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A two-dimensional map describing chaotic behaviour of an economic model has been stabilized on various periodic orbits by the use of Pyragas time-delayed feedback control. The method avoids fancy data processing used in the Ott–Grebogi–Yorke approach and is based solely on the plain measurement and time lag of a scalar signal which in our case is a value of sales of a firm following an active investment strategy (Behrens–Feichtinger model). We show that the application of this control method is very straightforward and one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the system's economic properties.

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... In addition, this phenomenon could threaten investment safety [7] . Over the past few decades, several high-dimensional economical models, such as the Behrens-Feichtinger model [8] , the Cournot-Puu system [9] , and other four-dimensional hyperchaotic financial systems [10,11] , have been introduced. ...

... In recent years, several studies have also introduced control methods to maintain specific characteristics and behaviours in hyperchaotic finance systems. Hołyst and Urbanowicz have proposed a time-delayed feedback technique for the control of chaotic behaviour of an economic model [8] . Yao et al. have developed a straight-line stabilization technique to achieve chaos control in economic models [22] . ...

... However, following a boundedly rational adjustment process, a dynamic of a duopoly game was obtained in [11][12][13][14]. Furthermore, the nonlinear dynamics of a duopoly game with delayed bounded rationality [15,16], heterogeneous players [17][18][19], and other aspects on nonlinear duopoly games [20][21][22] were studied. ...

... The third novelty is the stability of the dynamic system with respect to the entanglement level. The stability conditions of our dynamic system are different from those of [11][12][13][14][15][16][17][18][19][20][21][22], since the entanglement was added. Note that most of economic and management models were analyzed by the classical information theory. ...

We investigate the dynamics of a quantum duopoly game where the players use bounded rationality to adjust own decision. The stability conditions of the equilibrium points are analyzed. Furthermore, we present the numerical simulations to show the nonlinear behaviors: bifurcations, strange attractors, stability region.

... Precisely, because the chaos in market are not expected and are even harmful to the participants, certain methods should be adopted to suppress or eliminate the occurrence of bifurcations and chaos. Various methods for controlling chaos have been used in dynamical systems; the OGY method was presented by Ott et al. [40] and had been applied in the dynamic game model to control chaos [41,42], a modified straight-line stabilization method [12], adaptive control [13], time-delayed feedback method [43], and other feedback control methods [6][7][8] had also been studied for the chaos control in an economic model with homogeneous or heterogeneous expectations. It can be known from previous works that feedback and parameter variation are two Discrete Dynamics in Nature and Society effective methods [9,12,13,16,27,28,[40][41][42][43][44], to achieve chaos control. ...

... Various methods for controlling chaos have been used in dynamical systems; the OGY method was presented by Ott et al. [40] and had been applied in the dynamic game model to control chaos [41,42], a modified straight-line stabilization method [12], adaptive control [13], time-delayed feedback method [43], and other feedback control methods [6][7][8] had also been studied for the chaos control in an economic model with homogeneous or heterogeneous expectations. It can be known from previous works that feedback and parameter variation are two Discrete Dynamics in Nature and Society effective methods [9,12,13,16,27,28,[40][41][42][43][44], to achieve chaos control. Recently, a new control method called as control strategy of the state variables feedback and parameter variation was proposed [45] and had been used in the work of [8,13,26]. ...

Many works studied on complex dynamics of Cournot or Stackelberg games, but few references discussed a dynamic game model combined with the Cournot game phase and Stackelberg game phase. Under the assumption that R&D spillovers only flow from the R&D leader to the R&D follower, a duopoly Stackelberg-Cournot game with heterogeneous expectations is considered in this paper. Two firms with different R&D capabilities determine their R&D investments sequentially in the Stackelberg R&D phase and make output decisions simultaneously in the Cournot production phase. R&D spillovers, R&D investments, and technological innovation efficiency are introduced in our model. We find that: (i) the boundary equilibrium of the dynamic Stackelberg-Cournot duopoly system, where two players adopt boundedly rational expectation and naïve expectation, respectively, is unstable if the Nash equilibrium of the system is strictly positive. (ii) The Nash equilibrium of the dynamic Stackelberg-Cournot duopoly system, where two players adopt boundedly rational expectation and naïve expectation, respectively, is locally asymptotically stable only if the model parameters meet certain conditions. Especially, results indicate that small value of R&D spillovers or big value of output adjustment speed may yield bifurcations or even chaos. Numerical simulations are performed to exhibit maximum Lyapunov exponents, bifurcation diagrams, strange attractors, and sensitive dependence on initial conditions to verify our findings. It is also shown that the chaotic behaviors can be controlled with the state variables feedback and parameter variation method.

... Lyapunov exponents widely apply to the analysis of dynamic system qualitative behavior. They allow estimating trajectory behavior of various objects in physics [1], medicine [2][3][4], economy [5], astronomy [6]. LE determine on the basis of time series analysis most often. ...

... Write the solution of the system (1) as 0 ( ) ( , , ) X t t U t X (5) where X is the operator who is determined by matrixes , ...

Approach to the analysis of nonlinear dynamic systems structural identifiability (SI) under uncertainty is proposed. This approach has a difference from methods applied to SI estimation of dynamic systems in the parametrical space. Structural identifiability is interpreted as of the structural identification possibility a system nonlinear part. We show that the input should synchronize the system for the SI problem solution. The structural identifiability estimation method is based on the analysis of the framework special class. The input parameter effect on
the possibility of the SI estimation of the system is studied.

... Lyapunov exponents widely apply to the analysis of dynamic system qualitative behavior. They allow estimating trajectory behavior of various objects in physics [1], medicine [2][3][4], economy [5], astronomy [6]. LE determine on the basis of time series analysis most often. ...

... Write the solution of the system (1) as 0 ( ) ( , , ) X t t U t X (5) where X is the operator who is determined by matrixes , ...

Lyapunov exponents (LE) identification problem of dynamic systems with periodic coefficients is considered under uncertainty. LE identification is based on the analysis of framework special class describing dynamics of their change. Upper bound for the smallest LE and mobility limit for the large LE are obtained and the indicator set of the system is determined. The graphics criteria based on the analysis of framework special class features are proposed for an adequacy estimation of obtained LE estimations. The histogram method is applied to check for obtained estimation set. We show that the dynamic system can have the LE set.

... At the same time, theoretical tools were developed for effective chaos control, which, by small fine-tuning the parameters of system, made it possible to stabilize selected orbits embedded in a chaotic attractor and nudge the dynamics toward a desired trajectory. Examples applications of these tools can be found in [32][33][34][35][36][37][38][39][40]. The reviewed literature shows the relevance of chaos for economic models and contributed to development of advanced mathematical tools for study of complex nonlinear dynamical systems in economics, which continues up to now. ...

... It follows from condition (15) that the coefficient at Q 2 in the first inequality of (33) is positive and the coefficient at Q 2 in the second inequality of (33) is negative. Hence, we can reduce (33) to the following inequalities L(b, c, r, P ) ≤ Q 2 ≤ R(b, c, r, P ), (34) where L(b, c, r, P ) = 3P +2b+2 8P (b−2) > 0, ...

