This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other...This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other existing MMCSs, the most eye-catching point of the proposed MMCS is that the amplitude of the wing will enlarge towards the poles as the number of wings increases. Diverse coexisting attractors are numerically found in the MMCS, including chaos,quasi-period, and stable point. The circuits of the proposed memristor and MMCS are designed and the obtained results demonstrate their validity and reliability.展开更多
This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide...This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis,the Hopf bifurcation processes are proved to arise at certain equilibrium points.Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours;the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally,an analog electronic circuit is designed to physically realize the chaotic system;the existence of four-wing chaotic attractor is verified by the analog circuit realization.展开更多
A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can gener...A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincare maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.展开更多
This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz s...This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings (eight wings) of these novel attractors axe located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.展开更多
This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be gen...This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be generated by adding two triangular waves and a sign function. Some basic dynamical properties, such as equilibrium points, bifurcations, and phase diagrams, were studied. Furthermore, the system was experimentally confirmed by an electronic circuit. The circuit simulation results and numerical simulation results verified the feasibility of this method.展开更多
Due to uncertain push-pull action across boundaries between different attractive domains by random excitations, attractors of a dynamical system will drift in the phase space, which readily leads to colliding and mixi...Due to uncertain push-pull action across boundaries between different attractive domains by random excitations, attractors of a dynamical system will drift in the phase space, which readily leads to colliding and mixing with each other, so it is very difficult to identify irregular signals evolving from arbitrary initial states. Here, periodic attractors from the simple cell mapping method are further iterated by a specific Poincare map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations. The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure. From the positions and the variations of attractors in the phase space, the action mechanism of bounded noise excitation is studied in detail. Several numerical examples are employed to illustrate the present procedure. It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.展开更多
In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a si...In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.展开更多
A new three-dimensional (3D) continuous autonomous system with one parameter and three quadratic terms is presented firstly in this paper. Countless embedded trumpet-shaped chaotic attractors in two opposite directi...A new three-dimensional (3D) continuous autonomous system with one parameter and three quadratic terms is presented firstly in this paper. Countless embedded trumpet-shaped chaotic attractors in two opposite directions are generated from the system as time goes on. The basic dynamical behaviors of the strange chaotic system are investigated. Another more complex 3D system with the same capability of generating countless embedded trumpet-shaped chaotic attractors is also put forward.展开更多
A five-value memristor model is proposed,it is proved that the model has a typical hysteresis loop by analyzing the relationship between voltage and current.Then,based on the classical Liu-Chen system,a new memristor-...A five-value memristor model is proposed,it is proved that the model has a typical hysteresis loop by analyzing the relationship between voltage and current.Then,based on the classical Liu-Chen system,a new memristor-based fourdimensional(4D)chaotic system is designed by using the five-value memristor.The trajectory phase diagram,Poincare mapping,bifurcation diagram,and Lyapunov exponent spectrum are drawn by numerical simulation.It is found that,in addition to the general chaos characteristics,the system has some special phenomena,such as hidden homogenous multistabilities,hidden heterogeneous multistabilities,and hidden super-multistabilities.Finally,according to the dimensionless equation of the system,the circuit model of the system is built and simulated.The results are consistent with the numerical simulation results,which proves the physical realizability of the five-value memristor-based chaotic system proposed in this paper.展开更多
Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented.By applying impulsive control theory of the universal equation, the asympt...Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented.By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, and with it, the upper bond of the impulse interval for asymptotically stable control was given. Numerical results are presented, which are considered with important reference value for control of chaotic attractors.展开更多
A new three-dimensional (3D) system is constructed and a novel spherical chaotic attractor is generated from the system. Basic dynamical behaviors of the chaotic system are investigated respectively. Novel spherical...A new three-dimensional (3D) system is constructed and a novel spherical chaotic attractor is generated from the system. Basic dynamical behaviors of the chaotic system are investigated respectively. Novel spherical chaotic attractors can be generated from the system within a wide range of parameter values. The shapes of spherical chaotic attractors can be impacted by the variation of parameters. Finally, a simpler 3D system and a more complex 3D system with the same capability of generating spherical chaotic attractors are put forward respectively.展开更多
