fuzzy goal programming complementary slackness conditions and computational schemes abstract goal programming gp is an important category in linear programming and multiobjective versions which minimize the deviation among value of each objective function and its goal with satisfying other constraints of problem in this paper we deal with fuzzy goal programming fgp which is a generalized problem in vague nature where parameters relations and goals are given as fuzzy quantities with respect to imprecise data recently inuiguchi et al m inuiguchi j ramik t tanio m vlach satisficing solutions and duality in interval and fuzzy linear programming fuzzy sets and systems 135 2003 151 177 have proved some important dual theorems on fgp which are very close to the same crisp versions in continuation to their works based on fuzzy extensions of relations we get complementary slackness conditions for fgp which have an essential role in many optimization techniques moreover we obtain the most optimistic and pessimistic satisficing solutions employing these conditions and present a convex combination of them which covers the satisficing solutions for a fixed level of certainty our practical schemes implement an efficient interior point algorithm as sub procedure so they are polynomial time algorithms

on duality in linear programming under fuzzy environment a pair of linear primal dual problems is introduced under fuzzy environment and appropriate results are proved to establish the duality relationship between them possible extensions are also suggested

satisficing solutions and duality in interval and fuzzy linear programming in this paper we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions the necessary tools for dealing with such problems in this way we show that the class of crisp classical lp problems can be embedded into the class of flp ones moreover for flp problems we define the concept of duality and prove the weak and strong duality theorems further we define a class of interval linear programming problems as a special subclass of flp problems and apply the previous results to this special case

computational aspects of fuzzy arithmetics based on archimedean t norms for fuzzy multiplication and division of l r numbers we derive approximation formulas by including the exact results using l r numbers this permits an error control and meets user s ideas on tractable solutions better than the level sets approach does as potential application the resolution of linear systems is considered

closure properties of cellular automata concerning the power of one dimensional cellular automata recognizers ibarra and jiang have proved that real time cellular automata ca and linear time ca are equivalent if and only if real time ca is closed under reverse in this paper we investigate the question of equality of real time ca and linear time ca with respect to the operations of concatenation and cycle in particular we prove that if real time ca is closed under concatenation then real time ca is as powerful as linear time ca on the unary languages we also prove that the question of knowing whether real time ca is as powerful than linear time ca is equivalent to the question of whether real time ca is closed under cycle moreover in the case of two dimensional ca recognizers we investigate how restricted communication reduces the computational power in particular we show that real time ca and linear time ca with restricted variants of moore and von neumann neighborhoods are not closed under rotation furthermore they are not equivalent to real time ca with moore or von neumann neighborhoods

a labeling algorithm for the fuzzy assignment problem abstract this paper concentrates on the assignment problem where costs are not deterministic numbers but imprecise ones here the elements of the cost matrix of the assignment problem are subnormal fuzzy intervals with increasing linear membership functions whereas the membership function of the total cost is a fuzzy interval with decreasing linear membership function by the max min criterion suggested by bellman and zadeh the fuzzy assignment problem can be treated as a mixed integer nonlinear programming problem we show that this problem can usually be simplified into either a linear fractional programming problem or a bottleneck assignment problem here we propose an efficient algorithm based on the labeling method for solving the linear fractional programming case the algorithm begins with primal feasibility and proceeds to obtain dual feasibility while maintaining complementary slackness until the primal optimal solution is found the computational results show that the proposed labeling algorithm offers an effective and efficient way for handling the fuzzy assignment problem

