initial conditions and its future behavior is unpredictle in the long term. The Poincaré map of sstem () is obtained via selecting the sections ()

3 Entrop,, of = {(x, ) R = } and = {(, ) R x = }. As shown in Figure, the Poincaré map is a sheet of point set. It is consistent with the nature of chaos. x x x (c) (d) Figure. The butterfl attractor of sstem (): x ; x ; (c) x ; (d). t Figure. The time series of varile generated from initial conditions (,, ) (red color) and (,,.) (blue color).

4 Entrop,, of x Figure. The Poincaré maps of sstem () with crossing sections: ;.. The Stilit of Equilibria Suppose that parameters a, b, c, k are all positive real numbers. The equilibria of sstem () can be obtained b solving ẋ = ẏ = ż =. If k c, sstem () has onl one equilibrium O(,, k/c). If k < c, sstem () has three equilibria as follows: O(,, k/c), O ( (c k)/a, (c k)/b, ), O ( (c k)/a, (c k)/b, ). Proposition. Suppose that b > a >, k >, and the parameter c satisfies the following condition: k < c < k[(a + b ) + k(b a)] [(a + b) + k(b a)], () then the equilibria O and O of sstem () are asmptoticall stle. Proof. B lineariing the sstem () at the equilibrium, the Jacobian matrix is given b where H = a b x x x c. () B using λi H =, the corresponding characteristic equation evaluated at O, O is obtained as λ + w λ + w λ + w =, () w = (b a + k ), w = (a b)(c k ), w = (c k ).

5 Entrop,, of According to the Routh Hurwit criterion, the equilibria O, O are stle if all the roots of Equation () have negative real parts. This requires that w >, w >, w > and w w > w. It is eas to verif that w >, w >, w > if b > a >, k > and k < c < k. () To make w w > w, the parameter c should meet c < c = k[(a + b ) + k(b a)] [(a + b) + k(b a)]. () Since c < k k, then O, O are asmptoticall stle if b > a >, k >, < c < c. When the parameter c passes through the critical value c, then double Hopf bifurcation occur with two limit ccles branched from O, O and sstem () loses its stilit. Proposition. Suppose that b > a >, c >, k >, then: (i) the equilibrium O is unstle for c > and (ii) the equilibrium O is asmptoticall stle for c k. Proof. The characteristic equation evaluated at O is given b If c > k ; (λ + c)[c λ + (b a)c λ + k c ] =. () k, then Equation () has a root with a positive real part. Thus, O is unstle. If c < all the roots of Equation () have negative real parts, which implies that O is stle. When c = Equation () has three roots λ =, λ = a b, λ = c. Therefore, O is non-hperbolic equilibrium. It can be verified that O is asmptoticall stle b appling the center manifold theorem.. The Evolution of Multiple Attractors Detailed investigation of the complex dnamic behaviors of sstem () is presented in this section. Simulation experiments including bifurcation diagrams, phase portraits, Lapunov exponents, and Poincaré maps give a close and intuitive look at sstem (). There is a wealth of chaotic dnamics associated with the fractal properties of the attractor in sstem (). With the change of parameters, sstem () experiences stle state, periodic state and chaotic state. For different initial values, sstem () performs different tpes of attractors with independent domains of attraction... Dnamic Evolution with Parameter c Consider the dnamic evolution of sstem () with respect to parameter c under the given parameter conditions a =, b =, k =. The bifurcation diagrams of sstem () versus c (, ) are shown in Figure a, where the red color branch and blue color branch are ielded from initial values x = (,, ), x = (,, ), respectivel. The overlapped regions of the red color and blue color branches indicate that the trajectories of x, x eventuall tend to the same attractor, while the separated regions indicate that the trajectories of x, x tend to different attractors. Figure b is the Lapunov exponents of sstem () with initial value x. It shows that the sstem () experiences stle state, periodic state, chaotic state with the variation of c. When c (, ), sstem () is mono-stle as it has onl one stle equilibrium. When c (,.), sstem () performs bi-stilit with respect to the existence of two stle equilibria. As c increases across the critical value c =., sstem () occurs double Hopf bifurcation at the equilibria. When c (.,.), sstem () performs bi-periodicit. When c (.,.), sstem () changes into mono-periodic state. When c (.,.), sstem () ields two strange attractors from initial values x, x. When c (.,.) (.,.) (., ), sstem () has onl one chaotic attractor. Tle describes k, k,