Based on passive theory, this paper studies a hybrid chaotic dynamical system from the mathematics perspective to implement the control of system stabilization. According to the Jacobian matrix of the nonlinear system...Based on passive theory, this paper studies a hybrid chaotic dynamical system from the mathematics perspective to implement the control of system stabilization. According to the Jacobian matrix of the nonlinear system, the stabilization control region is gotten. The controller is designed to stabilize fast the minimum phase Lorenz-Chen chaotic system after equivalently transforming from chaotic system to passive system. The simulation results show that the system not only can be controlled at the different equilibria, but also can be transformed between the different chaotic attractors.展开更多
We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability...We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.展开更多
A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2...A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.展开更多
A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring tra- jectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical syste...A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring tra- jectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1>0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an alternative method to calculate λ1 has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.展开更多
The 1970-1985 day to day averaged pressure dataset of Shanghai and the extension method in phase space are used to calculate the correlation dimension D and the second-order Renyi entropy K2 of the approximation of Ko...The 1970-1985 day to day averaged pressure dataset of Shanghai and the extension method in phase space are used to calculate the correlation dimension D and the second-order Renyi entropy K2 of the approximation of Kolmogorov's entropy, the fractional dimension D = 7.7-7.9 and the positive value K2 - 0.1 are obtained. This shows that the attractor for the short-term weather evolution in the monsoon region of China exhibits a chaotic motion. The estimate of K2 yields a predictable time scale of about ten days. This result is in agreement with that obtained earlier by the dynamic-statistical approach.The effects of the lag time i on the estimate of D and K2 are investigated. The results show that D and K2 are convergent with respect to i. The day to day averaged pressure series used in this paper are treated for the extensive phase space with T = 5, the coordinate components are independent of each other; therefore, the dynamical character quantities of the system are stable and reliable.展开更多
In recent years,there are numerous studies on chaotic systems with special equilibrium curves having various shapes such as circle,butterfly,heart and apple.This paper describes a new 3-D chaotic dynamical system with...In recent years,there are numerous studies on chaotic systems with special equilibrium curves having various shapes such as circle,butterfly,heart and apple.This paper describes a new 3-D chaotic dynamical system with a capsule-shaped equilibrium curve.The proposed chaotic system has two quadratic,two cubic and two quartic nonlinear terms.It is noted that the proposed chaotic system has a hidden attractor since it has an infinite number of equilibrium points.It is also established that the proposed chaotic system exhibits multi-stability with two coexisting chaotic attractors for the same parameter values but differential initial states.A detailed bifurcation analysis with respect to variations in the system parameters is portrayed for the new chaotic system with capsule equilibrium curve.We have shown MATLAB plots to illustrate the capsule equilibrium curve,phase orbits of the new chaotic system,bifurcation diagrams and multi-stability.As an engineering application,we have proposed a speech cryptosystem with a numerical algorithm,which is based on our novel 3-D chaotic system with a capsule-shaped equilibrium curve.The proposed speech cryptosystem follows its security evolution and implementation on Field Programmable Gate Array(FPGA)platform.Experimental results show that the proposed encryption system utilizes 33%of the FPGA,while the maximum clock frequency is 178.28 MHz.展开更多
Recently, we received a letter from Prof. G. L. Oppo, which indicated that he had doubts about the transformation of the system in the article Chin. Phys. B 31 060503 (2022) and gave other considerations. After inspec...Recently, we received a letter from Prof. G. L. Oppo, which indicated that he had doubts about the transformation of the system in the article Chin. Phys. B 31 060503 (2022) and gave other considerations. After inspection, we found that there was a clerical error in the article. Based on this, we have made corrections and supplements to the original article.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 62366014 and 61961019)the Natural Science Foundation of Jiangxi Province, China (Grant No. 20232BAB202008)。
文摘This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other existing MMCSs, the most eye-catching point of the proposed MMCS is that the amplitude of the wing will enlarge towards the poles as the number of wings increases. Diverse coexisting attractors are numerically found in the MMCS, including chaos,quasi-period, and stable point. The circuits of the proposed memristor and MMCS are designed and the obtained results demonstrate their validity and reliability.