Cyclicity and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in the long run. We focus on an approach, associated with the identification of a deterministic endogenous mechanism of irregular fluctuations in the economy. Using of a mid-size firm model as an example, we demonstrate the use of effective analytical and numerical procedures for calculating the quantitative characteristics of its irregular limiting dynamics based on Lyapunov exponents, such as dimension and entropy. We use an analytical approach for localization of a global attractor and study limiting dynamics of the model. We estimate the Lyapunov exponents and get the exact formula for the Lyapunov dimension of the global attractor of this model analytically. With the help of delayed feedback control (DFC), the possibility of transition from irregular limiting dynamics to regular periodic dynamics is shown to solve the problem of reliable forecasting. At the same time, we demonstrate the complexity and ambiguity of applying numerical procedures to calculate the Lyapunov dimension along different trajectories of the global attractor, including unstable periodic orbits (UPOs).

... But, so far, we are unable to derive a general result relating to stability and the number n of competitors. Many researchers have also paid a great attention to the controlling dynamics of Cournot games, such as Agiza and Elsadany [16], Chen and Chen [17], Holyst and Urbanowicz [18], and Elabbasy et al. [19]. This paper presents a new Cournot duopoly game. ...

In Cournot model, when there are many competitions, the competitive equilibrium becomes chaotic. It is extremely difficult to derive the general equilibrium points. There is no previous research to explore a further problem with the general equilibrium points of n-contenders in Cournot model. In this paper, a general equilibrium Cournot game is proposed based on an inverse demand function. A market spatial structure model is built. Intermediate value theorem, as a realistic method, is introduced to handle a general competitive equilibrium in Cournot model. The number and stability in general equilibrium points are detected by means of celestial motion theory and spatial agglomeration competition model. The existence of general equilibrium points and the stability of Cournot equilibrium points, which are new and future complement of previously known results. Numerical simulations are given to support the research results.

... Several studies have been conducted for controlling chaos in economic systems [20][21][22]. The development of more accurate simplified linear representations of complex nonlinear supply chain models was studied in [23]. ...

A supply chain network (SCN) is a complex nonlinear system involving multiple entities. The policy of each entity in decision-making and the uncertainties of demand and supply (or production) significantly affect the complexity of its behavior. Although several studies have presented information about the measurement of chaos in the supply chain, there has not been an appropriate way to control the chaos in it. In this paper, the chaos control problem is considered for a SCN with a time-varying delay between its entities. The innovation of this paper is the more comprehensive modeling, analysis, and control of chaotic behavior in the system. The proposed model has a control center to determine the orders of entities and control their inventories. Customer demand is modeled as an unknown exogenous disturbance. A robust H∞ control method is designed to control its chaotic behavior in terms of a certain linear matrix inequality technique that can be readily solved using the MATLAB LMI toolbox. By using this technique and calculating the maximum Lyapunov exponent, decision parameters are determined in such a way that the behavior of the SCN is stable.

... d Phase portrait of eight chaotic attractor at ða; b; k; ; c 1 ; c 2 Þ ¼ ð0:870; 0:5; 2:4; 0:3; 0:1Þ and initial values ðq 1;0 ; q 2;0 Þ ¼ ð0:11; 0:12Þ adaptive control scheme has been presented and investigated using an economic chaotic system. Other approaches such as a time-delayed feedback control have been discussed in Holyst and Urbanowicz (2000). For more details and investigations of control methods, readers are advised to see literature (Ding et al. 2012;Elsadany et al. 2013;Ding et al. 2010Ding et al. , 2015Luo et al. 2003;Pu and Ma 2013). ...

The current paper introduces a duopolistic Stackelberg game under the assumption of differentiated products. The game is built based on a cost function that is nonlinear and depends on the quantities produced by firms and the announced plan products.
Indeed, there is a difference between the actual product and the plan product. Here, we try to confirm that the equilibrium production of a firm is no more than its announced plan product. The dynamic of the proposed game is described by a nonlinear discrete dynamical system with bounded rationality mechanism by which the time evolution of the competing firms is analyzed. The equilibrium point of the system is obtained and its local stability is discussed using the eigenvalues properties or Jury conditions. The discussion shows that the stability of equilibrium point is affected by the speed of adjustment parameter and some other cost parameters. Confirmation of this discussion is supported by some numerical simulations that lead to the existence of complex dynamic phenomena such as bifurcation and chaos. In addition, we are applied a feedback control scheme to suppress and overcome chaos existed and at the same time to force the system to go back to its stabilization behavior.

... Scholars have proposed many methods for chaos control, such as time-delayed feedback method [46], modified straight-line stabilization method [47]. In this section, the state feedback control method is used to control the system's chaos. ...

This paper studies a low-carbon dual-channel supply chain in which a manufacturer sells products through the direct channel and traditional channel, and a retailer sells products through the traditional channel. The manufacturer considers carbon emission reduction and has fairness concern behavior. The retailer provides sales service in the traditional channel and considers fairness concern behavior. The objective of this paper is to analyze the effects of different parameter values on the price stability and utility of the supply chain system emphatically using 2D bifurcation diagram, parameter plot basin, the basins of attraction, chaos attractor and sensitivity to the initial value, etc. The results find that the retailer’s fairness concern behavior shrinks the stability of the supply chain system more than that of the manufacturer’s fairness concern behavior. The system stability region decreases with the increase of carbon emission reduction level and the retailer’s fairness concern. The customers’ preference for the direct channel decreases the stable range of the direct channel, while it enlarges the stable range of the traditional channel. The supply chain system enters into chaos through flip bifurcation with the increase of price adjustment speed. In a stable state, the manufacture improving customer’s preference for the direct channel and the retailer choosing the appropriate fairness concern level can achieve the maximum utility separately. In a chaotic state, the average utilities of the manufacturer and retailer all decline, while that of the retailer declines even more. By selecting appropriate control parameter, the low-carbon dual-channel supply chain system can return to a stable state from chaos again. The research of this paper is of great significance to price decisions of participants and supply chain operation management.

... Recently, time-delayed feedback method has been also considered for chaos control in economic model (Holyst and Urbanowicz, 2000) and inducing or suppressing chaos in a double-well Duffing oscillator (Sun et al., 2006). Synchronisation of nonlinear systems is an important research area where the trajectories of a set of two systems called master and slave systems are synchronised with the help of some feedback controls (Arabyani and Nik, 2016). ...

In this research work, a novel 3D jerk chaotic system with one-quadratic nonlinearity and two-cubic nonlinearities is designed to generate complex chaotic signals. We show that the novel jerk chaotic system has a unique equilibrium at the origin, which is a saddle-focus and unstable. The Lyapunov exponents of the novel jerk chaotic system are obtained as L1 = 0.30899, L2 = 0 and L3 = -4.11304. The Kaplan-Yorke dimension of the novel jerk chaotic system is obtained as DKY = 2.0751. The qualitative properties of the novel jerk chaotic system are described in detail and MATLAB plots are shown. Next, we use backstepping control method to establish global chaos synchronisation of the identical novel jerk chaotic systems with unknown parameters. Next, an electronic circuit realisation of the novel jerk chaotic system is presented using MultiSIM to confirm the feasibility of the theoretical model. Finally, we present an application of the novel jerk chaotic system for voice encryption. The comparison between the MATLAB 2010 and MultiSIM 10.0 simulation results demonstrate the effectiveness of the proposed voice encryption scheme.

... Chaos control with time-delayed feedback method in an economical model has been applied by Holyst and Urbanowicz. 38 Adaptive control of a duopoly advertising game with heterogeneous firms has been investigated by Ding et al. 39 Elabbasy et al. 40 have considered a feedback control in their triopoly with heterogeneous players. In the work by Ding et al., 41 the authors have achieved chaos control in their multiteam Bertrand model. ...

