To seek for lower-dimensional chaotic systems that have complex topological attractor structure with simple algebraic system structure, a new chaotic system of three-dimensional autonomous ordinary differential equati...To seek for lower-dimensional chaotic systems that have complex topological attractor structure with simple algebraic system structure, a new chaotic system of three-dimensional autonomous ordinary differential equations is presented. The new system has simple algebraic structure, and can display a 2-scroll attractor with complex topological structure, which is different from the Lorenz＇s, Chen＇s and Lu¨＇s attractors. By introducing a linear state feedback controller, the system can be controlled to generate a hyperchaotic attractor. The novel chaotic attractor, hyperchaotic attractor and dynamical behaviors of corresponding systems are further investigated by employing Lyapunov exponent spectrum, bifurcation diagram, Poincar′e mapping and phase portrait, etc., and then verified by simulating an experimental circuit.展开更多
A new kind of secure communication system which combines the chaotic encryption means with the conventional encryption method is discussed. With the analysis results and the experiment data, the anti-attack ability of...A new kind of secure communication system which combines the chaotic encryption means with the conventional encryption method is discussed. With the analysis results and the experiment data, the anti-attack ability of this communication system is significantly improved compared to that of the either method. At the same time, a new method of chaotic synchronization is proposed. With a small mixed discrete chaotic signal, it is quickly to synchronize the communication and a good security performance is ensured.展开更多
A more general form of projective synchronization, so called linear generalized synchronization （LGS） is proposed, which includes the generalized projective synchronization （GPS） and the hybrid projective synchron...A more general form of projective synchronization, so called linear generalized synchronization （LGS） is proposed, which includes the generalized projective synchronization （GPS） and the hybrid projective synchronization （HPS） as its special cases, Based on the adaptive technique and Lyapunov stability theory, a general method for achieving the LGS between two chaotic or hyperehaotic systems with uncertain parameters in any scaling matrix is presented. Some numerical simulations are provided to show the effectiveness and feasibility of the proposed synchronization method.展开更多
A class of the hybrid chaotic sequences is presented. The generator of the sequences is given and realized by the digital method. The hybrid chaotic sequences exhibit good random properties that are very important for...A class of the hybrid chaotic sequences is presented. The generator of the sequences is given and realized by the digital method. The hybrid chaotic sequences exhibit good random properties that are very important for the performance of QS-CDMA system with RAKE receiver. The performance of the system is analyzed when the hybrid chaotic sequences are used as spreading codes in a QS-CDMA system with RAKE receiver and compared with those obtained for m-se-quences and logistic sequences. The results show that the hybrid chaotic sequences are a class of very promising spreading codes for QS-CDMA system.展开更多
A strategy for a novel concatenated chaotic communication system is presented. The transmitter system comprises chaotic turbo encoder and logistic CSK block in a serially concatenated form. Chaotic turbo code is capab...A strategy for a novel concatenated chaotic communication system is presented. The transmitter system comprises chaotic turbo encoder and logistic CSK block in a serially concatenated form. Chaotic turbo code is capable of reducing bit error rate (BER) of the chaotic system in the AWGN channel. Through the chaotic turbo encoder, the coded sequence, which has quasi-chaotic properties, will be transmitted into the logistic CSK block. Having a very sensitive dependence on initial conditions of the map, the logistic CSK block can also be taken as the chaotic authentication method. The receiver, which has logistic demodulation block and chaotic decoder, is a linear asymptotic approximation to the inverse of the transmitter system. A chaotic iterative soft-decision decoding algorithm is also developed based on conventional maximum A posteriori decoding algorithm. At last, a two-step authentication method of this chaotic system is also presented.展开更多
It is important to develop control techniques able to control not only known chaos but also chaotic systems with unknown parameters. This paper proposes a novel adaptive tracking control approach for identifying the u...It is important to develop control techniques able to control not only known chaos but also chaotic systems with unknown parameters. This paper proposes a novel adaptive tracking control approach for identifying the unknown parameters and controlling the chaos, which is not closely related to the particular chaotic system to be controlled. The global uniform boundedness of estimated parameters and the asymptotical stability of the tracking errors are proved by Lyapunov stability theory and LaSalleYoshizawa theorem. The suggested method enables stabilization of chaotic motion to a steady state ad well as tracking of any desired trajectory to be achieved in a systematic way. Computer simulation on a complex chaotic system illustrtes the effectiveness of the proposed control method.展开更多