6 Entrop,, of the attractors of sstem () with different values of c. The phase portraits in Figure illustrate the existence of different tpes of attractors in sstem (). Tle. Attractors of sstem () with different values of c. Value of c Equilibrium Point Tpe of Attractor Figure c =. Stle point: (,, ) A point attractor Figure a c =. c =. c =. c =. c =. c =. c =. Saddle node: (,,.) Stle point: (±., ±., ) Saddle node: (,,.) Saddle focus: (±., ±., ) Saddle node: (,,.) Saddle focus: (±., ±., ) Saddle node: (,,.) Saddle focus: (±., ±., ) Saddle node: (,,.) Saddle focus: (±., ±., ) Saddle node: (,,.) Saddle focus: (±., ±., ) Saddle node: (,,.) Saddle focus: (±., ±., ) A pair of point attractors A pair of limit ccles A smmetric limit ccle A pair of strange attractors A pair of limit ccles A smmetric limit ccle A butterfl strange attractor Figure b Figure c Figure d Figure e Figure f Figure g Figure h c Lapunov exponents c Figure. The bifurcation diagrams and Lapunov exponents of sstem () versus c (, ). O O O O x x.... x x.... Figure. Cont.

7 Entrop,, of O O O O O O x x.... (c) x x (d) O O O O x O x (e) x O x (f) O O O O x O x (g) x O x (h) Figure. The phase portraits of sstem () with: c =.; c =.; (c) c =.; (d) c =.; (e) c =.; (f) c =.; (g) c =.; (h) c =.... Dnamic Evolution with Parameter k The bifurcation diagrams of sstem () for parameters (a, b, c) = (,, ), k (, ) are shown in Figure a, where the red color branch and blue color branch are ielded from initial values x, x, respectivel. Obviousl, the state of sstem () changes from chaos to period and then to stle when parameter k increases from to. It also can be illustrated b the Lapunov exponents in Figure b. The maximum Lapunov exponent is positive with c (,.) (.,.) (.,.), negative with c (., ), and equal to ero with c (.,.) (.,.) (,.). For c =,,,, we can observe a strange attractor, a limit ccle, and a stle point of sstem (), with their phase portraits are shown in Figure. For c =,, we can observe two coexisting periodic attractors and two coexisting point attractors of sstem (), as shown in Figure.

8 Entrop,, of Lapunov exponents k k Figure. The bifurcation diagrams and Lapunov exponents of sstem () versus k (, ) (c)... (d) Figure. The phase portraits of sstem () with: k = ; k = ; (c) k = ; (d) k =.

9 Entrop,, of t (c). t (d) Figure. The coexisting attractors of sstem (): projections on x with k = ; time series of with k = ; (c) projections on x with k = ; (d) time series of with k =.. Electronic Circuit Realiation There are man works that are related to chaos based applications in the literature [ ]. Here, we will present the circuit realiation of sstem () for realisticall obtaining its chaotic attractors. The numerical simulation in Figure e displas two coexisting strange attractors in sstem () for (a, b, c, k) = (,,., ) and initial conditions (±, ±, ). For circuit realiation of this state of sstem (), we need to refrain from saturation of circuit elements, and the effective wa to achieve this goal is to reduce the voltage values of the circuit via scaling the variles of sstem (). In the process of scaling, we assume (X, Y, Z) = (x,, /) and then the scaled sstem is obtained as Ẋ = ax YZ, Ẏ = by + XZ, Ż = cz + XYZ + k. () Figure gives the new phase portraits of the scaled sstem () for (a, b, c, k) = (,,., ). Evidentl, the scaling process does not cause fundamental changes to the sstem (), but just limits the variles to a smaller region Ω = {(x,, ) x, (, ), (, )}. The circuit diagram of sstem () raised b the OrCAD-PSpice programme (OrCAD., OrCAD compan, Hillsboro, OR, USA) is presented in Figure. It has three input (or output) signals with respect to the variles X, Y, Z, and the operations between signals realied via the basic electronic materials including resistors, capacitor, TL operational amplifiers (op-amps), and AD multipliers. B fixing R = R = KΩ, R = KΩ, R = KΩ, R = R = KΩ, R = KΩ, R = KΩ, R = KΩ, C = C = C = nf, Vn = V, Vp = V and executing the circuit on electronic card shown in Figure, we can obtain the outputs of circuit in the oscilloscope. The oscilloscope graphics in Figure show good consistenc with the numerical simulations in

10 Entrop,, of Figure. Hence, we can come to a conclusion that the coexisting chaotic attractors in sstem () are phsicall obtained. x x (c) Figure. The phase portraits of the scaled sstem () for (a, b, c, k) = (,,., ): x ; x ; (c). Figure. The circuit diagram of sstem ().