This article investigates the dynamics of a Cournot triopoly game whose demand function is characterized by log-concavity. The game is formed using the bounded rationality approach. The existence and local stability of steady states of the game are analyzed. We find that an increase in the game parameters out of the stability region destabilizes the Cournot–Nash steady state. We confirm our obtained results using some numerical simulation. The simulation shows the consistence with the theoretical analysis and displays new and interesting dynamic behaviors, including bifurcation diagrams, phase portraits, maximal Lyapunov exponent, and sensitive dependence on initial conditions. Finally, a feedback control scheme is adopted to overcome the uncontrollable behavior of the game’s system occurred due to chaos.

... This means that, existing chaos in a financial system can lead the market to bog down or to make it out of control [32,33]. In recent decades, various economical models were discovered and analyzed in different high-dimensional systems, e.g., the Behrens-Feichtinger model [34], the Cournot-Puu system [35], the 4D hyperchaotic financial system [36] and some certain kinds of chaotic economic model. Memory and history of the system play a crucial role to describe the financial processes [37]. ...

This manuscript mainly focuses on the construction, dynamic analysis and control of a new fractional-order financial system. The basic dynamical behaviors of the proposed system are studied such as the equilibrium points and their stability, Lyapunov exponents, bifurcation diagrams, phase portraits of state variables and the intervals of system parameters. It is shown that the system exhibits hyperchaotic behavior for a number of system parameters and fractional-order values. To stabilize the proposed hyperchaotic fractional system with uncertain dynamics and disturbances, an efficient adaptive sliding mode controller technique is developed. Using the proposed technique, two hyperchaotic fractional-order financial systems are also synchronized. Numerical simulations are presented to verify the successful performance of the designed controllers.

... Recently, time-delayed feedback method has been also considered for chaos control in economic model (Holyst and Urbanowicz, 2000) and inducing or suppressing chaos in a double-well Duffing oscillator (Sun et al., 2006). Synchronisation of nonlinear systems is an important research area where the trajectories of a set of two systems called master and slave systems are synchronised with the help of some feedback controls (Arabyani and Nik, 2016). ...

... Many methods for the chaos control have been proposed, such as time-delayed feedback method [18], modified straight-line stabilization method [19], OGY method [20], and pole placement method [21]. In this section, feedback control method proposed by Elabbasy et al. [1] is used, so the controlled system is given by ...

The impact of inaccurate demand beliefs on dynamics of a Triopoly game is studied. We suppose that all the players make their own estimations on possible demand with errors. A dynamic Triopoly game with such demand belief is set up. Based on this model, existence and local stable region of the equilibriums are investigated by 3D stable regions of Nash equilibrium point. The complex dynamics, such as bifurcation scenarios and route to chaos, are displayed in 2D bifurcation diagrams, in which e1 and α are negatively related to each other. Basins of attraction are investigated and we found that the attraction domain becomes smaller with the increase in price modification speed, which indicates that all the players’ output must be kept within a certain range so as to keep the system stable. Feedback control method is used to keep the system at an equilibrium state.

... Kemudian, sistem keuangan tiga dimensi yang rumit serta risiko dari analisis sistem dinamis yang dibangun untuk mengatur pasar keuangan telah terbukti dapat dikendalikan secara efektif [2] [3]. Demikian pula pada [4]dengan menggunakan metode delayed feedback control (DFC) dan membuktikan bahwa sistem keuangan yang rumit dapat distabilkan pada berbagai orbit periodik. ...

... There are many methods to control the supply chain from the chaos state to the stable state such as the modified straight-line stabilization method [38], the time-delayed feedback method [39], and the OGY method [40]. In this section, the state feedback control method was used to delay or eliminate the chaos in the supply chain system. ...

In this paper, we developed a dynamic price game model for a low-carbon, closed-loop supply chain system in which (1) the manufacturer had fairness concern and carbon emission reduction (CER) behaviors, and market share and profit maximization were their objectives, and (2) the retailer showed fairness concern behaviors in market competition and provided service input to reduce return rates. The retailer recycled old products from customers, and the manufacturer remanufactured the recycled old products. The effects of different parameter values on the stability and utility of the dynamic price game model were determined through analysis and numerical simulation. Results found that an increasing customer loyalty to the direct marketing channel decreased the stable region of the manufacturer’s price adjustment and increase that of the retailer. The stable region of the system shrank with an increase of CER and the retailer’s service level, which expanded with return rates. The dynamic system entered into chaos through flip bifurcation with the increase of price adjustment speed. In the chaotic state, the average utilities of the manufacturer and retailer all declined, while that of the retailer declined even more. Changes to parameter values had a great impact on the utilities of the manufacturer and retailer. By selecting appropriate control parameters, the dynamic system can return to a stable state from chaos again. The research of this paper is of great significance to participants’ price decision-making and supply chain operation management.

... 11 There are many contributions that have applied control of chaos in non-linear dynamic economic models. Holyst (1996);Holyst, Hagel, Haag and Weidlich (1996);Holyst, Hagel, and Haag (1997); Holyst and Urbanowicz (2000); Ahmed and Hassan (2000); Salarieh and Alasty (2009), and Chen and Chen (2007) applies the control of chaos to microeconomic models of business sector competition. Haag, Hagel and Sigg (1997) use techniques of control to stabilize urban economic system. ...

Complexity is one of the most important characteristic properties of the economic behaviour. The new field of knowledge called Chaotic Dynamic Economics borns precisely with the objective of understanding, structuring and explaining in an endogenous way such complexity. In this paper, and after scanning the principal concepts and techniques of the chaos theory, we analyze, principally, the different areas of Economic Science from the point of view of complexity and chaos, the main and most recent researches, and the present situation about the results and possibilities of achieving an useful application of those techniques and concepts in our field.

... Введение Д анная статья является продолжением цикла работ [3,4,6,9,[18][19][20][21] ( Zvyagintsev, 2015;Zvyagintsev, 2019;Loskutov, 2010;Petryakov, 2017;Holyst, Urbanowicz, 2000;Holyst, Urbanowicz, Zebrowska, 2001, посвященных контролю хао-тичных процессов в экономике и способам подавления рыночного хаоса. В качестве объекта исследования рассмотрим динамику валют-ного курса для пары доллар/рубль. ...

В статье рассматривается задача по разработке превентивных мер, направленных на предотвращение хаотичных скачков валютного курса. Предложена экономико-математическая модель, которая позволяет проводить анализ возможностей по нейтрализации высокой волатильности валютного курса и способствует выводу курсовой динамики на устойчивую периодическую траекторию. Важнейшая особенность данной модели заключается в получении с ее помощью практических рекомендаций по предотвращению краткосрочных кризисных явлений путем своевременных превентивных мер. Разработанные в статье методы довольно удобны с практической точки зрения и легко адаптируются к рыночным реалиям. Практическое применение сконструированной модели показано на примере статистических данных для среднемесячного курса американского доллара к российскому рублю. Полученная модель позволяет дополнить комплекс антикризисных мер для валютного рынка.

... We know that chaos is not desirable in a real economic system and it is often hoped to be controlled so that the dynamic system could run in a stable status. In this section, we show that the usually used method of time-delayed feedback control (e.g., [32,[36][37][38]) can also be used to control the chaos in our model. ...