As known to all that Henon chaotic system is not appropriate for generating the key-streams because it has non-uniformly distributed output signal, a new key-stream generation scheme based on Henon chaotic system is p...As known to all that Henon chaotic system is not appropriate for generating the key-streams because it has non-uniformly distributed output signal, a new key-stream generation scheme based on Henon chaotic system is presented. In order to get the key-streams with good statistics and long enough cycle length, a specific method for dividing the enon attractor into numerous non-overlapping sub-regions, and a new one-to-one mapping strategy between the divided sub-regions and elements of dynamically generated matrix consisting of O＇s and l＇s are proposed. Experimental results demonstrate that the generated key-streams are with long enough cycle length and very sensitive to the initial values and secret keys. For example, key-streams with the cycle length of 10^32 can easily be obtained. Moreover, even if the fluctuation to the initial values or secret keys is as small as 10^- 14 uncorrelated key-streams will be generated. Experimental results also demonstrate that the generated key-streams have good randomness and they can pass all the standard criteria specified in FIPS PUB 140^-2 with no less than 98%.展开更多
The synchronization problem under two cases is considered. One is that the bound on the uncertainty existing in the controller is known, the other is that the bound is unknown. In the latter case, the simple adaptatio...The synchronization problem under two cases is considered. One is that the bound on the uncertainty existing in the controller is known, the other is that the bound is unknown. In the latter case, the simple adaptation laws for upper bound on the norm of the uncertainty is proposed. Using this adaptive upper bound, a variable structure control is designed. The proposed method does not guarantee the convergence of the adaptive upper bound to the real one but makes the system asymptotically stable.展开更多
Chaos synchronization of systems with perturbations was investigated.A generic nonlinear control scheme to realize chaos synchronization of systems was proposed.This control scheme is flexible and practicable,and give...Chaos synchronization of systems with perturbations was investigated.A generic nonlinear control scheme to realize chaos synchronization of systems was proposed.This control scheme is flexible and practicable,and gives more freedom in designing controllers in order to achieve some desired performance.With the aid of Lyapunov stability theorem and partial stability theory,two cases were presented:1) Chaos synchronization of the system without perturbation or with vanishing perturbations;2) The boundness of the error state for the system with nonvanishing perturbations satisfying some conditions.Finally,several simulations for Lorenz system were provided to verify the effectiveness and feasibility of our method.Compared numerically with the existing results of linear feedback control scheme,the results are sharper than the existing ones.展开更多
In this paper,we first propose a memristive chaotic system and implement it by circuit simulation.The chaotic dynamics and various attractors are analysed by using phase portrait,bifurcation diagram,and Lyapunov expon...In this paper,we first propose a memristive chaotic system and implement it by circuit simulation.The chaotic dynamics and various attractors are analysed by using phase portrait,bifurcation diagram,and Lyapunov exponents.In particular,the system has robust chaos in a wide parameter range and the initial value space,which is favourable to the security communication application.Consequently,we further explore its application in image encryption and present a new scheme.Before image processing,the external key is protected by the Grain-128a algorithm and the initial values of the memristive system are updated with the plain image.We not only perform random pixel extraction and masking with the chaotic cipher,but also use them as control parameters for Brownian motion to obtain the permutation matrix.In addition,multiplication on the finite field GF(2^(8))is added to further enhance the cryptography.Finally,the simulation results verify that the proposed image encryption scheme has better performance and higher security,which can effectively resist various attacks.展开更多
It is well known that the periodic performance of spread spectrum sequence heavily affects the correlative and secure characteristics of communication systems. The chaotic binary sequence is paid more and more attenti...It is well known that the periodic performance of spread spectrum sequence heavily affects the correlative and secure characteristics of communication systems. The chaotic binary sequence is paid more and more attention since it is one kind of applicable spread spectrum sequences. However, there are unavoidable short cyclic problems for chaotic binary sequences in finite precision. The chaotic binary sequence generating methods are studied first. Then the short cyclic behavior of the chaotic sequences is analyzed in detail, which are generated by quantification approaches with finite word-length. At the same time, a chaotic similar function is defined for presenting the cyclic characteristics of the sequences. Based on these efforts, an improved method with scrambling control for generating chaotic binary sequences is proposed. To quantitatively describe the improvement of periodic performance of the sequences, an orthogonal estimator is also defined. Some simulating results are provided. From the theoretical deduction and the experimental results, it is concluded that the proposed method can effectively increase the period and raise the complexity of the chaotic sequences to some extent.展开更多