11 Entrop,, of Figure. The experimental circuit of sstem (). (c) (d) (e) (f) Figure. The phase portraits of two coexisting attractors of sstem () on the oscilloscope for ( a, b, c, k) = (,,., ): (a,b) x ; (c,d) x ; (e,f).. S-Box Design and Its Performance Analsis This section aims to raise a new chaotic S-Box algorithm b appling sstem (). In the algorithm design, first random number generation is performed and then S-Box is produced. The S-Box generation algorithm pseudo code is shown in Algorithm. For estlishing the S-Box production algorithm, we first input the parameters (a, b, c, k) = (,,., ) and initial value (x,, ) = (,, ) of the sstem and then the float number outputs are produced. In order to generate more random outputs in the analsis of the chaotic sstem, we select an appropriate step interval h and used it as a sample value. More random sequences are obtained b setting the appropriate step interval h =.. Sstem () is solved b using the RK algorithm with the initial conditions and the specified sampling value, and time series are obtained. In our designed chaotic S-Box algorithm, the outputs of, phases of sstem () are used. Float number values ( bits) obtained from these phases are converted to a binar sstem. B taking bits from the low significance parts (LSB) of the -bit number sequences generated from both phases, these values are XORed. The obtained new -bit value is converted to a decimal

12 Entrop,, of number. This value is discarded if the decimal number was previousl generated and included in the S-Box, if not produced before, it is added to the S-Box. In this wa, this process continues until the distinct values (between and ) are obtained on the S-Box. The generated S-Box is shown in Tle. Algorithm The S-Box generation algorithm pseudo code. : Start; : Inputting parameters and initial value of the sstem; : Sampling with step interval h; : i =, S-Box = []; : while (i < ) do : Solving sstem with RK algorithm and obtaining time series (, ); : Convert float to binar number; : Take LSB- bit value from RNG phase; : Convert binar to decimal number ( bit) : if (Is there decimal value in S-Box = es) then : Go step. : else(is there decimal value in S-Box = no) : Sbox[i] decimal value; : i++; : end : end : S-Box reshape(sbox,,); : Read to use chaos-based S-Box; : End. Tle. The chaotic S-Box of sstem (). In order to determine that the produced S-Boxes are robust and strong against attack, some performance tests are applied. We mainl focus on these tests: nonlinearit, outputs bit independence criterion (BIC), strict avalanche criterion (SAC), and differential approach probilit (DP). In addition, the comparisons of the performance between this new S-Box and the existing chaotic S-Box proposed b Chen [], Khan [], Wang [], Okanak [], Jakimoski [], Hussain [], Tang [] are presented in Tle. Nonlinearit is regarded to be the most core part of all the performance tests. The nonlinearit values of the S-Box ielded b sstem () are obtained as,,,,,, and. Accordingl, its average value, minimum value and maximum value are computed as, and.

13 Entrop,, of B comparing the nonlinearit of other S-Box shown in Tle, we can claim that the new S-Box is better than others in some measure. Tle. The comparison of different chaotic S-Boxes (BIC: bit independence criterion; SAC: strict avalanche criterion; DP: differential approach probilit). S-Box Nonlinearit BIC SAC BIC-SAC Min Avg Max Nonlinearit Min Avg Max Proposed S-Box..... Chen []..... Khan []..... Wang []..... Okanak [] Jakimoski [] Hussain [] Tang [] DP SAC is put forward b Webster et al. []. Generall speaking, the estlishment of SAC implies a possibilit that half of each output bit will be changed with the change of a single bit. Tle tells the average, minimum and maximum SAC values of the new S-Box as.,.,.. Evidentl, the average value of the new S-Box is close to the ideal value.. BIC is also an important criterion found b Webster et al. []. It can partiall measure the securit of crptosstems. The set of vectors generated b reversing one bit of the open text is tested to be independent of all the pairs of avalanche variles. While the relation between avalanches is measured, varile pairs are necessar to calculate the correlation value []. BIC-SAC and BIC-Nonlinearit values are calculated when the BIC value is calculated. When the values in Tle are examined, the BIC-SAC values are calculated as follows: average value., minimum value. and maximum value.. The average value is almost equal to the optimum value.. DP is another performance index for testing the S-Box, which is estlished b Biham et al. []. In this analsis, the XOR distribution balance between the input and output bits of the S-Box is determined. The ver close probilit of XOR distribution between input and output bits often indicates the ilit to resist the differential attack of the S-Box. The low DP value suggests that the S-Box is more resistant to attack. The minimum and maximum DP values of the new S-Box are determined as. and. From Tle, we know that the DP value of the new S-Box is the same as the S-Boxes presented b Tang and Okanak. After testing the performance of the new S-Box b using some important indices and comparing with other S-Boxes, we can determine that the new S-Box generated b sstem () has better performance than other S-Boxes. Thus, it will be more suitle for attack resistant and strong encrption.. Conclusions A special chaotic sstem with multiple attractors was studied in this letter. The complex dnamic behaviors of the sstem were mainl presented b numerical simulations. Bifurcation diagrams and phase portraits indicated that the sstem exhibits a pair of point attractors, a pair of periodic attractors, and a pair of strange attractors with the variation of sstem parameters. In addition, an electronic circuit was designed for realiing the chaotic attractors of the sstem. Moreover, a new S-Box was generated b appling the chaotic sstem, and the performance evaluation and comparison of the S-Box were presented. It showed that the new S-Box has better performance than some existing S-Boxes. Actuall, the stud of chaotic sstem with multiple attractors is of recent interest. More important issues corresponding to this topic will be addressed in our future paper.