Two different time delay structures for the dynamical Cournot game with two heterogeneous players are considered in this paper, in which a player is assumed to make decision via his marginal profit with time delay and another is assumed to adjust strategy according to the delayed price. The dynamics of both players output adjustments are analyzed and simulated. The time delay for the marginal profit has more influence on the dynamical behaviors of the system while the market price delay has less effect, and an intermediate level of the delay weight for the marginal profit can expand the stability region and thus promote the system stability. It is also shown that the system may lose stability due to either a period-doubling bifurcation or a Neimark-Sacker bifurcation. Numerical simulations show that the chaotic behaviors can be stabilized by the time-delayed feedback control, and the two different delays play different roles on the system controllability: the delay of the marginal profit has more influence on the system control than the delay of the market price.

... There are many methods available for chaos control such as open-loop control based on periodic system excitation, closed-loop control such as OGY method (Ott et al., 1990) and also time-delayed feedback control such as Pyragas (1992) method, etc. Recently, time-delayed feedback method has been also considered for chaos control in economic model (Holyst and Urbanowicz, 2000) and inducing or suppressing chaos in a double-well Duffing oscillator (Sun et al., 2006). ...

... Du et al. [11] suppressed the chaos of an economic model by virtue of phase space compression. For more related works on this theme, one can see [12][13][14][15]. ...

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.

... Since Ott, Grebogi, and Yorke [6], in 1990, pointed out the existence of many unstable periodic orbits (UPOs) embedded in chaotic attractors that raise the possibility of using very small external forces to obtain various types of regular behavior, Pyragas [7] in 1992, proposed a so-called "Delayed Feedback Control (DFC)" idea that an appropriate continuous controlling signal formed from the difference between the current state and the delayed state is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit, several techniques were devised for controlling chaos [8][9][10][11][12][13][14][15][16][17][18][19][20][21] during the past years and applied to various systems [22][23][24][25][26][27][28][29][30][31]. It is worth noting that in spite of the enormous number of applications among the chaos control, very few rigorous results are so far available. ...

Open-loop dynamic characteristics of an underactuated system with nonholonomic constraints, such as a horizontal bar gymnastic robot, show the chaotic nature due to its nonlinearity. This chapter deals with the stabilization problems of periodic motions for the giant swing motion of gymnastic robot using chaos control methods. In order to make an extension of the chaos control method and apply it to a new practical use, some stabilization control strategies were proposed, which were, based on the idea of delayed feedback control (DFC), devised to stabilize the periodic motions embedded in the movements of the gymnastic robot. Moreover, its validity has been investigated by numerical simulations. First, a method named as prediction-based DFC was proposed for a two-link gymnastic robot using a Poincar section. Meanwhile, a way to calculate analytically the error transfer matrix and the input matrix that are necessary for discretization was investigated. Second, an improved DFC method, multiprediction delayed feedback control, using a periodic gain, was extended to a four-link gymnastic robot. A set of plural Poincare maps were defined with regard to the original continuous-time system as a T-periodic discrete-time system. Finally, some simulation results showed the effectiveness of the proposed methods.

... In order to avoid this risk, it is very necessary to choose appropriate adjustment parameters to keep the system in a stable state. Scholars have proposed many methods for chaos control, such as modified straight-line stabilization method [61], time-delay feedback method [62], OGY method [63], and parameter adaptation method [64]. In this section, the delayed feedback control (DFC) method [65] is used to control the system's chaos. is control method has been widely used in many documents, such as [55,66]. ...

This paper constructs a supply chain model composed of a manufacturer and a recycler. The manufacturer’s CSR and the recycler’s fairness concerns are introduced to the benchmark model in turn, and the optimal decision-making problems under different models are studied and compared. The findings show that the manufacturer’s utility will increase and the recycler’s utility will decrease when the manufacturer undertakes CSR within a reasonable range. The optimal utility of manufacturers does not change, and the utility of the recycler is affected by the proportion of CSR undertaken by the manufacturer when the recycler considers fairness concerns. Based on the CSR and fairness model, this paper constructs a dynamic decision system of production quantity and eco-innovation effort. We analyze the influence of adjustment speed on the dynamic decision system and obtain the conditions required to maintain system stability. The research conclusion indicates that with the increase of adjusting parameters, the system gradually appears chaotic state from a stable state and the chaotic state of the system has a negative impact on the utility of manufacturer and recycler. In order to avoid chaos in the system, this paper uses the delayed feedback method to control the system.

... However, the technical criterions and pricing levels of modern agricultural business agents represented by professional cooperatives are strictly controlled by the industrial associations and government departments, and the price adjustment process has typical characteristics of delayed feedback control [22][23][24][25][26]. erefore, the price competition model with quality difference, bounded rationality, and delayed feedback control is more conducive to the complexity analysis for price competition of agricultural products with regional brands [27][28][29][30][31]. Based on the theories of signal transmission and public goods, this paper explores the behavior characteristics of heterogeneous agents and demonstrates the origin of chaos in price competition of agricultural products with regional brands. en, this paper establishes the Salop circular market model with bounded rationality and delayed feedback, simulates the complexity in price competition of agricultural products with regional brands, and explains the 2 Discrete Dynamics in Nature and Society pricing control mechanism according to the variation of parameters [32]. ...

The disordered price competition among various agricultural business agents leads to chaos of agricultural products’ prices, which makes it difficult for customers to form stable price expectation and correct brand cognition, restricting the sales of agricultural products with regional brands. Based on the Salop circular market model with bounded rationality and delayed feedback, this paper discusses the complexity in price competition of agricultural products with regional brands. It is found that when the price adjustment speed of agricultural business agents exceeds the stability region, the pricing system of agricultural products with regional brands would appear the phenomenon of periodic bifurcation or chaos. The delayed feedback controlling mechanism of price adjustment could make the pricing system in the chaos state turn to the equilibrium state. Therefore, the price fluctuation of agricultural products with regional brands needs reasonable control from the industrial associations and government departments.

... In the following ten years, the research on chaos control and synchronization has been booming, which has rapidly become an important hotspot in the field of chaos research. For example, linear state feedback control [16], sliding mode control [17], and adaptive Lyapunov control [18] are widely used. Recently, the chaos control in biological systems has been studied. ...

The dynamics behavior of a discrete-time three-species food chain model is investigated. By using bifurcation theory, it is shown that the equilibrium point of the system loses its stability, and the system undergoes Neimark–Sacker bifurcation, which leads to chaos as the parameter changes. The chaotic motion is controlled on the stable periodic period-1 orbit using the implementation of the hybrid control strategy. The factor affecting the control time of chaos is also studied. Numerical simulations are consistent with the theoretical analysis. The results of this research prove that the chaos control method can be extended to the higher-dimensional biological model and can be realized.

... It is well known that chaos in financial systems may lead to the destabilization of the economy and also to a financial crisis [11]. Thus, the market can be out of control [17,29]. So, it is necessary to study the global stabilization of financial chaotic systems. ...

Synchronization of chaotic dynamical systems with fractional order is receiving great attention in recent literature because of its applications in a variety of fields including optics, secure communications of analog and digital signals, and cryptographic systems. In this paper, chaos control of a new financial system, and chaos synchronization between two identical financial systems, and non-identical financial systems with integer and fractional order are investigated. Chaos control is based on a linear feedback controller for stabilizing chaos to unstable equilibrium. In addition, chaos synchronization, not only between two identical new chaotic financial systems, but also between the new financial system and an another financial system given in the literature is realized by using active control technique. The synchronization is done for integer and fractional order in each case. It is shown that chaotic behavior can be controlled easily to any unstable equilibrium point of the new financial system. Also, it is observed that synchronization is enhanced when the fractional order increases and approximates to one. Numerical simulations are used to verify the proposed methods.