基金Project supported by the National Natural Science Foundation of China(Grant Nos 60774088 and 10772135)the Foundation of the Application Base and Frontier Technology Research Project of Tianjin,China (Grant Nos 07JCZDJC09600,08JCZDJC21900 and 08JCZDJC18600)the Tianjin Key Laboratory for Control Theory & Applications in Complicated Industry Systems of China
文摘This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis,the Hopf bifurcation processes are proved to arise at certain equilibrium points.Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours;the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally,an analog electronic circuit is designed to physically realize the chaotic system;the existence of four-wing chaotic attractor is verified by the analog circuit realization.
文摘A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincare maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.
文摘This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings (eight wings) of these novel attractors axe located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.
基金Project supported by the National Natural Science Foundation of China (Grant No. 60972069)the Science and Technology Foundation of the Education Department of Chongqing (Grant No. KJ090513)
文摘This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be generated by adding two triangular waves and a sign function. Some basic dynamical properties, such as equilibrium points, bifurcations, and phase diagrams, were studied. Furthermore, the system was experimentally confirmed by an electronic circuit. The circuit simulation results and numerical simulation results verified the feasibility of this method.
基金supported by the National Natural Science Foundation of China (10672140,11072213)
文摘Due to uncertain push-pull action across boundaries between different attractive domains by random excitations, attractors of a dynamical system will drift in the phase space, which readily leads to colliding and mixing with each other, so it is very difficult to identify irregular signals evolving from arbitrary initial states. Here, periodic attractors from the simple cell mapping method are further iterated by a specific Poincare map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations. The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure. From the positions and the variations of attractors in the phase space, the action mechanism of bounded noise excitation is studied in detail. Several numerical examples are employed to illustrate the present procedure. It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.
文摘In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.
基金supported by the Science Research Foundation of Liaoning Provincial Education Department,China(Grant No.L2013229)
文摘A new three-dimensional (3D) continuous autonomous system with one parameter and three quadratic terms is presented firstly in this paper. Countless embedded trumpet-shaped chaotic attractors in two opposite directions are generated from the system as time goes on. The basic dynamical behaviors of the strange chaotic system are investigated. Another more complex 3D system with the same capability of generating countless embedded trumpet-shaped chaotic attractors is also put forward.
基金supported by the National Natural Science Foundation of China(Grant No.61203004)the Natural Science Foundation of Heilongjiang Province,China(Grant No.F201220)the Heilongjiang Provincial Natural Science Foundation of Joint Guidance Project(Grant No.LH2020F022).
文摘A five-value memristor model is proposed,it is proved that the model has a typical hysteresis loop by analyzing the relationship between voltage and current.Then,based on the classical Liu-Chen system,a new memristor-based fourdimensional(4D)chaotic system is designed by using the five-value memristor.The trajectory phase diagram,Poincare mapping,bifurcation diagram,and Lyapunov exponent spectrum are drawn by numerical simulation.It is found that,in addition to the general chaos characteristics,the system has some special phenomena,such as hidden homogenous multistabilities,hidden heterogeneous multistabilities,and hidden super-multistabilities.Finally,according to the dimensionless equation of the system,the circuit model of the system is built and simulated.The results are consistent with the numerical simulation results,which proves the physical realizability of the five-value memristor-based chaotic system proposed in this paper.
文摘Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented.By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, and with it, the upper bond of the impulse interval for asymptotically stable control was given. Numerical results are presented, which are considered with important reference value for control of chaotic attractors.