Chaotic mixing is regarded as an effective method to improve the mixing effect in the laminar stirred tank.To generate chaos in flow field in stirred tank,many efforts including aperiodic movement,asystematic installa...Chaotic mixing is regarded as an effective method to improve the mixing effect in the laminar stirred tank.To generate chaos in flow field in stirred tank,many efforts including aperiodic movement,asystematic installation impeller,etc.,were made to save mixing time and energy consumption.The computational fluid dynamics of chaotic flow field and the hyperchaotic controlling method as well as macro-instability were reviewed.Multiple flow field coupling is important in the turbulent mixing region of stirred tank.The development trends of flow field coupling and hyperchaotic controlling in chaotic mixing were prospected in energy saving operation.展开更多
Two problems are proposed. The first one is the noise decontamination of chaotic carriers using a deterministic approach to reconstruct pseudo trajectories, the second is the design of communications schemes with chao...Two problems are proposed. The first one is the noise decontamination of chaotic carriers using a deterministic approach to reconstruct pseudo trajectories, the second is the design of communications schemes with chaotic carriers. After presenting our deterministic noise decontamination algorithm, conventional chaos shift keying (CSK) communication system is applied. The difference of Euclidean distance between noisy trajectory and decontaminated trajectory in phase space could be utilized to non-coherently detect the sent symbol simply and effectively. It is shown that this detection method can achieve the bit error rate performance comparable to other non-coherent systems.展开更多
Pulse Doppler(PD) fuze is widely used in current battlefield. However, with the threat of repeater jamming, especially digital radio frequency memory technology, the deficiency in the anti-repeater jamming of a tradit...Pulse Doppler(PD) fuze is widely used in current battlefield. However, with the threat of repeater jamming, especially digital radio frequency memory technology, the deficiency in the anti-repeater jamming of a traditional PD fuze increasingly emerges. Therefore, a repeater jamming suppression method for a PD fuze based on identity(ID) recognition and chaotic encryption is proposed. Every fuze has its own ID which is encrypted with different chaotic binary sequences in every pulse period of the transmitted signal. The thumbtack-shaped ambiguity function shows a good resolution and distance cutoff characteristic. The ability of anti-repeater jamming is emphatically analyzed, and the results at different signal-to-noise ratio(SNR) show a strong anti-repeater jamming ability and range resolution that the proposed method possesses. Furthermore, the anti-repeater jamming ability is influenced by processing gain, bit error rate(BER) and correlation function. The simulation result validates the theoretical analysis, it shows the proposed method can significantly improve the anti-repeater jamming ability of a PD fuze.展开更多
Considering chaotic time series multi-step prediction, multi-step direct prediction model based on partial least squares （PLS） is proposed in this article, where PLS, the method for predicting a set of dependent var...Considering chaotic time series multi-step prediction, multi-step direct prediction model based on partial least squares （PLS） is proposed in this article, where PLS, the method for predicting a set of dependent variables forming a large set of predictors, is used to model the dynamic evolution between the space points and the corresponding future points. The model can eliminate error accumulation with the common single-step local model algorithm~ and refrain from the high multi-collinearity problem in the reconstructed state space with the increase of embedding dimension. Simulation predictions are done on the Mackey-Glass chaotic time series with the model. The satisfying prediction accuracy is obtained and the model efficiency verified. In the experiments, the number of extracted components in PLS is set with cross-validation procedure.展开更多
A novel chaotic search method is proposed,and a hybrid algorithm combining particle swarm optimization(PSO) with this new method,called CLSPSO,is put forward to solve 14 integer and mixed integer programming problems....A novel chaotic search method is proposed,and a hybrid algorithm combining particle swarm optimization(PSO) with this new method,called CLSPSO,is put forward to solve 14 integer and mixed integer programming problems.The performances of CLSPSO are compared with those of other five hybrid algorithms combining PSO with chaotic search methods.Experimental results indicate that in terms of robustness and final convergence speed,CLSPSO is better than other five algorithms in solving many of these problems.Furthermore,CLSPSO exhibits good performance in solving two high-dimensional problems,and it finds better solutions than the known ones.A performance index(PI) is introduced to fairly compare the above six algorithms,and the obtained values of(PI) in three cases demonstrate that CLSPSO is superior to all the other five algorithms under the same conditions.展开更多
基金supported by the National Natural Science Foundation of China (60971090)the Natural Science Foundation of Jiangsu Province (BK 2009105)
文摘To seek for lower-dimensional chaotic systems that have complex topological attractor structure with simple algebraic system structure, a new chaotic system of three-dimensional autonomous ordinary differential equations is presented. The new system has simple algebraic structure, and can display a 2-scroll attractor with complex topological structure, which is different from the Lorenz＇s, Chen＇s and Lu¨＇s attractors. By introducing a linear state feedback controller, the system can be controlled to generate a hyperchaotic attractor. The novel chaotic attractor, hyperchaotic attractor and dynamical behaviors of corresponding systems are further investigated by employing Lyapunov exponent spectrum, bifurcation diagram, Poincar′e mapping and phase portrait, etc., and then verified by simulating an experimental circuit.