... Later, another threedimensional financial chaotic risk dynamic system was constructed for management process of financial markets and proved to be controlled effectively [6]. Then Holyst and Urbanowicz [7] used the method of delayed feedback control (DFC) and proved that a chaotic financial system can be stabilized on various periodic orbits [8]. Reference [9] discussed the complex behaviors of a financial system with time-delayed feedback by numerical simulation. ...

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

... (3)(4)(5)(6)(7) There are many applications of chaotic synchronization, such as secure communication, (8,9) spread spectrum communication, (10) and information compression and storage. (11) As a result, chaos control and synchronization methods, such as the drive-response method, (12) feedback method, (13) pulse control method, (14) and so forth, (15,16) have been receiving intensive attention and been extensively explored. ...

... Some chaos control methods have been applied to the supply chain, such as modified straight-line stabilization method [39], time-delayed feedback method [40], OGY method [41], and the parameter adaptation method [42] In this section, the parameter adaptation method is adopted to control the market prices of participants. Based on dynamic system (15), the controlled system can be expressed as follows: ...

Considering firm’s innovation input of green products and channel service, this paper, in dynamic environment, studies a dynamic price game model in a dual-channel green supply chain and focuses on the effect of parameter changing on the pricing strategies and complexity of the dynamic system. Using dynamic theory, the complex behaviors of the dynamic system are discussed; besides, the parameter adaptation method is adopted to restrain the chaos phenomenon. The conclusions are as follows: the stable scope of the green supply chain system enlarges with decision makers’ risk-aversion level increasing and decreases with service value increasing; excessive adjustment of price parameters will make the green supply chain system fall into chaos with a large entropy value; the attraction domain of initial prices shrinks with price adjustment speed increasing and enlarges with the channel service values raising. As the dynamic game model system is in a chaotic state, the profit of the manufacturer will be damaged, while the efficiency of the retailer will be improved. The system would be kept at a stable state and casts off chaos by the parameter adaptation method. Results are significant for the manager to make reasonable price decision.

... Kopel [26] uses the chaotic target method to control chaos of a monopoly output adjustment model. Hołyst and Urbanowicz [27] use the delayed feedback control method (DFC) to control chaos in a duopoly investment model. Wieland and Westerhoff [28] applies the OGY method and DFC separately to stabilizing chaos in an exchange rate dynamic model. ...

Finance is the core of modern economy. The security and stability of the financial system is the key to stable economic and social development. During the operation of the financial system, financial chaos such as the severe turbulence of the financial market and the financial crisis occurred due to deterministic instability, which brought a great negative impact on economic growth and social stability. For the financial chaotic system, an intermittent feedback controller is designed in this paper. By adjusting the controller parameters, the financial system can be controlled from chaotic to periodic evolution. First, the dynamic equations and controllers of the financial system are analyzed and the range of values of the controller parameters is theoretically obtained. Then, the influence of parameters on the system is studied, and the feasibility of the proposed method is proved by numerical simulation. Finally, the practical significance of the controller on the macrocontrol of the financial crisis is discussed. It is theoretically proven that when the financial crisis comes, the financial system can be stabilized more quickly through appropriate control methods.

... The primary mechanism applied to suppress chaos is localization with the help of small perturbations in the system, or introduction of UPO controls embedded in a chaotic attractor. Holyst et al. (Holyst et al., 1996;Holyst and Urbanowicz, 2000;Holyst et al., 2001) showed that applying the Pyragas time-delayed feedback control to the microeconomic Behrens-Feichtinger model can facilitate an easy switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the system's economic properties. Kopel (Kopel, 1997), using a model of evolutionary market dynamics, demonstrated how chaotic behaviour can be controlled by making small changes in a parameter that is accessible to the decision makers. ...

Control and stabilization of irregular and unstable behavior of dynamic systems (including chaotic processes) are interdisciplinary problems of interest to a variety of scientific fields and applications. One effective method for computation of unstable periodic orbits (UPOs) is the unstable delay feedback control (UDFC) approach, suggested by K. Pyragas (Pyragas, 1992). This paper proposes the application of the Pyragas' time-delay feedback control method within framework of economic models. Using the control methods allows to improve forecasting of dynamics for unstable economic processes and offers opportunities for governments, central banks, and other policy makers to modify the behaviour of the economic system to achieve its best performance. We consider this method through the example of the Shapovalov model, by describing the dynamics of a mid-size firm. The results demonstrate chaos suppression in the Shapovalov model with certain values of its parameters, using UDFC method.

... Ma and Tu [32] established a class of complex dynamic macroeconomic systems and studied the effect of time delay on savings rate and dynamic financial stability. Holyst and Urbanowicz [33] have shown that the chaotic attractor of the financial model can be stabilized in a periodic track by using Pyragas delayed feedback control. In addition, Ma and Chen [14] added the delayed feedback to the three variables of financial system and gave some results on the existence of Hopf bifurcation and the effect of delayed feedback. ...

The complex chaotic dynamics and multistability of financial system are some important problems in micro- and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. The system undergoes Hopf bifurcation near the equilibrium point. The stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. The theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.

... Motivated by this background, modeling, control and synchronization of chaotic/hyperchaotic financial systems have been active topics of study to be considered by many researchers. In [3] , the chaotic behaviour of an economical model was controlled by using a timedelayed feedback control scheme. In [4] , the chaos phenomenon in a Cournot duopoly model was controlled by employing an adaptive parameter-tuning strategy. ...

This paper mainly focuses on the analysis of a hyperchaotic financial system as well as its chaos control and synchronization. The phase diagrams of the above system are plotted and its dynamical behaviours like equilibrium points, stability, hyperchaotic attractors and Lyapunov exponents are investigated. In order to control the hyperchaos, an efficient optimal controller based on the Pontryagin’s maximum principle is designed and an adaptive controller established by the Lyapunov stability theory is also implemented. Furthermore, two identical financial models are globally synchronized by using an interesting adaptive control scheme. Finally, a fractional economic model is introduced which can also generate hyperchaotic attractors. In this case, a linear state feedback controller together with an active control technique are used in order to control the hyperchaos and realize the synchronization, respectively. Numerical simulations verifying the theoretical analysis are included.

... There are many methods available for chaos control such as open-loop control based on periodic system excitation, closed-loop control such as OGY method (Ott et al., 1990) and also time-delayed feedback control such as Pyragas (1992) method, etc. Recently, time-delayed feedback method has been also considered for chaos control in economic model (Holyst and Urbanowicz, 2000) and inducing or suppressing chaos in a double-well Duffing oscillator (Sun et al., 2006). ...

This work announces an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities. A qualitative analysis of the properties of the novel 4-D hyperchaotic system is presented. A special feature of our novel hyperchaotic system is that it has three equilibrium points of which two are unstable and one is locally asymptotically stable. The Lyapunov exponents of the novel hyperchaotic system are obtained as L1 = 1.5146, L2 = 0.2527, L3 = 0 and L4 = −12.7626. The Kaplan-Yorke dimension of the novel hyperchaotic system is derived as DKY = 3.1385. Next, this work describes the design of an adaptive controller for the global hyperchaos synchronisation of identical novel hyperchaotic systems with unknown parameters. MATLAB simulations are shown to describe all the main results derived in this work.