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables linear programming problems with trapezoidal fuzzy variables fvlp have recently attracted some interest some methods have been developed for solving these problems by introducing and solving certain auxiliary problems here we apply a linear ranking function to order trapezoidal fuzzy numbers then we establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results in particular we prove that the auxiliary problem is indeed the dual of the fvlp problem having established the dual problem the results will then follow as natural extensions of duality results for linear programming problems with crisp data finally using the results we develop a new dual algorithm for solving the fvlp problem directly making use of the primal simplex tableau this algorithm will be useful for sensitivity or post optimality analysis when using primal simplex tableaus

adaptive control and synchronization of a novel hyperchaotic system with uncertain parameters in this paper the problems of control and synchronization of a novel hyperchaotic system with uncertain parameters are studied based on the lyapunov stability theory and the adaptive control theory this uncertain hyperchaotic system is suppressed to its unstable equilibrium furthermore synchronization between two identical uncertain hyperchaotic systems is achieved by proposing an adaptive control law and a parameter estimation update law numerical simulations are presented to demonstrate the analytical results

a shortest path problem on a network with fuzzy arc lengths abstract we concentrate on a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length introducing an order relation between fuzzy numbers based on fuzzy min a nondominated path or pareto optimal path from the specified node to every other node is defined an algorithm for solving the problem is developed on the basis of the multiple labeling method for a multicriteria shortest path as a result a number of nondominated paths can be obtained and is offered to a decision maker however a number of nondominated paths derived from large scale network may be too numerous for him to choose a preferable path due to this situation we propose a method to reduce the number of paths according to a possibility level the proposed algorithm is numerically evaluated on large scale random networks

combinatorial algorithms for the minimum interval cost flow problem the aim of minimum the interval cost flow problem micfp is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy demands at certain nodes from available supplies at other nodes where there exists some vague in vector cost of problem interval cost is a common event in uncertainty environments where statistical data are applied moreover they almost play an essential role in fuzzy programming specially in the case of using their cuts in this paper a complete order on intervals is defined and efficient combinatorial algorithms for micfp are proposed digital simulation results show the performance of the proposed algorithms compared with real scenarios

a dual approach to solve the fuzzy linear programming problem abstract a concept of fuzzy objective based on the fuzzification principle is presented in accordance with this concept the fuzzy linear mathematical programming problem is easily solved a relationship of duality among fuzzy constraints and fuzzy objectives is given the dual problem of a fuzzy linear programming problem is also defined

duality in fuzzy linear programming with possibility and necessity relations a class of fuzzy linear programming flp problems with fuzzy coefficients based on fuzzy relations is introduced the concepts of feasible and a b maximal and minimal solutions are defined the class of crisp classical lp problems can be embedded into the class of flp ones moreover for flp problems a new concept of duality is introduced and the weak and strong duality theorems are derived the introduced concepts and results are illustrated and discussed on a simple numerical example

interactive fuzzy programming for multilevel linear programming problems this paper presents interactive fuzzy programming for multilevel linear programming problems in fuzzy programming for multilevel linear programming problems recently developed by lai et al since the fuzzy goals are determined for both an objective function and decision variables at the upper level undesirable solutions are produced when these fuzzy goals are inconsistent in order to overcome such problems after eliminating the fuzzy goals for decision variables interactive fuzzy programming for multilevel linear programming problems is presented in our interactive method after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers at the upper level with considerations of overall satisfactory balance among all levels illustrative numerical examples for two level and for three level linear programming problems are provided to demonstrate the feasibility of the proposed method

fuzzy linear regression with spreads unrestricted in sign abstract by the use of actual examples it is shown that when the two trends namely the fuzzy spread trend and the center line or modal trend are in conflict or inconsistent the results of the fuzzy linear regression flr model based on the approach of tanaka and coworkers frequently misinterprets the data to avoid this misinterpretation a modification of the flr procedure by allowing the spreads of the parameter to be unrestricted in sign is proposed it is shown that by the use of this unrestricted fuzzy linear regression uflr model correct prediction of the trends are obtained

recognition of poly slender context free languages by trellis automata abstract a poly slender context free language is a context free language whose number of words of length n is polynomially bounded its structure has been thoroughly characterized by ilie rozenberg and salomaa thanks to this characterization we show that every poly slender context free language is recognizable by a trellis automaton if the languages of words with as many a as b and at most a constant number of alternations between a and b are recognizable by trellis automata then we construct a family of trellis automata that recognizes such languages