文摘A new kind of secure communication system which combines the chaotic encryption means with the conventional encryption method is discussed. With the analysis results and the experiment data, the anti-attack ability of this communication system is significantly improved compared to that of the either method. At the same time, a new method of chaotic synchronization is proposed. With a small mixed discrete chaotic signal, it is quickly to synchronize the communication and a good security performance is ensured.
基金the National Natural Science Foundation of China (60574045 10661006).
文摘A more general form of projective synchronization, so called linear generalized synchronization （LGS） is proposed, which includes the generalized projective synchronization （GPS） and the hybrid projective synchronization （HPS） as its special cases, Based on the adaptive technique and Lyapunov stability theory, a general method for achieving the LGS between two chaotic or hyperehaotic systems with uncertain parameters in any scaling matrix is presented. Some numerical simulations are provided to show the effectiveness and feasibility of the proposed synchronization method.
基金This project was supported by the National Defense Key Laboratory Foundation (99JS04.8.1. DZ02 24).
文摘A class of the hybrid chaotic sequences is presented. The generator of the sequences is given and realized by the digital method. The hybrid chaotic sequences exhibit good random properties that are very important for the performance of QS-CDMA system with RAKE receiver. The performance of the system is analyzed when the hybrid chaotic sequences are used as spreading codes in a QS-CDMA system with RAKE receiver and compared with those obtained for m-se-quences and logistic sequences. The results show that the hybrid chaotic sequences are a class of very promising spreading codes for QS-CDMA system.
基金This project was supported by the National Natural Science Foundation of China (No. 60072028).
文摘A strategy for a novel concatenated chaotic communication system is presented. The transmitter system comprises chaotic turbo encoder and logistic CSK block in a serially concatenated form. Chaotic turbo code is capable of reducing bit error rate (BER) of the chaotic system in the AWGN channel. Through the chaotic turbo encoder, the coded sequence, which has quasi-chaotic properties, will be transmitted into the logistic CSK block. Having a very sensitive dependence on initial conditions of the map, the logistic CSK block can also be taken as the chaotic authentication method. The receiver, which has logistic demodulation block and chaotic decoder, is a linear asymptotic approximation to the inverse of the transmitter system. A chaotic iterative soft-decision decoding algorithm is also developed based on conventional maximum A posteriori decoding algorithm. At last, a two-step authentication method of this chaotic system is also presented.
文摘It is important to develop control techniques able to control not only known chaos but also chaotic systems with unknown parameters. This paper proposes a novel adaptive tracking control approach for identifying the unknown parameters and controlling the chaos, which is not closely related to the particular chaotic system to be controlled. The global uniform boundedness of estimated parameters and the asymptotical stability of the tracking errors are proved by Lyapunov stability theory and LaSalleYoshizawa theorem. The suggested method enables stabilization of chaotic motion to a steady state ad well as tracking of any desired trajectory to be achieved in a systematic way. Computer simulation on a complex chaotic system illustrtes the effectiveness of the proposed control method.