基金Project supported by the Science Research Foundation of Liaoning Provincial Education Department,China(Grant No.L2013229)
文摘A new three-dimensional (3D) system is constructed and a novel spherical chaotic attractor is generated from the system. Basic dynamical behaviors of the chaotic system are investigated respectively. Novel spherical chaotic attractors can be generated from the system within a wide range of parameter values. The shapes of spherical chaotic attractors can be impacted by the variation of parameters. Finally, a simpler 3D system and a more complex 3D system with the same capability of generating spherical chaotic attractors are put forward respectively.
基金Project supported by the National Natural Science Foundation of China(Grant No60702023)Natural Science Foundation of Zhejiang Province,China(Grant No Y104414)
文摘Based on passive theory, this paper studies a hybrid chaotic dynamical system from the mathematics perspective to implement the control of system stabilization. According to the Jacobian matrix of the nonlinear system, the stabilization control region is gotten. The controller is designed to stabilize fast the minimum phase Lorenz-Chen chaotic system after equivalently transforming from chaotic system to passive system. The simulation results show that the system not only can be controlled at the different equilibria, but also can be transformed between the different chaotic attractors.
基金National Natural Science Foundation of China(Grant Nos.11672257,11632008,11772306,and 11972173)the Natural Science Foundation of Jiangsu Province of China(Grant No.BK20161314)+1 种基金the 5th 333 High-level Personnel Training Project of Jiangsu Province of China(Grant No.BRA2018324)the Excellent Scientific and Technological Innovation Team of Jiangsu University.
文摘We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.
文摘A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.
文摘A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring tra- jectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1>0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an alternative method to calculate λ1 has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.
文摘The 1970-1985 day to day averaged pressure dataset of Shanghai and the extension method in phase space are used to calculate the correlation dimension D and the second-order Renyi entropy K2 of the approximation of Kolmogorov's entropy, the fractional dimension D = 7.7-7.9 and the positive value K2 - 0.1 are obtained. This shows that the attractor for the short-term weather evolution in the monsoon region of China exhibits a chaotic motion. The estimate of K2 yields a predictable time scale of about ten days. This result is in agreement with that obtained earlier by the dynamic-statistical approach.The effects of the lag time i on the estimate of D and K2 are investigated. The results show that D and K2 are convergent with respect to i. The day to day averaged pressure series used in this paper are treated for the extensive phase space with T = 5, the coordinate components are independent of each other; therefore, the dynamical character quantities of the system are stable and reliable.
基金funded by the Center for Research Excellence,Incubation Management Center,Universiti Sultan Zainal Abidin via an internal grant UniSZA/2021/SRGSIC/07.
文摘In recent years,there are numerous studies on chaotic systems with special equilibrium curves having various shapes such as circle,butterfly,heart and apple.This paper describes a new 3-D chaotic dynamical system with a capsule-shaped equilibrium curve.The proposed chaotic system has two quadratic,two cubic and two quartic nonlinear terms.It is noted that the proposed chaotic system has a hidden attractor since it has an infinite number of equilibrium points.It is also established that the proposed chaotic system exhibits multi-stability with two coexisting chaotic attractors for the same parameter values but differential initial states.A detailed bifurcation analysis with respect to variations in the system parameters is portrayed for the new chaotic system with capsule equilibrium curve.We have shown MATLAB plots to illustrate the capsule equilibrium curve,phase orbits of the new chaotic system,bifurcation diagrams and multi-stability.As an engineering application,we have proposed a speech cryptosystem with a numerical algorithm,which is based on our novel 3-D chaotic system with a capsule-shaped equilibrium curve.The proposed speech cryptosystem follows its security evolution and implementation on Field Programmable Gate Array(FPGA)platform.Experimental results show that the proposed encryption system utilizes 33%of the FPGA,while the maximum clock frequency is 178.28 MHz.
文摘Recently, we received a letter from Prof. G. L. Oppo, which indicated that he had doubts about the transformation of the system in the article Chin. Phys. B 31 060503 (2022) and gave other considerations. After inspection, we found that there was a clerical error in the article. Based on this, we have made corrections and supplements to the original article.