extended fuzzy linear models and least squares estimates least squares regression of the fuzzy linear model is extended to overcome and interpret the occurrence of negative spreads 1 the idea is to introduce best fit difference models which exploit both the hukuhara difference and the l2 metric distance the fuzzy models use lr fuzzy numbers fitted models are compared by using the coefficient of determination in a similar way to its use in classical statistical least squares fitting the non lr fuzzy case is also considered

two dimensional cellular automata and their neighborhoods we investigate how the choice of the neighborhood can influence the computation ability of two dimensional cellular automata we present also a strict inclusion between low and high complexity classes of two dimensional cellular automata

fuzzy programming approach to multi level programming problems multi level programming is characterized as mathematical programming to solve decentralized planning problems we have considered a multi level linear programming problem and applied fuzzy mathematical programming fmp approach to obtain the solution of the system we have suggested fmp method for the minimization of the objectives using linear membership functions fmp is a supervised search procedure supervised by the higher level decision maker dm the higher level dm provides the preferred values of decision variables under his control to enable the lower level dm to search for his optimum in a wider feasible space and the bounds of his objective function to direct the lower level dm to search for his solutions in the right direction

duality theory in fuzzy mathematical programming problems with fuzzy coefficients in this paper the notions of subgradient subdifferential and differential with respect to convex fuzzy mappings are investigated which provides the basis for the fuzzy extremum problem theory we consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and or sufficient optimality conditions furthermore the concept of saddle points and minimax theorems under fuzzy environment is discussed the results obtained are used to formulate the lagrangian dual of fuzzy programming under certain fuzzy convexity assumptions kkt conditions for fuzzy programming are derived and the perturbed convex fuzzy programming is considered finally these results are applied to fuzzy linear programming and fuzzy quadratic programming

duality of fuzzy mc2 linear programming a constructive approach based on decision makers goal seeking and compromise behavior this paper proposes a constructive approach to duality of a fuzzy multiple criteria and multiple constraint level mc2 linear programming problem for a given mc2 problem both its fuzzy primal and dual programs are defined as the problems with fuzzy optimal objectives we first use the mc2 simplex method to locate sets of potential solutions of maximum and minimum mc2 problems for primal and dual programs respectively then we construct separate membership functions for primal and dual programs for each program a set of fuzzy potential solutions is determined in terms of the membership function and primal potential bases with certain ranges of decision parameters finally using the primal and dual membership functions we construct several fuzzy weak duality functions for any pair of primal and dual fuzzy potential solutions

a parametric representation of fuzzy numbers and their arithmetic operators abstract direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex implementation of the extension principle is equivalent to solving a nonlinear programming problem to overcome this difficulty many applications limit the membership functions to certain shapes usually either triangular fuzzy numbers tfn or trapezoidal fuzzy numbers trfn then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers thus making the calculations trivial unfortunately the tfn shape is not closed under multiplication and division the result of these operators is a polynomial membership function and the triangular shape only approximates the actual result the linear approximation can be quite poor and may lead to incorrect results when used in engineering applications we analyze this problem and propose six parameters which define parameterized fuzzy numbers pfn of which tfns are a special case we provide the methods for performing fuzzy arithmetic and show that the pfn representation is closed under the arithmetic operations the new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as tfns the new method has better accuracy and similar computational speed to using tfns and appears to have benefits when used in engineering applications

duality in fuzzy multi criteria and multi constraint level linear programming a parametric approach this paper presents a parametric approach for duality in fuzzy multi criteria and multi constraint level linear programming mc2lp which extends fuzzy linear programming approaches first the mc2 simplex method is used to solve the crisp prima dual mc2lp pair and then through these crisp formulations separate membership functions are constructed for fuzzy primal and dual program by considering the corresponding primal and dual decisions for each program a set of fuzzy potential solutions is determined in terms of the membership function and the related optimal solution with a certain range of decision parameters finally using the primal and dual membership functions a fuzzy weak duality function is obtained for any pair of primal and dual fuzzy potential solutions

