This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other...This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other existing MMCSs, the most eye-catching point of the proposed MMCS is that the amplitude of the wing will enlarge towards the poles as the number of wings increases. Diverse coexisting attractors are numerically found in the MMCS, including chaos,quasi-period, and stable point. The circuits of the proposed memristor and MMCS are designed and the obtained results demonstrate their validity and reliability.展开更多
We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability...We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.展开更多
It has been documented that a cyclic three-neuron-based neural network with resistive synaptic weights cannot exhibit chaos.Towards this end,a memristive cyclic three-neuron-based neural network is presented using a m...It has been documented that a cyclic three-neuron-based neural network with resistive synaptic weights cannot exhibit chaos.Towards this end,a memristive cyclic three-neuron-based neural network is presented using a memristive weight to substitute a resistive weight.The memristive cyclic neural network always has five equilibrium points within the parameters of interest,and their stability analysis shows that they are one index-2 saddle-focus,two index-1 saddle-foci,and two stable node-foci,respectively.Dynamical analyses are performed for the memristive cyclic neural network by several numerical simulation methods.The results demonstrate that the memristor synapse-based neural network with the simplest cyclic connection can not only exhibit chaos,but also present global coexisting attractors composed of stable points and unstable periodic or chaotic orbits under different initial conditions.Besides,with the designed implementation circuit,Multisim circuit simulations and hardware experiments are executed to validate the numerical simulations.展开更多
Purpose-The purpose of this paper is to investigate coexisting attractors,chaos control and synchronization in a self-exciting homopolar dynamo system in this paper.Design/methodology/approach-Two single controllers a...Purpose-The purpose of this paper is to investigate coexisting attractors,chaos control and synchronization in a self-exciting homopolar dynamo system in this paper.Design/methodology/approach-Two single controllers are designed and added to the proposed 3D autonomous chaotic system,and its stability at zero equilibrium point is guaranteed by applying an appropriate control signal based on the Lyapunov stability theory.Findings-Numerical simulations reveal that the proposed 3D dynamo system exhibits periodic oscillations,double-scroll chaotic attractors and coexisting attractors.Finally,a single controller is designed for the global asymptotic synchronization of a unidirectionally coupled identical 3D autonomous chaotic system.Originality/value-The derived results of this paper are new and complement some earlier works.The innovation concludes two points in this paper;coexisting attractors are foundthe and an appropriate control signal based on the Lyapunov stability theory is established.The ideas of this paper can be applied to investigate some other homopolar dynamo systems.展开更多
This paper introduces a four-dimensional (4D) segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an ap...This paper introduces a four-dimensional (4D) segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcation and pitchfork bifurcation occur in the system. The ultimate bound is also estimated. Some numerical investigations are also exploited to demonstrate and visualize the corresponding theoretical results.展开更多
Research on discrete memristor-based neural networks has received much attention.However,current research mainly focuses on memristor–based discrete homogeneous neuron networks,while memristor-coupled discrete hetero...Research on discrete memristor-based neural networks has received much attention.However,current research mainly focuses on memristor–based discrete homogeneous neuron networks,while memristor-coupled discrete heterogeneous neuron networks are rarely reported.In this study,a new four-stable discrete locally active memristor is proposed and its nonvolatile and locally active properties are verified by its power-off plot and DC V–I diagram.Based on two-dimensional(2D)discrete Izhikevich neuron and 2D discrete Chialvo neuron,a heterogeneous discrete neuron network is constructed by using the proposed discrete memristor as a coupling synapse connecting the two heterogeneous neurons.Considering the coupling strength as the control parameter,chaotic firing,periodic firing,and hyperchaotic firing patterns are revealed.In particular,multiple coexisting firing patterns are observed,which are induced by different initial values of the memristor.Phase synchronization between the two heterogeneous neurons is discussed and it is found that they can achieve perfect synchronous at large coupling strength.Furthermore,the effect of Gaussian white noise on synchronization behaviors is also explored.We demonstrate that the presence of noise not only leads to the transition of firing patterns,but also achieves the phase synchronization between two heterogeneous neurons under low coupling strength.展开更多