The dynamic model of price competition in which processes of strategic interaction between companies on an imperfect competition market are described with the game-theoretic approach and methods of nonlinear dynamics. The pricing dynamics for the companies is modeled with difference equations (mappings). We study the stability of the fixed point of the price mapping. Results of our numerical modeling have shown the existence of periodic and chaotic solutions in the price competition model. We present intra-company adaptation mechanisms based on changing the prices in a way proportional to the rate of change in the companies’ profits; this lets us reduce the prices to a local Nash equilibrium and stabilize the chaotic dynamics of the market.

Lyapunov exponents (LE) are an effective tool for analyzing the qualitative characteristics of dynamic systems. Identifiability, recoverability and detectability problem of Lyapunov exponents not studied. This problem is actual. We propose an approach for verifying identifiability, recoverability and detectability. The approach bases on the analysis of geometric frameworks depending on the structural properties coefficient of the system. The structural properties coefficient reflects the change in Lyapunov exponents, and geometric frameworks are a source for deciding on the type of indicators. We obtain conditions for the complete detectability of Lyapunov exponents. These conditions guarantee the receipt of indicators set. We propose a criterion of σ-detectability with a level of υ-non-recoverability and give a method to evaluate it. We propose the method for verifying the adequacy of the Lyapunov exponents set. The permissible mobility border of the largest Lyapunov exponent obtains.

This paper is devoted to introduce and study a Stackelberg game consisting of three competed firms. The three firms are classified as a leader which is the first firm and the other two firms are called the followers. A linear inverse demand function is used. In addition a quadratic cost based on an actual and announced quantities is adopted. Based on bounded rationality, a three-dimensional discrete dynamical system is constructed. For the system, the backward induction is used to solve the system and to get Nash equilibrium. The obtained results are shown that Nash equilibrium is unique and its stability is affected by the system's parameters by which the system behaves chaotically due to bifurcation and chaos appeared. Some numerical experiments are performed to portrays such chaotic behavior. A control scheme is used to return the system back to its stability state and is supported by some simulations.

The human cerebral cortex can be separated into cortical areas forming a clustered network structure. We build two different clustered networks, where one network is based on a healthy brain and the other according to a brain affected by a neurodegenerative process. Each cortical area has a subnetwork with small-world properties. We verify that both networks exhibit rich-club organisation and phase synchronisation. Due to the fact that neuronal synchronisation can be related to brain diseases, we consider the delayed feedback control as a method to suppress synchronous behaviours. In this work, it is presented that depending on the feedback parameters, intensity and time delay, phase synchronisation in both networks can be suppressed. Therefore, one of our main results is to show that delayed feedback control can be used to suppress undesired synchronous behaviours not only in the healthy brain, but also in the brain marked by neurodegenerative processes.

Information and Communication Technologies (ICTs) impact the communities in which we live and the way individuals, business, government and civil society interact and develop. Simultaneously, all sectors have shown increased interest in the concept of social capital and the role it could play in building stronger communities, increasing economic productivity and contributing to regional rejuvenation. Thus, ICTs and social capital concept interlink with each other in a variety of ways. The purpose of this paper is to investigate the relationship between ICTs and social capital through the study of the relative’s disparities among Italian regions. This paper provides an operational definition of the concepts of ICT and social capital and presents consistent evidence on the geography of this relationship in Italy. The statistical and geographical analysis, based on non-linear clustering with self-organizing map (SOM) neural networks, are performed to analyse the performance of Italian regions in the period 2006–2013. The results show the isolation of Southern Italian regions. In particular, we found that ICTs may not promote social capital, that is, ICTs could not play a decisive role in creating and developing social capital. These results prompt the formulation of new policies for Italian regions.

The present contribution contains an original interpretation of the logistic map popularized by the biologist Robert May in 1976. This map is potentially a powerful AI tool based on a deterministic methodology having a double possibility to be applied in economics. The first application is to investigate the intrinsic complexity of real economic phenomena characterized by endogenous non-linear dynamics. The second application is to determine results, typical of a normative science, useful for suggesting policy decisions aimed to avoid chaos and unpredictability in the real economic system. In the first type of application, the logistic map can be used as an AI tool of forecasting (for previsions of bifurcations, cycles and chaos). In the second, the logistic map can be considered as an AI tool for policy makers in order to deduce the analytical conditions that ensure the economic system to be sufficiently far away from chaos and uncontrollability.

Bounded rationality, asymmetric information and spillover effects are widespread in the economic market, and had been studied extensively in oligopoly games, but few references discussed incomplete information in a duopoly market with rationality expectations. Considering the positive externalities brought by the spillover effect between enterprises in a cluster, a duopoly Bertrand game with bounded rationality and asymmetric information is proposed in this paper. In our model, a firm with private information, high or low marginal cost, is introduced. Interestingly, our theoretical analysis reveals that: (1) In a dynamic duopoly Bertrand game with perfect rationality and asymmetric information, the equilibrium price is positively correlated with product substitution rate and the probability of a high marginal cost, while it is negatively correlated with the cluster spillover. (2) In a dynamic duopoly Bertrand game with asymmetric information and adaptive expectation adopted by both firms, the Nash equilibrium prices are always asymptotically stable. (3) In a dynamic duopoly Bertrand game with heterogenous expectation and asymmetric information, where two firms use adaptive expectation and boundedly rational expectation respectively, the Nash equilibrium prices are locally stable under certain conditions. Furthermore, results indicate that, high product substitution rate or large probability of high marginal cost for firm 2 with private information may make the market price unstable, bifurcating or even falling into chaos, while high technology spillover is conducive to stabilize the market by contrast. It is also shown that the chaos can be controlled by a hybrid control strategy with the state variables feedback and parameter variation. Our research has an important theoretical and practical significance to the price competition in oligopoly markets.

This chapter is of big interest to those readers, who look for three subjects of investigation: neural networks theory; nonlinear oscillations; and development of results for discontinuous almost periodicity to continuous dynamics and chaos. Thus, we will demonstrate that the technique developed in our papers can be applied to the construction of concrete discontinuous almost periodic functions, and the functions being postsynaptic currents generate continuous almost periodic motions of neural networks and chaotic dynamics with prescribed properties. Apparently, the results can be of use in the analysis of brain activity, for robotics, and artificial intelligence. Thus, one can say that a new stage of the theory of discontinuous almost periodicity and its applications to the neural networks investigation is upcoming.

The model of market pricing in duopoly describing the prices dynamics as a two-dimensional map is presented. It is shown that the fixed point of the map coincides with the local Nash-equilibrium price in duopoly game. There have been numerically identified a bifurcation of the fixed point, shown the scheme of transition from periodic to chaotic mode through a doubling period. To ensure the sustainability of local Nashequilibrium price the controlling chaos mechanism has been proposed. This mechanism allows to harmonize the economic interests of the firms and to form the balanced pricing policy.

This paper investigates the dynamical behaviors of a duopoly model with two content providers (CPs). Competition between two CPs is assumed to take place in terms of their pricing decisions and the credibility of content they offer. According to the CPs’ rationality level, we consider a scenario where both CPs are bounded rational. Each CP in any period uses the marginal profit observed from the previous period to choose its strategies. We compute explicitly the steady states of the dynamical system induced by bounded rationality, and establish a necessary and sufficient condition for stability of its Nash equilibrium (NE). Numerical simulations show that if some parameters of the model are varied, the stability of the NE point is lost and the complex (periodic or chaotic) behavior occurs. The chaotic behavior of the system is stabilized on the NE point by applying control.