evaluate fuzzy optimization problems based on biobjective programming problems abstract the embedding theorem shows that the set of fuzzy numbers can be embedded into a banach space inspired by this embedding theorem we propose a solution concept of the fuzzy optimization problem by solving a biobjective optimization problem which is transformed from the original fuzzy optimization problem using the embedding theorem we then also consider the fuzzy optimization problem with fuzzy coefficients i e the coefficients are assumed to be fuzzy numbers under a setting of core values of fuzzy numbers we show that the optimal solution of its corresponding crisp optimization problem the usual optimization problem is also an optimal solution of the original fuzzy optimization problem

chaos synchronization for a class of nonlinear oscillators with fractional order abstract in this paper a special kind of nonlinear chaotic oscillator the qi oscillator is studied in detail since such systems are shown to possess a relatively wide spectral bandwidth it is considerably beneficial to practical engineering in the secure communication field the chaos synchronization problem of the fractional order qi oscillators coupled in a master slave pattern is examined by applying three different kinds of methods the nonlinear feedback method the one way coupling method and the method based on the state observer suitable synchronization conditions are derived by using the lyapunov stability theory and most importantly a sufficient and necessary synchronization condition for the case with fractional order between 1 and 2 is presented results of numerical simulations validate the effectiveness and applicability of the proposed schemes

two dimensional cellular automata and deterministic on line tessalation automata in this work we consider the relationships between the classes of two dimensional languages defined by deterministic on line tessellation automata and by real time two dimensional cellular automata with moore and von neumann neighborhood we generalize the result known for one dimensional cellular automata to two dimensional cellular automata with von neumann neighborhood the class of real time cellular automata is closed under rotation of 180 if and only if real time cellular automata is equivalent to linear time cellular automata

hopf bifurcation analysis and ultimate bound estimation of a new 4 d quadratic autonomous hyper chaotic system based on lorenz system a new four dimensional quadratic autonomous hyper chaotic attractor is presented in this paper it can generate double wing chaotic and hyper chaotic attractors with only one equilibrium point several properties of the new system are investigated using lyapunov exponents spectrum bifurcation diagram and phase portraits using the center manifold and normal form theories the local dynamics the stability and hopf bifurcation at the equilibrium point are analyzed to obtain the ellipsoidal ultimate bound the ultimate bound of the proposed system is theoretically estimated using lagrange multiplier method lagrangian function and local maximizer point by properly choosing p and q matrices an estimation of the ultimate bound region as a function of the lagrange coefficient is obtained using local maximizer point and reduced hessian matrix to demonstrate the evolution of the bifurcation and to show the ultimate bound region numerical simulations are provided

a fuzzy approach to hierarchical multiobjective programming problems and its application to an industrial pollution control problem in this paper we focus on hierarchical multiobjective programming problems where multiple decision makers in a hierarchical organization have their own multiple objective functions and propose an interactive algorithm based on the dual decomposition method to obtain the satisfactory solution which reflects not only the hierarchical relationships between multiple decision makers but also their own preferences for their objective functions in the proposed algorithm assuming that each of the decision makers has fuzzy goals for his her objective functions corresponding membership functions are elicited from the decision makers in their subjective manner in order to deal with hierarchical multiobjective programming problems a new kind of pareto optimality concept in membership spaces is defined and the concept of decision powers of multiple decision makers in a hierarchical decision structure is introduced the proposed algorithm is applied to the industrial pollution control problem in osaka city in japan in order to show the efficiency of the proposed method