We investigated the synchronization dynamics of a coupled neuronal system composed of two identical Chay model neurons. The Chay model Showed coexisting period-1 and period-2 bursting patterns as a parameter and initi...We investigated the synchronization dynamics of a coupled neuronal system composed of two identical Chay model neurons. The Chay model Showed coexisting period-1 and period-2 bursting patterns as a parameter and initial values are varied. We simulated multiple periodic and chaotic bursting patterns with non-(NS), burst phase (BS), spike phase (SS), complete (CS), and lag synchronization states. When the coexisting behavior is near period-2 bursting, the transitions of synchronization states of the coupled system follows very complex transitions that begins with transitions between BS and SS, moves to transitions between CS and SS, and to CS. Most initial values lead to the CS state of period-2 bursting while only a few lead to the CS state of period-I bursting. When the coexisting behavior is near period-1 bursting, the transitions begin with NS, move to transitions between SS and BS, to transitions between SS and CS, and then to CS. Most initial values lead to the CS state of period-1 bursting but a few lead to the CS state of period-2 bursting. The BS was identified as chaos synchronization. The patterns for NS and transitions between BS and SS are insensitive to initial values. The patterns for transitions between CS and SS and the CS state are sensitive to them. The number of spikes per burst of non-CS bursting increases with increasing coupling strength. These results not only reveal the initial value- and parameter- dependent synchronization transitions of coupled systems with coexisting behaviors, but also facilitate interpretation of various bursting patterns and synchronization transitions generated in the nervous system with weak coupling strength.展开更多
The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper.The critical parameter of stable and unstable states is calculated by identifying the eige...The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper.The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix.Besides,complex dynamical behaviors are investigated in the system,such as coexisting attractors,hidden attractors,coexisting bifurcation modes,intermittent chaos,and multistability.From the theoretical analyses and numerical simulations,it is found that there are four different kinds of transient transition behaviors in the memcapacitive system.Finally,field programmable gate array(FPGA)is used to implement the proposed chaotic system.展开更多
Synaptic crosstalk is a prevalent phenomenon among neuronal synapses,playing a crucial role in the transmission of neural signals.Therefore,considering synaptic crosstalk behavior and investigating the dynamical behav...Synaptic crosstalk is a prevalent phenomenon among neuronal synapses,playing a crucial role in the transmission of neural signals.Therefore,considering synaptic crosstalk behavior and investigating the dynamical behavior of discrete neural networks are highly necessary.In this paper,we propose a heterogeneous discrete neural network(HDNN)consisting of a three-dimensional KTz discrete neuron and a Chialvo discrete neuron.These two neurons are coupled mutually by two discrete memristors and the synaptic crosstalk is considered.The impact of crosstalk strength on the firing behavior of the HDNN is explored through bifurcation diagrams and Lyapunov exponents.It is observed that the HDNN exhibits different coexisting attractors under varying crosstalk strengths.Furthermore,the influence of different crosstalk strengths on the synchronized firing of the HDNN is investigated,revealing a gradual attainment of phase synchronization between the two discrete neurons as the crosstalk strength decreases.展开更多
By introducing a discrete memristor and periodic sinusoidal functions,a two-dimensional map with coexisting chaos and hyperchaos is constructed.Various coexisting chaotic and hyperchaotic attractors under different Ly...By introducing a discrete memristor and periodic sinusoidal functions,a two-dimensional map with coexisting chaos and hyperchaos is constructed.Various coexisting chaotic and hyperchaotic attractors under different Lyapunov exponents are firstly found in this discrete map,along with which other regimes of coexistence such as coexisting chaos,quasiperiodic oscillation,and discrete periodic points are also captured.The hyperchaotic attractors can be flexibly controlled to be unipolar or bipolar by newly embedded constants meanwhile the amplitude can also be controlled in combination with those coexisting attractors.Based on the nonlinear auto-regressive model with exogenous inputs(NARX)for neural network,the dynamics of the memristive map is well predicted,which provides a potential passage in artificial intelligencebased applications.展开更多
Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are...Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.展开更多