We establish dynamic game models in a low-carbon supply chain consisting of a single manufacturer and a single retailer with social preference. This study investigates the complex dynamic characteristics of pricing decision and carbon abatement strategy in the supply chain and focuses on the impact of the retailer’s social preference on pricing decision, carbon emission abatement strategy, profits, supply chain coordination, and complexity of dynamic models. We find that adjustment parameters of pricing and carbon emission abatement should be maintained in a certain range; otherwise, the system will be unstable and even chaotic through period double bifurcation or wave shape chaos. A higher social preference of the retailer is always beneficial to carbon abatement and the manufacturer and helps maintain the stability of the supply chain system. However, the impact on the long-term profitability of the supply chain is related to the state of the system. Compared with the setting of a centralized decision, the optimal carbon abatement strategy and supply chain profit in a decentralized decision are always less than those in a centralized setting, regardless of whether the retailer has social preference. Therefore, a side-payment self-executing contract is designed to coordinate the supply chain and achieve Pareto improvement. The coordination mechanism proposed in this study not only leads to Pareto improvement but also increases the stability of the supply chain system. Finally, this study enlightens management in operating a low-carbon supply chain.

Control and stabilization of irregular and unstable behavior of dynamic systems (including chaotic processes) are interdisciplinary problems of interest to a variety of scientific fields and applications. Using the control methods allows improvements in forecasting the dynamics of unstable economic processes and offers opportunities for governments, central banks, and other policy makers to modify the behaviour of the economic system to achieve its best performance. One effective method for control of chaos and computation of unstable periodic orbits (UPOs) is the unstable delay feedback control (UDFC) approach, suggested by K. Pyragas. This paper proposes the application of the Pyragas’ method within framework of economic models. We consider this method through the example of the Shapovalov model, by describing the dynamics of a mid-size firm. The results demonstrate that suppressing chaos is capable in the Shapovalov model, using the UDFC method.

Cyclicality and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in the long run. We focus on an approach, associated with the identification of a deterministic endogenous mechanism of irregular fluctuations in the economy. Using of a mid-size firm model as an example, we demonstrate the use of effective analytical and numerical procedures for calculating the quantitative characteristics of its irregular limiting dynamics based on Lyapunov exponents, such as dimension and entropy. We use an analytical approach for localization of a global attractor and study limiting dynamics of the model. We estimate the Lyapunov exponents and get the exact formula for the Lyapunov dimension of the global attractor of this model analytically. With the help of delayed feedback control (DFC), the possibility of transition from irregular limiting dynamics to regular periodic dynamics is shown to solve the problem of reliable forecasting. At the same time, we demonstrate the complexity and ambiguity of applying numerical procedures to calculate the Lyapunov dimension along different trajectories of the global attractor, including unstable periodic orbits (UPOs).

The supply chain network is a complex nonlinear system that may have a chaotic behavior. This network involves multiple entities that cooperate to meet customers demand and control network inventory. Although there is a large body of research on measurement of chaos in the supply chain, no proper method has been proposed to control its chaotic behavior. Moreover, the dynamic equations used in the supply chain ignore many factors that affect this chaotic behavior. This paper offers a more comprehensive modeling, analysis, and control of chaotic behavior in the supply chain. A supply chain network with a centralized decision-making structure is modeled. This model has a control center that determines the order of entities and controls their inventories based on customer demand. There is a time-varying delay in the supply chain network, which is equal to the maximum delay between entities. Robust control method with linear matrix inequality technique is used to control the chaotic behavior. Using this technique, decision parameters are determined in such a way as to stabilize network behavior.

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An economic system which exhibits chaotic behaviour has been stabilized on various periodic orbits by use of the Ott-Grebogi-Yorke method. This procedure has been recently applied to controlling chaotic phenomena in physical, chemical and biological systems. We adopt this method successfully for Feichtinger's generic model of two competing firms with asymmetrical investment strategies. We show that the application of this control method to the particular economic process considered brings a substantial advantage: one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the system's economic properties. Numerical simulations are presented in order to illustrate the effectiveness of the whole procedure.

We present a technique for stabilizing unstable periodic orbits in low-dimensional dynamical systems that allows for control over a large domain of parameters. The technique uses a continuous feedback loop incorporating information from many previous states of the system in a form closely related to the amplitude of light reflected from a Fabry-Pérot interferometer. We demonstrate that the approach is well suited for pratical implementation in fast systems by stabilizing a chaotic diode resonator driven at 10.1 MHz.

The performance of time-delayed feedback control is studied by linear stability analysis. Analytical approximations for the resulting eigenvalue spectrum are proposed. Our investigations demonstrate that eigenbranches that develop from the stable Lyapunov exponents of the free system also have a strong influence on the control properties, either by hybridization or by a crossing of branches which interchanges the role of the leading eigenvalue. Our findings are confirmed by numerical analysis of two particular examples, the Toda and the Rossler models. More important is the verification by actual electronic circuit experiments. Here, the observed reduction of control domains can be attributed to these additional eigenvalue branches. The investigations lead to a thorough analytical understanding of the stability properties in time-delayed feedback systems.

The Pyragas method for controlling chaos is investigated in detail from the experimental as well as theoretical point of view. We show by an analytical stability analysis that the revolution around an unstable periodic orbit governs the success of the control scheme. Our predictions concerning the transient behaviour of the control signal are confirmed by numerical simulations and an electronic circuit experiment. Comment: 4 pages, REVTeX, 4 eps-figures included Phys. Rev. Lett., in press also available at http://athene.fkp.physik.th-darmstadt.de/public/wolfram.html

This paper is to report the observation that when the popular time-delayed feedback strategy is used for control purpose, it may actually create unwanted bifurcations. Hopf bifurcation created by delayed feedback control is the main concern of this article, but some other types of bifurcations are also observed to exist in such delayed-feedback control systems. The observations are illustrated by computer simulations.

This paper investigates robust stability of the extended delayed-feedback control method [Phys. Rev. E 50 (1994) 3245] from the viewpoint of control theory. The feedback-gain range where control can be achieved for this method is wider than that of the original delayed-feedback control method. We provide a procedure to design the extended delayed-feedback controller with a minimum amount of a priori knowledge for chaotic systems including a parameter uncertainty. Some numerical experiments are given to check the results.

Two methods of chaos control with a small time continuous perturbation are proposed. The stabilization of unstable periodic orbits of a chaotic system is achieved either by combined feedback with the use of a specially designed external oscillator, or by delayed self-controlling feedback without using of any external force. Both methods do not require an a priori analytical knowledge of the system dynamics and are applicable to experiment. The delayed feedback control does not require any computer analyses of the system and can be particularly convenient for an experimental application. On leave of absence from Institute of Semiconductor Physics, 2600 Vilnius, Lithuania.

This Letter proposes an observer-based delayed-feedback control method that overcomes an inherent weak point of the well-known delayed-feedback control. This method employs a state observer that estimates difference between a system state and a desired unstable fixed point without using location of the point. The estimated difference is used for state feedback control. We give a systematic procedure to design the observer and the controller. To check the theoretical results, some numerical examples are presented.