a three level optimization method for fuzzy large scale multiobjective nonlinear programming problems abstract in general mathematical models to reflect real world complicated decision situations are often formulated as largescale programming problems which include a large number of variables and or constraints furthermore considering the fuzziness of human s subjective judgement in real world decision making it is necessary to deal with the large scale programming problems in the fuzzy environment from such a point of view in this paper under the assumption that the decision maker dm has fuzzy goal for each of the objective functions and the corresponding membership function may be nonlinear a 3 level optimization method is proposed for large scale multiobjective nonlinear programming problems ls monlps with the block angular structure to obtain the compromise solution of the dm based on the proposed algorithm the computer program is developed in fortran language and applied to the numerical example to clarify the efficiency of the proposed algorithm

relating the power of cellular arrays to their closure properties there are many fundamental open problems concerning cellular arrays ca s for example 1 is the class of real time ca languages closed under reversal concatenation 2 2 are linear time ca s more powerful than real time ca s 3 3 are nonlinear time ca s more powerful than linear time ca s 4 4 does one way communication reduce the computing power of a ca although some of these problems appear to be easier to resolve than the others e g problem 1 seems easier than 2 no solution to any of these problems is forthcoming in this paper we investigate the relationships among these problems as well as prove some positive results concerning ca s we show a a the class of real time ca languages is closed under reversal if and only if linear time ca s are equivalent to real time ca s b if ca s are more powerful than ca s restricted to one way data communication i e one way ca s then nonlinear time ca s are more powerful than linear time ca s c c if the class of real time ca languages is closed under reversal then it is also closed under concatenation in the case of unary ca languages we show that the class is closed under concatenation we also show that the languagel 0n1m m n 0 mdividesn is a real time ca language disproving a conjecture of bucher and culik

a fuzzy dual decomposition method for large scale multiobjective nonlinear programming problems abstract in this paper we propose a fuzzy dual decomposition method for large scale multiobjective nonlinear programming problems ls monlps with the block angular structure by considering the vague nature of human judgements we assume that the decision maker dm may have a fuzzy goal for each of the objective functions in the ls monlp after eliciting the corresponding membership function for each of the objective functions through the interaction with the dm an extended primal problem and the corresponding extended dual problem are formulated then a two level optimization algorithm for the extended dual problem is proposed for deriving the compromise solution for the dm to the ls monlp based on the proposed algorithm fortran programs are developed and an illustrative numerical example is demonstrated

duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on different solution concepts under a general setting of partial ordering defined on the set of all fuzzy numbers the duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on two solution concepts for primal problem and three solution concepts for dual problem are derived in this paper those solution concepts are inspired by the nondominated solution concept employed in multiobjective programming problems since the ordering among fuzzy numbers introduced in this paper is a partial ordering not a total ordering we also provide a concept of fuzzy scalar inner product for fuzzy numbers then the fuzzy valued lagrangian function and the fuzzy valued lagrangian dual function are proposed via the concept of fuzzy scalar product under these settings the dual problem is formulated and the objective function of this dual problem is a point to set fuzzy valued function in this case three solution concepts for dual problem are proposed the duality theorems and saddle point optimality conditions can be naturally elicited based on these three solution concepts

chaotic and hyperchaotic attractors of a complex nonlinear system in this paper we introduce a complex nonlinear hyperchaotic system which is a five dimensional system of nonlinear autonomous differential equations this system exhibits both chaotic and hyperchaotic behavior and its dynamics is very rich based on the lyapunov exponents the parameter values at which this system has chaotic hyperchaotic attractors periodic and quasi periodic solutions and solutions that approach fixed points are calculated the stability analysis of these fixed points is carried out the fractional lyapunov dimension of both chaotic and hyperchaotic attractors is calculated some figures are presented to show our results hyperchaos synchronization is studied analytically as well as numerically and excellent agreement is found

duality theorems in fuzzy mathematical programming problems based on the concept of necessity the fuzzy valued lagrangian function for the fuzzy mathematical programming problem via the concept of the fuzzy scalar inner product is proposed a solution concept of fuzzy optimization problems which is essentially similar to the notion of pareto solution in multiobjective programming problems is also introduced by ranking the fuzzy numbers using the necessity indices under these settings the weak and strong duality theorems can be elicited we show that the primal and dual fuzzy mathematical programming problems have no duality gap under suitable convexity assumptions for fuzzy valued functions