In a global dynamic analysis,the coexisting attractors and their basins are the main tools to understand the system behavior and safety.However,both basins and attractors can be drastically influenced by uncertainties...In a global dynamic analysis,the coexisting attractors and their basins are the main tools to understand the system behavior and safety.However,both basins and attractors can be drastically influenced by uncertainties.The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors.Accordingly,analytical and numerical tools for calculation of nondeterministic global structures,namely attractors and basins,are proposed.First,based on the definition of the Perron-Frobenius,Koopman and Foias linear operators,a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases.In this context,the stochastic basins of attraction and attractors’distributions replace the usual basin and attractor concepts.Then,numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method.Sample results of the methodology are presented for a canonical dynamical system.展开更多
Unlike dissipative systems,conservative systems do not have attractors and no attractor reconstruction occurs.Therefore,these systems are more suitable for application in image encryption.On the basis of above appoint...Unlike dissipative systems,conservative systems do not have attractors and no attractor reconstruction occurs.Therefore,these systems are more suitable for application in image encryption.On the basis of above appoints,here we develop and propose a conservative system with infinite chaotic-like attractors.The conservative and chaotic characteristics and coexistence chaotic-like attractors are studied using Lyapunov exponents,Poincare maps,and numerical simulation.The results show that the coexistence of chaotic-like attractors has a more complex structure and dynamic behaviour than traditional ones.Additionally,the developed system is further used to design an encryption system for a digital image.Using the coexistence chaotic-like attractor sequence to scramble and diffuse the image can destroy the correlation of adjacent pixels and hide the information of all pixels.The feasibility and security of the encryption scheme are demonstrated through the analysis of key space,histogram,information entropy,key sensitivity and pixel correlation.展开更多
The generalized dynamic Tullock contest model with two homogeneous participants is established, in which both players have the same valuation of winning rewards and losing rewards. Firstly, the unique symmetric equili...The generalized dynamic Tullock contest model with two homogeneous participants is established, in which both players have the same valuation of winning rewards and losing rewards. Firstly, the unique symmetric equilibrium point of the system is obtained by calculation and its local stability condition is given based on the Jury criterion. Then, two paths of the system from stability to chaos, namely flip bifurcation and Neimark-Sacker bifurcation, are analyzed by using the two-dimensional parametric bifurcation diagram. Meanwhile, the abundant Arnold tongues in the two-dimensional parametric bifurcation diagram are analyzed. Finally, the phenomenon of multistability of the system is illustrated through the basin of attraction, and the contact bifurcation occurs during the evolution of the basin of attraction with varying parameters.展开更多
A two-SBT-memristor-based chaotic circuit was proposed. The stability of the equilibrium point was studied by theoretical analysis. The close dependence of the circuit dynamic characteristics on its initial conditions...A two-SBT-memristor-based chaotic circuit was proposed. The stability of the equilibrium point was studied by theoretical analysis. The close dependence of the circuit dynamic characteristics on its initial conditions and circuit parameters was investigated by utilizing Lyapunov exponents spectra, bifurcation diagrams, phase diagrams, and Poincaré maps. The analysis showed that the circuit system had complex dynamic behaviors, such as stable points, period, chaos, limit cycles,and so on. In particular, the chaotic circuit produced the multistability phenomenon, such as coexisting attractors and coexisting periods.展开更多
Multiplex networks have drawn much attention since they have been observed in many systems,e.g.,brain,transport,and social relationships.In this paper,the nonlinear dynamics of a multiplex network with three neural gr...Multiplex networks have drawn much attention since they have been observed in many systems,e.g.,brain,transport,and social relationships.In this paper,the nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied.The stability and bifurcation of the network equilibrium are discussed,and interesting neural activities of the network are explored.Based on the neuron circuit,transfer function circuit,and time delay circuit,a circuit platform of the network is constructed.It is shown that delayed couplings play crucial roles in the network dynamics,e.g.,the enhancement and suppression of the stability,the patterns of the synchronization between networks,and the generation of complicated attractors and multi-stability coexistence.展开更多