基金Foundation item： Proj ects（61172184, 61173147） supported by the National Natural Science Foundation of China Project（12JJ6062） supported by Natural Science Foundation of Hunan Province, China+1 种基金 Project（121gpy31） supported by the Fundamental Research Funds for the Central Universities of China Project supported by the State Key Laboratory of Information Security （Institute of Software, Chinese Academy of Sciences）, China
文摘As known to all that Henon chaotic system is not appropriate for generating the key-streams because it has non-uniformly distributed output signal, a new key-stream generation scheme based on Henon chaotic system is presented. In order to get the key-streams with good statistics and long enough cycle length, a specific method for dividing the enon attractor into numerous non-overlapping sub-regions, and a new one-to-one mapping strategy between the divided sub-regions and elements of dynamically generated matrix consisting of O＇s and l＇s are proposed. Experimental results demonstrate that the generated key-streams are with long enough cycle length and very sensitive to the initial values and secret keys. For example, key-streams with the cycle length of 10^32 can easily be obtained. Moreover, even if the fluctuation to the initial values or secret keys is as small as 10^- 14 uncorrelated key-streams will be generated. Experimental results also demonstrate that the generated key-streams have good randomness and they can pass all the standard criteria specified in FIPS PUB 140^-2 with no less than 98%.
文摘The synchronization problem under two cases is considered. One is that the bound on the uncertainty existing in the controller is known, the other is that the bound is unknown. In the latter case, the simple adaptation laws for upper bound on the norm of the uncertainty is proposed. Using this adaptive upper bound, a variable structure control is designed. The proposed method does not guarantee the convergence of the adaptive upper bound to the real one but makes the system asymptotically stable.
基金Projects(61075065,60774045,U1134108) supported by the National Natural Science Foundation of ChinaProject(20110162110041) supported by the Ph.D Programs Foundation of Ministry of Education of ChinaProject(CX2011B086) supported by Hunan Provincial Innovation Foundation For Postgraduate,China
文摘Chaos synchronization of systems with perturbations was investigated.A generic nonlinear control scheme to realize chaos synchronization of systems was proposed.This control scheme is flexible and practicable,and gives more freedom in designing controllers in order to achieve some desired performance.With the aid of Lyapunov stability theorem and partial stability theory,two cases were presented:1) Chaos synchronization of the system without perturbation or with vanishing perturbations;2) The boundness of the error state for the system with nonvanishing perturbations satisfying some conditions.Finally,several simulations for Lorenz system were provided to verify the effectiveness and feasibility of our method.Compared numerically with the existing results of linear feedback control scheme,the results are sharper than the existing ones.
基金This work was supported by the National Natural Science Foundation of China(61203004)the Natural Science Foundation of Heilongjiang Province(F201220)the Heilongjiang Provincial Natural Science Foundation of Joint Guidance Project(LH2020F022).
文摘In this paper,we first propose a memristive chaotic system and implement it by circuit simulation.The chaotic dynamics and various attractors are analysed by using phase portrait,bifurcation diagram,and Lyapunov exponents.In particular,the system has robust chaos in a wide parameter range and the initial value space,which is favourable to the security communication application.Consequently,we further explore its application in image encryption and present a new scheme.Before image processing,the external key is protected by the Grain-128a algorithm and the initial values of the memristive system are updated with the plain image.We not only perform random pixel extraction and masking with the chaotic cipher,but also use them as control parameters for Brownian motion to obtain the permutation matrix.In addition,multiplication on the finite field GF(2^(8))is added to further enhance the cryptography.Finally,the simulation results verify that the proposed image encryption scheme has better performance and higher security,which can effectively resist various attacks.
基金This work was partly supported by the National Natural Science Foundation of China （11501581）, the Project by Central South Uni- versity （502042032）, and the China Postdoctoral Science Foundation （2015M570683）.
基金the National Natural Science Foundation of China (60572075)the Natural Science Researching Project for Jiangsu Universities (05KJD510177).
文摘It is well known that the periodic performance of spread spectrum sequence heavily affects the correlative and secure characteristics of communication systems. The chaotic binary sequence is paid more and more attention since it is one kind of applicable spread spectrum sequences. However, there are unavoidable short cyclic problems for chaotic binary sequences in finite precision. The chaotic binary sequence generating methods are studied first. Then the short cyclic behavior of the chaotic sequences is analyzed in detail, which are generated by quantification approaches with finite word-length. At the same time, a chaotic similar function is defined for presenting the cyclic characteristics of the sequences. Based on these efforts, an improved method with scrambling control for generating chaotic binary sequences is proposed. To quantitatively describe the improvement of periodic performance of the sequences, an orthogonal estimator is also defined. Some simulating results are provided. From the theoretical deduction and the experimental results, it is concluded that the proposed method can effectively increase the period and raise the complexity of the chaotic sequences to some extent.