fuzzy multi level minimum cost flow problems both multiple objectives and multiple hierarchies minimum cost flow problems with fuzzy costs and fuzzy capacities in the arcs are investigated to reduce the complexity a possibility programming is used to handle the vagueness in the parameters fuzzy approach can considerably simplify the problem and thus a fairly general multi level problem can be solved reasonably easily in spite of the fact that the multi level problem is np hard and very difficult to solve several numerical examples are considered to illustrate the approach

interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters in this paper we present interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters using the level sets of fuzzy parameters the corresponding nonfuzzy two level linear fractional programming problem is introduced in our interactive method after determining fuzzy goals of decision makers at both levels a satisfactory solution is derived efficiently by updating a minimal satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels the satisfactory solution well balanced between both levels is easily computed by combined use of the bisection method the phase one of the simplex method and the variable transformation method by charnes and cooper an illustrative numerical example for two level linear fractional programming problems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method

the solution and duality of imprecise network problems duality properties have been investigated by many researchers in the recent literature they are introduced in this paper for a fully fuzzified version of the minimal cost flow problem which is a basic model in network flow theory this model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes first we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper by applying hukuhara s difference approximated and exact multiplication and wu s scalar production we exhibit the flow in network models then we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified mcfp to acquire fuzzy optimal flows to give duality theorems we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms finally we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision making problems

optimality and duality in nonsmooth multiobjective optimization involving generalized type i functions abstract a nonsmooth multiobjective optimization problem involving generalized type i vectorvalued functions is considered karush kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be an efficient or properly efficient solution duality theorems are proved for wolfe type and mond weir type duals under the generalized type i assumptions

possible and necessary optimality tests in possibilistic linear programming problems in this paper two kinds of optimality concepts are introduced into fuzzy linear programming problems in order to discuss the rationality of an obtained solution one is possible optimality evaluating the possibility degree to which the solution is optimal the other the dual concept is necessary optimality evaluating the necessity degree to which the solution is optimal when possibility distributions are defined by closed intervals degrees of possible and necessary optimalities become binary necessary and sufficient conditions for possible and necessary optimalities are induced in this case extending the results methods of calculating the degrees of possible and necessary optimalities are proposed some numerical examples are given to demonstrate the proposed methods

stability and hopf bifurcation of a lorenz like system hopf bifurcation is one of the important dynamical behaviors it could often cause some phenomena such as quasiperiodicity and intermittency consequently chaos will happen due to such dynamical behaviors since chaos appears in the lorenz like system to understand the dynamics of such system hopf bifurcation will be explored in this paper first the stability of equilibrium points is presented then hopf bifurcation of the lorenz like system is investigated by applying the normal form theory the conditions guaranteeing the hopf bifurcation are derived further the direction of hopf bifurcation and the stability of bifurcating periodic solutions are also presented finally numerical simulations are given to verify the theoretical analysis it is found that hopf bifurcation could happen when conditions are satisfied the stable bifurcating periodic orbit is displayed chaos will also happen when parameter further increases

adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters this paper investigates adaptive synchronization between two novel different hyperchaotic systems with partly uncertain parameters based on the lyapunov stability theorem and the adaptive control theory synchronization between these two hyperchaotic systems is achieved by proposing a new adaptive controller and a parameter estimation update law numerical simulations are presented to demonstrate the analytical results

analysis of a new chaotic system abstract this paper reports a new three dimensional continuous quadratic autonomous chaotic system modified from the lorenz system in which each equation contains a single quadratic cross product term which is different from the lorenz rossler chen lu systems basic properties of the new system are analyzed by means of lyapunov exponent spectrum and bifurcation diagrams analysis results show that this system has complex dynamics with some interesting characteristics