Based on bounded rationality,this paper established a price game model of dual channel supply chain composed of manufacturers and retailers.According to the eigenvalue of Jacobi matrix and Jury criterion,the stability...Based on bounded rationality,this paper established a price game model of dual channel supply chain composed of manufacturers and retailers.According to the eigenvalue of Jacobi matrix and Jury criterion,the stability of the equilibrium point is analyzed,and then the dynamic evolution process under the parameters of price adjustment speed and retailer's service input is studied through stability region,bifurcation diagram,maximum Lyapunov exponent diagram and attraction basin.The results show that the system enters chaos through flip and Neimark-Sacker bifurcation,and increase of price adjustment speed and service input value will make the system produce more dynamic behavior.In addition,it can be found that the impact of service input value on itself is much greater than that on manufacturers.Secondly,when adjustment speed is selected as bifurcation parameter,the change curve of sales price is inconsistent,in which the change of retailers mainly remains in periodic state,while manufacturers will gradually enter chaos.Finally,studies the evolution of attraction basin in which three kinds of attractors coexist.In particular,coexistence of boundary attractors and internal attractors increases the complexity of system.Therefore,enterprises need to carefully adjust parameters of the game model to control the stability of system and maintain the long-term stability of market competition.展开更多
An autonomous five-dimensional(5D)system with offset boosting is constructed by modifying the well-known three-dimensional autonomous Liu and Chen system.Equilibrium points of the proposed autonomous 5D system are fou...An autonomous five-dimensional(5D)system with offset boosting is constructed by modifying the well-known three-dimensional autonomous Liu and Chen system.Equilibrium points of the proposed autonomous 5D system are found and its stability is analyzed.The proposed system includes Hopf bifurcation,periodic attractors,quasi-periodic attractors,a one-scroll chaotic attractor,a double-scroll chaotic attractor,coexisting attractors,the bistability phenomenon,offset boosting with partial amplitude control,reverse period-doubling,and an intermittency route to chaos.Using a field programmable gate array(FPGA),the proposed autonomous 5D system is implemented and the phase portraits are presented to check the numerical simulation results.The chaotic attractors and coexistence of the attractors generated by the FPGA implementation of the proposed system have good qualitative agreement with those found during the numerical simulation.Finally,a sound data encryption and communication system based on the proposed autonomous 5D chaotic system is designed and illustrated through a numerical example.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 62366014 and 61961019)the Natural Science Foundation of Jiangxi Province, China (Grant No. 20232BAB202008)。
文摘This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other existing MMCSs, the most eye-catching point of the proposed MMCS is that the amplitude of the wing will enlarge towards the poles as the number of wings increases. Diverse coexisting attractors are numerically found in the MMCS, including chaos,quasi-period, and stable point. The circuits of the proposed memristor and MMCS are designed and the obtained results demonstrate their validity and reliability.
基金National Natural Science Foundation of China(Grant Nos.11672257,11632008,11772306,and 11972173)the Natural Science Foundation of Jiangsu Province of China(Grant No.BK20161314)+1 种基金the 5th 333 High-level Personnel Training Project of Jiangsu Province of China(Grant No.BRA2018324)the Excellent Scientific and Technological Innovation Team of Jiangsu University.
文摘We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.
基金supported by the National Natural Science Foundation of China(Grant Nos.62201094,62271088 and 12172066)the Natural Science Foundation of Jiangsu Province,China(Grant No.BK20210850)+1 种基金the Scientific Research Foundation of Jiangsu Provincial Education Department,China(Grant No.22KJB510001)。
文摘It has been documented that a cyclic three-neuron-based neural network with resistive synaptic weights cannot exhibit chaos.Towards this end,a memristive cyclic three-neuron-based neural network is presented using a memristive weight to substitute a resistive weight.The memristive cyclic neural network always has five equilibrium points within the parameters of interest,and their stability analysis shows that they are one index-2 saddle-focus,two index-1 saddle-foci,and two stable node-foci,respectively.Dynamical analyses are performed for the memristive cyclic neural network by several numerical simulation methods.The results demonstrate that the memristor synapse-based neural network with the simplest cyclic connection can not only exhibit chaos,but also present global coexisting attractors composed of stable points and unstable periodic or chaotic orbits under different initial conditions.Besides,with the designed implementation circuit,Multisim circuit simulations and hardware experiments are executed to validate the numerical simulations.
