Pub Date : 2023-12-30DOI: 10.1142/s0218127423300392
K. Thamilmaran, T. Thamilvizhi, S. Kumarasamy, Premraj Durairaj
In this study, we investigate the occurrence of dragon-king extreme events in a three-dimensional autonomous Shimizu–Morioka oscillator. We observe that the bounded chaotic oscillations transition into large amplitude extreme events at a critical value of the system control parameter triggered by an interior crisis. These extreme events exhibit a unique distribution characterized by the probability distribution function. We performed laboratory experiments and conducted rigorous numerical simulations on the Shimizu–Morioka oscillator to validate our findings. The results from both approaches are in excellent agreement and confirm extreme behavior in this autonomous system. Our study represents the first comprehensive investigation of extreme events in the Shimizu–Morioka oscillator, integrating experimental observations and numerical simulations. Also, we observed the dragon-king extreme events in both experimental and numerical studies. These findings enhance our understanding of extreme events and their potential applications in chaos-based dynamical systems, contributing to advancing this field.
{"title":"Experimental Observation of Extreme Events in the Shimizu Morioka Oscillator","authors":"K. Thamilmaran, T. Thamilvizhi, S. Kumarasamy, Premraj Durairaj","doi":"10.1142/s0218127423300392","DOIUrl":"https://doi.org/10.1142/s0218127423300392","url":null,"abstract":"In this study, we investigate the occurrence of dragon-king extreme events in a three-dimensional autonomous Shimizu–Morioka oscillator. We observe that the bounded chaotic oscillations transition into large amplitude extreme events at a critical value of the system control parameter triggered by an interior crisis. These extreme events exhibit a unique distribution characterized by the probability distribution function. We performed laboratory experiments and conducted rigorous numerical simulations on the Shimizu–Morioka oscillator to validate our findings. The results from both approaches are in excellent agreement and confirm extreme behavior in this autonomous system. Our study represents the first comprehensive investigation of extreme events in the Shimizu–Morioka oscillator, integrating experimental observations and numerical simulations. Also, we observed the dragon-king extreme events in both experimental and numerical studies. These findings enhance our understanding of extreme events and their potential applications in chaos-based dynamical systems, contributing to advancing this field.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 44","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139141438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-30DOI: 10.1142/s0218127423501870
Yu-Han Zhang, Tao Zhang
This paper considers a dual-channel closed-loop supply chain (CLSC) consisting of a manufacturer who wholesales new products through the traditional retail channel and distributes remanufactured products via a direct (online) channel established by himself. Two dynamical Stackelberg game models are developed based on the assumption that the retailer is an adaptive player, and the manufacturer is a bounded rational player who may adopt a delay decision. The existence and locally asymptotic stability of the Nash equilibrium are examined. Moreover, the impacts of key parameters on the complexity characteristics of the models and the performance of chain members are studied by numerical simulation. The results reveal that the excessively fast price adjustment speeds of the manufacturer, the larger consumers’ discount perception for the remanufactured products, and the consumers’ preference for the direct channel have a strong destabilizing effect on the Nash equilibrium’s stability. Furthermore, the delay decision implemented by the manufacturer could be a stabilizing or destabilizing factor for the system. The manufacturer will tolerate a considerable profit reduction while the retailer gains more profits when the dual-channel CLSC system enters periodic cycles and chaotic motions.
{"title":"Dynamic Analysis of a Dual-Channel Closed-Loop Supply Chain with Heterogeneous Players and a Delay Decision","authors":"Yu-Han Zhang, Tao Zhang","doi":"10.1142/s0218127423501870","DOIUrl":"https://doi.org/10.1142/s0218127423501870","url":null,"abstract":"This paper considers a dual-channel closed-loop supply chain (CLSC) consisting of a manufacturer who wholesales new products through the traditional retail channel and distributes remanufactured products via a direct (online) channel established by himself. Two dynamical Stackelberg game models are developed based on the assumption that the retailer is an adaptive player, and the manufacturer is a bounded rational player who may adopt a delay decision. The existence and locally asymptotic stability of the Nash equilibrium are examined. Moreover, the impacts of key parameters on the complexity characteristics of the models and the performance of chain members are studied by numerical simulation. The results reveal that the excessively fast price adjustment speeds of the manufacturer, the larger consumers’ discount perception for the remanufactured products, and the consumers’ preference for the direct channel have a strong destabilizing effect on the Nash equilibrium’s stability. Furthermore, the delay decision implemented by the manufacturer could be a stabilizing or destabilizing factor for the system. The manufacturer will tolerate a considerable profit reduction while the retailer gains more profits when the dual-channel CLSC system enters periodic cycles and chaotic motions.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 38","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139141711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-30DOI: 10.1142/s0218127423501869
Yizi Cheng, Fuhong Min
In this paper, a type of modified dual memristive Shinriki oscillator is constructed with a flux-controlled absolute-type memristor and a voltage-controlled generic memristor, and the proposed oscillator with abundant dynamical behaviors, including the multistability and antimonotonicity, is comprehensively studied through dynamical distribution graphs, bifurcation diagrams, Lyapunov exponents and phase portraits. It is found that the passive/active state of memristor, which means different characteristics in the [Formula: see text]–[Formula: see text] domain with positive and negative parameters of the elements, can affect the state of the oscillator. For example, if the memristor is active, the oscillator will change more frequently in the multistable region. Also, it is noted that, for inherent initial-related symmetry and circuit structures with duality, both phenomena have strong symmetric characteristics and opposite evolution trends modulated by values of corresponding components. Especially, the bubbles, which are symmetric about parameters with duality and own complex evolution laws, have rarely been explored in previous works. In addition, the memristive oscillator is modularized based on field programmable gate array (FPGA) technology, and the multiple coexisting attractors are captured, which verifies the accuracy of the numerical results.