基金supported by Natural Science Foundation of China(20806095)ScientificResearch Foundation for the Returned Overseas Chinese Scholars,State Education Ministry(教外司留20091341-2)+1 种基金China Postdoctoral Science Foundation(20080430747)National High Technology Research and Development Program of China(2008AA0312)
文摘Chaotic mixing is regarded as an effective method to improve the mixing effect in the laminar stirred tank.To generate chaos in flow field in stirred tank,many efforts including aperiodic movement,asystematic installation impeller,etc.,were made to save mixing time and energy consumption.The computational fluid dynamics of chaotic flow field and the hyperchaotic controlling method as well as macro-instability were reviewed.Multiple flow field coupling is important in the turbulent mixing region of stirred tank.The development trends of flow field coupling and hyperchaotic controlling in chaotic mixing were prospected in energy saving operation.
文摘Two problems are proposed. The first one is the noise decontamination of chaotic carriers using a deterministic approach to reconstruct pseudo trajectories, the second is the design of communications schemes with chaotic carriers. After presenting our deterministic noise decontamination algorithm, conventional chaos shift keying (CSK) communication system is applied. The difference of Euclidean distance between noisy trajectory and decontaminated trajectory in phase space could be utilized to non-coherently detect the sent symbol simply and effectively. It is shown that this detection method can achieve the bit error rate performance comparable to other non-coherent systems.
基金National Natural Science Foundation of China under Grant No. 61973037 and No. 61673066。
文摘Pulse Doppler(PD) fuze is widely used in current battlefield. However, with the threat of repeater jamming, especially digital radio frequency memory technology, the deficiency in the anti-repeater jamming of a traditional PD fuze increasingly emerges. Therefore, a repeater jamming suppression method for a PD fuze based on identity(ID) recognition and chaotic encryption is proposed. Every fuze has its own ID which is encrypted with different chaotic binary sequences in every pulse period of the transmitted signal. The thumbtack-shaped ambiguity function shows a good resolution and distance cutoff characteristic. The ability of anti-repeater jamming is emphatically analyzed, and the results at different signal-to-noise ratio(SNR) show a strong anti-repeater jamming ability and range resolution that the proposed method possesses. Furthermore, the anti-repeater jamming ability is influenced by processing gain, bit error rate(BER) and correlation function. The simulation result validates the theoretical analysis, it shows the proposed method can significantly improve the anti-repeater jamming ability of a PD fuze.
文摘Considering chaotic time series multi-step prediction, multi-step direct prediction model based on partial least squares （PLS） is proposed in this article, where PLS, the method for predicting a set of dependent variables forming a large set of predictors, is used to model the dynamic evolution between the space points and the corresponding future points. The model can eliminate error accumulation with the common single-step local model algorithm~ and refrain from the high multi-collinearity problem in the reconstructed state space with the increase of embedding dimension. Simulation predictions are done on the Mackey-Glass chaotic time series with the model. The satisfying prediction accuracy is obtained and the model efficiency verified. In the experiments, the number of extracted components in PLS is set with cross-validation procedure.
基金Projects(50275150,61173052) supported by the National Natural Science Foundation of ChinaProject(14FJ3112) supported by the Planned Science and Technology of Hunan Province,ChinaProject(14B033) supported by Scientific Research Fund Education Department of Hunan Province,China
文摘A novel chaotic search method is proposed,and a hybrid algorithm combining particle swarm optimization(PSO) with this new method,called CLSPSO,is put forward to solve 14 integer and mixed integer programming problems.The performances of CLSPSO are compared with those of other five hybrid algorithms combining PSO with chaotic search methods.Experimental results indicate that in terms of robustness and final convergence speed,CLSPSO is better than other five algorithms in solving many of these problems.Furthermore,CLSPSO exhibits good performance in solving two high-dimensional problems,and it finds better solutions than the known ones.A performance index(PI) is introduced to fairly compare the above six algorithms,and the obtained values of(PI) in three cases demonstrate that CLSPSO is superior to all the other five algorithms under the same conditions.