文摘Purpose-The purpose of this paper is to investigate coexisting attractors,chaos control and synchronization in a self-exciting homopolar dynamo system in this paper.Design/methodology/approach-Two single controllers are designed and added to the proposed 3D autonomous chaotic system,and its stability at zero equilibrium point is guaranteed by applying an appropriate control signal based on the Lyapunov stability theory.Findings-Numerical simulations reveal that the proposed 3D dynamo system exhibits periodic oscillations,double-scroll chaotic attractors and coexisting attractors.Finally,a single controller is designed for the global asymptotic synchronization of a unidirectionally coupled identical 3D autonomous chaotic system.Originality/value-The derived results of this paper are new and complement some earlier works.The innovation concludes two points in this paper;coexisting attractors are foundthe and an appropriate control signal based on the Lyapunov stability theory is established.The ideas of this paper can be applied to investigate some other homopolar dynamo systems.
基金supported by the National Natural Science Foundation of China(Grant No.11671149)
文摘This paper introduces a four-dimensional (4D) segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcation and pitchfork bifurcation occur in the system. The ultimate bound is also estimated. Some numerical investigations are also exploited to demonstrate and visualize the corresponding theoretical results.
基金Project supported by the National Natural Science Foundations of China(Grant Nos.62171401 and 62071411).
文摘Research on discrete memristor-based neural networks has received much attention.However,current research mainly focuses on memristor–based discrete homogeneous neuron networks,while memristor-coupled discrete heterogeneous neuron networks are rarely reported.In this study,a new four-stable discrete locally active memristor is proposed and its nonvolatile and locally active properties are verified by its power-off plot and DC V–I diagram.Based on two-dimensional(2D)discrete Izhikevich neuron and 2D discrete Chialvo neuron,a heterogeneous discrete neuron network is constructed by using the proposed discrete memristor as a coupling synapse connecting the two heterogeneous neurons.Considering the coupling strength as the control parameter,chaotic firing,periodic firing,and hyperchaotic firing patterns are revealed.In particular,multiple coexisting firing patterns are observed,which are induced by different initial values of the memristor.Phase synchronization between the two heterogeneous neurons is discussed and it is found that they can achieve perfect synchronous at large coupling strength.Furthermore,the effect of Gaussian white noise on synchronization behaviors is also explored.We demonstrate that the presence of noise not only leads to the transition of firing patterns,but also achieves the phase synchronization between two heterogeneous neurons under low coupling strength.
基金supported by the National Natural Science Foundation of China(Grant Nos.11372224 and 11402039)the Fundamental Research Funds for Central Universities designated to Tongji University(Grant No.1330219127)
文摘We investigated the synchronization dynamics of a coupled neuronal system composed of two identical Chay model neurons. The Chay model Showed coexisting period-1 and period-2 bursting patterns as a parameter and initial values are varied. We simulated multiple periodic and chaotic bursting patterns with non-(NS), burst phase (BS), spike phase (SS), complete (CS), and lag synchronization states. When the coexisting behavior is near period-2 bursting, the transitions of synchronization states of the coupled system follows very complex transitions that begins with transitions between BS and SS, moves to transitions between CS and SS, and to CS. Most initial values lead to the CS state of period-2 bursting while only a few lead to the CS state of period-I bursting. When the coexisting behavior is near period-1 bursting, the transitions begin with NS, move to transitions between SS and BS, to transitions between SS and CS, and then to CS. Most initial values lead to the CS state of period-1 bursting but a few lead to the CS state of period-2 bursting. The BS was identified as chaos synchronization. The patterns for NS and transitions between BS and SS are insensitive to initial values. The patterns for transitions between CS and SS and the CS state are sensitive to them. The number of spikes per burst of non-CS bursting increases with increasing coupling strength. These results not only reveal the initial value- and parameter- dependent synchronization transitions of coupled systems with coexisting behaviors, but also facilitate interpretation of various bursting patterns and synchronization transitions generated in the nervous system with weak coupling strength.
基金Project supported by the National Natural Science Foundation of China(Grant No.51377124)the Science Fund for New Star of Youth Science and Technology of Shaanxi Province,China(Grant No.2016KJXX-40).
文摘The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper.The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix.Besides,complex dynamical behaviors are investigated in the system,such as coexisting attractors,hidden attractors,coexisting bifurcation modes,intermittent chaos,and multistability.From the theoretical analyses and numerical simulations,it is found that there are four different kinds of transient transition behaviors in the memcapacitive system.Finally,field programmable gate array(FPGA)is used to implement the proposed chaotic system.
基金Project supported by the Key Projects of Hunan Provincial Department of Education(Grant No.23A0133)the Natural Science Foundation of Hunan Province(Grant No.2022JJ30572)the National Natural Science Foundations of China(Grant No.62171401).