Depending upon the assumed ordering policy, the classical Beer Distribution Model shows a great variety of complex dynamic behaviors, including limit cycle oscillations of different periodicities and deterministic chaos. This article presents an overview of these phenomena. The ordering policy is expressed in terms of two parameters that measure the fraction of the supply line and the fraction of the anticipated shipments accounted for in the placement of orders. Our results indicate that, in certain regions of parameter space, any neighborhood to a given solution will contain qualitatively different solutions. The complexity of the system is revealed in two- and three-dimensional reconstructions of chaotic attractors as well as in Poincaré sections obtained from such reconstructions.

The problem of control of chaos in a microeconomical model describing two competing firms with asymmetrical investment strategies is studied. Cases when both firms try to perform the control simultaneously or when noise is present are considered. For the first case the resulting control efficiency depends on the system parameters and on the maximal values of perturbations of investment parameters for each firm. Analytic calculations and numerical simulations show that competition in the control leads to ‘parasitic’ oscillations around the periodic orbit that can destroy the expected stabilization effect. The form of these oscillations is dependent on non-linear terms describing the motion around periodic orbits. An analytic condition for stable behaviour of the oscillation (i.e. the condition for control stability) is found. The values of the mean period of these oscillations is a decreasing function of the amplitude of investment perturbation of the less effective firm. On the other hand, amplitudes of market oscillations are increasing functions of this parameter. In the presence of noise the control can be also successful provided the amplitude of allowed investment changes is larger than some critical threshold which is proportional to the maximal possible noise value. In the case of an unbounded noise, the time of laminar epochs is always finite but their mean length increases with the amplitude of investment changes. Computer simulations are in very good agreement with analytical results obtained for this model.

A simple Keynesian macroeconomic disequilibrium model with rationing is considered. This paper investigates the influence of different expectations hypotheses (regarding next period's goods prices) on the model's dynamic behavior when intertemporal substitution effects exist. While the bifurcation behavior is not qualitatively influenced when expectations are formed according to adaptive expectations or an unweighted-averages hypothesis, the dynamic behavior of the model can be changed in an essential way when pattern-recognition expectations are assumed. So-called perfect cyclic expectations can turn a previously chaotically moving economy into a system with a low-periodic motion in particular cases.

General features of stability domains for time-delayed feedback control exist, which can be predicted analytically. We clarify, why the control scheme with a single delay term can only stabilise orbits with short periods or small Lyapunov exponents, and derive a quantitative estimate. The limitation can be relaxed by employing multiple delay terms. In particular, the extended time delay autosynchronisation method is investigated in detail. Analytic calculations are in good agreement with results of numerical simulations and with experimental data from a nonlinear diode resonator.

In the Cont–Bouchaud model [cond-mat/9712318] of stock markets, percolation clusters act as buying or selling investors and their statistics controls that of the price variations. Rather than fixing the concentration controlling each cluster connectivity artificially at or close to the critical value, we propose that clusters shatter and aggregate continuously as the concentration evolves randomly, reflecting the incessant time evolution of groups of opinions and market moods. By the mechanism of “sweeping of an instability” [Sornette, J. Phys. I 4, 209(1994)], this market model spontaneously exhibits reasonable power-law statistics for the distribution of price changes and accounts for the other important stylized facts of stock market price fluctuations.

Two theorems on limitations in controlling chaos by delayed feedback control are proved. The results are as follows: (1) If the linear variational equation about the target hyperbolic unstable periodic orbit (UPO) has an odd number of real characteristic multipliers which is greater than unity, the UPO can never be stabilized with any value of the feedback gain. (2) If all the characteristic exponents of the variational equation are different from each other and at least one of them is real and positive, then the UPO can never be stabilized with any feedback gain matrix of the form diag(k,…, k). Both theorems are proved on the basis of Floquet theory. The result of the first theorem is also explained intuitively using bifurcation theory.

Bifurcation theory shows that policy adaptation and the rational expectations hypothesis of macro-economics can be used to explain unpredictability, rapid changes in solution structure and chaos in decision problems with uncertainty. Structural errors lead to catastrophic instability and forecasts become irrelevant. Short evaluation horizon and the application of measures designed to give quick response give multiperiodicity and chaos. Finally, wrong interpretations of the context lead to global bifurcations, laminar drift (complacency) and chaotic bursting. The discrete map representing these dynamics is of interest in its own right. It is non-invertible and displays bifurcation behaviour not commonly seen in systems derived from physical considerations.

It is shown that one can convert a chaotic attractor to any one of a large number of possible attracting time-periodic motions by making only {ital small} time-dependent perturbations of an available system parameter. The method utilizes delay coordinate embedding, and so is applicable to experimental situations in which {ital a} {ital priori} analytical knowledge of the system dynamics is not available. Important issues include the length of the chaotic transient preceding the periodic motion, and the effect of noise. These are illustrated with a numerical example.

Recently, Pyragas [1993] proposed a control method called delayed
feedback control to stabilize unstable periodic orbits in nonlinear
continuous-time systems. In this letter, we consider delayed
state-feedback control in discrete-time systems. We show that a fixed
point can not be stabilized by the delayed state-feedback control if the
linearized system around the fixed point has an odd number of real
eigenvalues greater than one. Moreover, in one- and two-dimensional
systems, we show necessary and sufficient conditions for the local
stabilization by the delayed state-feedback control

- E Moseklide
- E R Larsen

E. Moseklide, E.R. Larsen, System Dyn. Rev. 4 (1988) 131.

- H W Lorenz
- M Lohmann

H.W. Lorenz, M. Lohmann, Chaos, Solitons Fractals 7 (1996) 2135.

- W Just
- E Reibold
- K Kacperski
- P Fronczak
- J A Ho Lyst
- H Benner

W. Just, E. Reibold, K. Kacperski, P. Fronczak, J.A. Ho lyst, H. Benner, Phys. Rev. E. 61 (2000) 5045.

- H W Lorenz

H.W. Lorenz, Nonlinear Dynamical Equations and Chaotic Economy, Springer, Berlin, 1993.

- J E S Socolar
- D W Sukow
- D J Gauthier

J.E.S. Socolar, D.W. Sukow, D.J. Gauthier, Phys. Rev. E 50 (1994) 2343.

- G Chen
- J Lu
- B Nicholas
- S M Ranganathan

G. Chen, J. Lu, B. Nicholas, S.M. Ranganathan, Int. J. Bifurc. Chaos 9 (1999) 287.

- D Stau Er
- D Sornette

D. Stau er, D. Sornette, Physica A 271 (1999) 496.

- J A Ho Lyst
- T Hagel
- G Haag

J.A. Ho lyst, T. Hagel, G. Haag, Chaos, Solitons Fractals 8 (1997) 1489.

- E Ott
- C Grebogi
- J A Yorke

E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196.

- W Just
- E Reibold
- H Benner
- K Kacperski
- P Fronczak
- J Ho
- Lyst

W. Just, E. Reibold, H. Benner, K. Kacperski, P. Fronczak, J. Ho lyst, Phys. Lett. A 254 (1999) 158.

- H Haken

H. Haken, Synergetics. An Introduction, Springer, Heidelberg, 1983; Advanced Synergetics, Springer,
Heidelberg, 1983.

- W Just
- T Bernard
- M Ostheimer
- E Reibold
- H Benner

W. Just, T. Bernard, M. Ostheimer, E. Reibold, H. Benner, Phys. Rev. Lett. 78 (1997) 203.

- H Haken

H. Haken, Synergetics. An Introduction, Springer, Heidelberg, 1983; Advanced Synergetics, Springer,
Heidelberg, 1983.

- T Ushio

T. Ushio, IEEE CAS-I 43 (1996) 815.