14 Entrop,, of Acknowledgments: This work was supported b the National Natural Science Foundation of China (No., ),the Jiangxi Natural Science Foundation of China (No. BAB) and Sakara Universit Scientific Research Projects Unit (No. ---, ---). Author Contributions: Qiang Lai designed the stud and wrote the paper. Akif Akgul and Ünal Çavuşoğlu contributed to the experiment and algorithm design. Chunbiao Li and Guanghui Xu partiall undertook the theoretical analsis and simulation work of the paper. All authors read and approved the manuscript. Conflicts of Interest: The authors declare no conflict of interest. References. Loren, E.N. Deterministic non-periodic flow. J. Atmos. Sci.,,.. Rössler, O.E. An equation for continuous chaos. Phs. Lett. A,,.. Chen, G.; Ueta, T. Yet another chaotic attractor. Int. J. Bifurc. Chaos,,.. Lü, J.; Chen, G. A new chaotic attractor coined. Int. J. Bifurc. Chaos,,.. Guan, Z.H.; Lai, Q.; Chi, M.; Cheng, X.M.; Liu, F. Analsis of a new three-dimensional sstem with multiple chaotic attractors. Nonlinear Dn.,,.. Jafari, S.; Sprott, J.C.; Molaie, M. A simple chaotic flow with a plane of equilibria. Int. J. Bifurc. Chaos,,.. Lai, Q.; Guan, Z.H.; Wu, Y.; Liu, F.; Zhang, D. Generation of multi-wing chaotic attractors from a Loren-like sstem. Int. J. Bifurc. Chaos,,.. Wang, X.; Chen, G. Constructing a chaotic sstem with an number of equilibria. Nonlinear Dn.,,.. Riecke, H.; Roxin, A.; Madruga, S.; Solla, S.A. Multiple attractors, long chaotic transients, and failure in small-world networks of excitle neurons. Chaos,,.. Schwart, J.L.; Grimault, N.; Hupe, J.M.; Moore, B.C.; Pressniter, D. Multistilit in perception: Binding sensor modalities, an overview. Philos. Trans. R. Soc. B,,.. Yuan, G.; Wang, Z. Nonlinear mechanisms for multistilit in microring lasers. Phs. Rev. A,,.. Li, C.B.; Sprott, J.C. Varile-boostle chaotic flows. Optik,, -.. Li, C.B.; Sprott, J.C. Multistilit in the Loren sstem: A broken butterfl. Int. J. Bifurc. Chaos,,.. Li, C.B.; Sprott, J.C.; Hu, W.; Xu, Y. Infinite multistilit in a self-reproducing chaotic sstem. Int. J. Bifurc. Chaos,,.. Li, C.B.; Sprott, J.C.; Xing, H. Hpogenetic chaotic jerk flows. Phs. Lett. A,,.. Li, C.B.; Sprott, J.C.; Xing, H. Constructing chaotic sstems with conditional smmetr. Nonlinear Dn.,,.. Kengne, J.; Tekoueng, Z.N.; Tamba, V.K.; Negou, A.N. Periodicit, chaos, and multiple attractors in a memristor-based Shinriki s circuit. Chaos,,.. Kengne, J.; Tekoueng, Z.N.; Fotsin, H.B. Dnamical analsis of a simple autonomous jerk sstem with multiple attractors. Nonlinear Dn.,,.. Bao, B.C.; Jiang, T.; Xu, Q.; Chen, M.; Wu, H.; Hu, H.Y. Coexisting infinitel man attractors in active band-pass filter-based memristive circuit. Nonlinear Dn.,,.. Xu, Q.; Lin, Y.; Bao, B.C.; Chen, M. Multiple attractors in a non-ideal active voltage-controlled memristor based Chua s circuit. Chaos Solitons Fractals,,.. Lai, Q.; Chen, S. Research on a new D autonomous chaotic sstem with coexisting attractors. Optik,,.. Lai, Q.; Chen, S. Coexisting attractors gnerated from a new D smooth chaotic sstem. Int. J. Control Autom. Sst.,,.. Lai, Q.; Chen, S. Generating multiple chaotic atractors from Sprott B sstem. Int. J. Bifurc. Chaos,,.. Wei, Z.C.; Yu, P.; Zhang, W.; Yao, M. Stud of hidden attractors, multiple limit ccles from Hopf bifurcation and boundedness of motion in the generalied hperchaotic Rinovich sstem. Nonlinear Dn.,,.

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