文摘Synaptic crosstalk is a prevalent phenomenon among neuronal synapses,playing a crucial role in the transmission of neural signals.Therefore,considering synaptic crosstalk behavior and investigating the dynamical behavior of discrete neural networks are highly necessary.In this paper,we propose a heterogeneous discrete neural network(HDNN)consisting of a three-dimensional KTz discrete neuron and a Chialvo discrete neuron.These two neurons are coupled mutually by two discrete memristors and the synaptic crosstalk is considered.The impact of crosstalk strength on the firing behavior of the HDNN is explored through bifurcation diagrams and Lyapunov exponents.It is observed that the HDNN exhibits different coexisting attractors under varying crosstalk strengths.Furthermore,the influence of different crosstalk strengths on the synchronized firing of the HDNN is investigated,revealing a gradual attainment of phase synchronization between the two discrete neurons as the crosstalk strength decreases.
基金Project supported by the National Natural Science Foundation of China(Grant No.61871230)the Natural Science Foundation of Jiangsu Province,China(Grant No.BK20181410)the Postgraduate Research and Practice Innovation Project of Jiangsu Province,China(Grant No.SJCX210350).
文摘By introducing a discrete memristor and periodic sinusoidal functions,a two-dimensional map with coexisting chaos and hyperchaos is constructed.Various coexisting chaotic and hyperchaotic attractors under different Lyapunov exponents are firstly found in this discrete map,along with which other regimes of coexistence such as coexisting chaos,quasiperiodic oscillation,and discrete periodic points are also captured.The hyperchaotic attractors can be flexibly controlled to be unipolar or bipolar by newly embedded constants meanwhile the amplitude can also be controlled in combination with those coexisting attractors.Based on the nonlinear auto-regressive model with exogenous inputs(NARX)for neural network,the dynamics of the memristive map is well predicted,which provides a potential passage in artificial intelligencebased applications.
基金Project supported by the National Natural Science Foundation of China (Grant No 10275053)
文摘Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.
基金support of the Brazil-ian research agencies,the National Council for Scientific and Technological Development (CNPq)(Nos. 301355/2018-5 and 200198/2022-0)FAPERJ-CNE (No. E-26/202.711/2018)+1 种基金FAPERJ Nota 10 (No. E-26/200.357/2020)CAPES (Finance code 001 and 88881.310620/2018-01)。
文摘In a global dynamic analysis,the coexisting attractors and their basins are the main tools to understand the system behavior and safety.However,both basins and attractors can be drastically influenced by uncertainties.The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors.Accordingly,analytical and numerical tools for calculation of nondeterministic global structures,namely attractors and basins,are proposed.First,based on the definition of the Perron-Frobenius,Koopman and Foias linear operators,a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases.In this context,the stochastic basins of attraction and attractors’distributions replace the usual basin and attractor concepts.Then,numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method.Sample results of the methodology are presented for a canonical dynamical system.
基金the china Macedonia Intergovernmental Scientific and Technological Cooperation Project[grant number[2019]22:6-8].
文摘Unlike dissipative systems,conservative systems do not have attractors and no attractor reconstruction occurs.Therefore,these systems are more suitable for application in image encryption.On the basis of above appoints,here we develop and propose a conservative system with infinite chaotic-like attractors.The conservative and chaotic characteristics and coexistence chaotic-like attractors are studied using Lyapunov exponents,Poincare maps,and numerical simulation.The results show that the coexistence of chaotic-like attractors has a more complex structure and dynamic behaviour than traditional ones.Additionally,the developed system is further used to design an encryption system for a digital image.Using the coexistence chaotic-like attractor sequence to scramble and diffuse the image can destroy the correlation of adjacent pixels and hide the information of all pixels.The feasibility and security of the encryption scheme are demonstrated through the analysis of key space,histogram,information entropy,key sensitivity and pixel correlation.
基金National Natural Science Foundation of China (No. 61863022)China Postdoctoral Science Foundation,China (No. 2017M623276)。
文摘The generalized dynamic Tullock contest model with two homogeneous participants is established, in which both players have the same valuation of winning rewards and losing rewards. Firstly, the unique symmetric equilibrium point of the system is obtained by calculation and its local stability condition is given based on the Jury criterion. Then, two paths of the system from stability to chaos, namely flip bifurcation and Neimark-Sacker bifurcation, are analyzed by using the two-dimensional parametric bifurcation diagram. Meanwhile, the abundant Arnold tongues in the two-dimensional parametric bifurcation diagram are analyzed. Finally, the phenomenon of multistability of the system is illustrated through the basin of attraction, and the contact bifurcation occurs during the evolution of the basin of attraction with varying parameters.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 61703247 and 61703246)the Qingdao Science and Technology Plan Project, China (Grant No. 19-6-2-2-cg)the Elite Project of Shandong University of Science and Technology, and the Taishan Scholar Project of Shandong Province of China
文摘A two-SBT-memristor-based chaotic circuit was proposed. The stability of the equilibrium point was studied by theoretical analysis. The close dependence of the circuit dynamic characteristics on its initial conditions and circuit parameters was investigated by utilizing Lyapunov exponents spectra, bifurcation diagrams, phase diagrams, and Poincaré maps. The analysis showed that the circuit system had complex dynamic behaviors, such as stable points, period, chaos, limit cycles,and so on. In particular, the chaotic circuit produced the multistability phenomenon, such as coexisting attractors and coexisting periods.
基金Project supported by the National Natural Science Foundation of China(Nos.11872169 and 11472097)the Fundamental Research Funds for the Central Universities of China(No.B200202114)the Natural Science Foundation of Jiangsu Province of China(No.BK20191295)。
文摘Multiplex networks have drawn much attention since they have been observed in many systems,e.g.,brain,transport,and social relationships.In this paper,the nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied.The stability and bifurcation of the network equilibrium are discussed,and interesting neural activities of the network are explored.Based on the neuron circuit,transfer function circuit,and time delay circuit,a circuit platform of the network is constructed.It is shown that delayed couplings play crucial roles in the network dynamics,e.g.,the enhancement and suppression of the stability,the patterns of the synchronization between networks,and the generation of complicated attractors and multi-stability coexistence.
基金supported by the Foundation of a Hundred Youth Talents Training Program of Lanzhou Jiaotong University,the Innovation Fund Project of Colleges and Universities in Gansu Province under Grant No.2021A-040.
文摘Based on bounded rationality,this paper established a price game model of dual channel supply chain composed of manufacturers and retailers.According to the eigenvalue of Jacobi matrix and Jury criterion,the stability of the equilibrium point is analyzed,and then the dynamic evolution process under the parameters of price adjustment speed and retailer's service input is studied through stability region,bifurcation diagram,maximum Lyapunov exponent diagram and attraction basin.The results show that the system enters chaos through flip and Neimark-Sacker bifurcation,and increase of price adjustment speed and service input value will make the system produce more dynamic behavior.In addition,it can be found that the impact of service input value on itself is much greater than that on manufacturers.Secondly,when adjustment speed is selected as bifurcation parameter,the change curve of sales price is inconsistent,in which the change of retailers mainly remains in periodic state,while manufacturers will gradually enter chaos.Finally,studies the evolution of attraction basin in which three kinds of attractors coexist.In particular,coexistence of boundary attractors and internal attractors increases the complexity of system.Therefore,enterprises need to carefully adjust parameters of the game model to control the stability of system and maintain the long-term stability of market competition.
文摘An autonomous five-dimensional(5D)system with offset boosting is constructed by modifying the well-known three-dimensional autonomous Liu and Chen system.Equilibrium points of the proposed autonomous 5D system are found and its stability is analyzed.The proposed system includes Hopf bifurcation,periodic attractors,quasi-periodic attractors,a one-scroll chaotic attractor,a double-scroll chaotic attractor,coexisting attractors,the bistability phenomenon,offset boosting with partial amplitude control,reverse period-doubling,and an intermittency route to chaos.Using a field programmable gate array(FPGA),the proposed autonomous 5D system is implemented and the phase portraits are presented to check the numerical simulation results.The chaotic attractors and coexistence of the attractors generated by the FPGA implementation of the proposed system have good qualitative agreement with those found during the numerical simulation.Finally,a sound data encryption and communication system based on the proposed autonomous 5D chaotic system is designed and illustrated through a numerical example.