{"title":"Symmetry, Multistability and Antimonotonicity of a Shinriki Oscillator with Dual Memristors","authors":"Yizi Cheng, Fuhong Min","doi":"10.1142/s0218127423501869","DOIUrl":"https://doi.org/10.1142/s0218127423501869","url":null,"abstract":"In this paper, a type of modified dual memristive Shinriki oscillator is constructed with a flux-controlled absolute-type memristor and a voltage-controlled generic memristor, and the proposed oscillator with abundant dynamical behaviors, including the multistability and antimonotonicity, is comprehensively studied through dynamical distribution graphs, bifurcation diagrams, Lyapunov exponents and phase portraits. It is found that the passive/active state of memristor, which means different characteristics in the [Formula: see text]–[Formula: see text] domain with positive and negative parameters of the elements, can affect the state of the oscillator. For example, if the memristor is active, the oscillator will change more frequently in the multistable region. Also, it is noted that, for inherent initial-related symmetry and circuit structures with duality, both phenomena have strong symmetric characteristics and opposite evolution trends modulated by values of corresponding components. Especially, the bubbles, which are symmetric about parameters with duality and own complex evolution laws, have rarely been explored in previous works. In addition, the memristive oscillator is modularized based on field programmable gate array (FPGA) technology, and the multiple coexisting attractors are captured, which verifies the accuracy of the numerical results.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 6","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139140992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-30DOI: 10.1142/s0218127423300409
Y. Miino, T. Ueta
This study investigates a structurally unstable synchronization phenomenon observed in the two-coupled Izhikevich neuron model. As the result of varying the system parameter in the region of parameter space close to where the unstable synchronization is observed, we find significant changes in the stability of its periodic motion. We derive a discrete-time dynamical system that is equivalent to the original model and reveal that the unstable synchronization in the continuous-time dynamical system is equivalent to border-collision bifurcations in the corresponding discrete-time system. Furthermore, we propose an objective function that can be used to obtain the parameter set at which the border-collision bifurcation occurs. The proposed objective function is numerically differentiable and can be solved using Newton’s method. We numerically generate a bifurcation diagram in the parameter plane, including the border-collision bifurcation sets. In the diagram, the border-collision bifurcation sets show a novel bifurcation structure that resembles the “strike-slip fault” observed in geology. This structure implies that, before and after the border-collision bifurcation occurs, the stability of the periodic point discontinuously changes in some cases but maintains in other cases. In addition, we demonstrate that a border-collision bifurcation set successively branches at distinct points. This behavior results in a tree-like structure being observed in the border-collision bifurcation diagram; we refer to this structure as a border-collision bifurcation tree. We observe that a periodic point disappears at the border-collision bifurcation in the discrete-time dynamical system and is simultaneously replaced by another periodic point; this phenomenon corresponds to a change in the firing order in the continuous-time dynamical system.
{"title":"Structurally Unstable Synchronization and Border-Collision Bifurcations in the Two-Coupled Izhikevich Neuron Model","authors":"Y. Miino, T. Ueta","doi":"10.1142/s0218127423300409","DOIUrl":"https://doi.org/10.1142/s0218127423300409","url":null,"abstract":"This study investigates a structurally unstable synchronization phenomenon observed in the two-coupled Izhikevich neuron model. As the result of varying the system parameter in the region of parameter space close to where the unstable synchronization is observed, we find significant changes in the stability of its periodic motion. We derive a discrete-time dynamical system that is equivalent to the original model and reveal that the unstable synchronization in the continuous-time dynamical system is equivalent to border-collision bifurcations in the corresponding discrete-time system. Furthermore, we propose an objective function that can be used to obtain the parameter set at which the border-collision bifurcation occurs. The proposed objective function is numerically differentiable and can be solved using Newton’s method. We numerically generate a bifurcation diagram in the parameter plane, including the border-collision bifurcation sets. In the diagram, the border-collision bifurcation sets show a novel bifurcation structure that resembles the “strike-slip fault” observed in geology. This structure implies that, before and after the border-collision bifurcation occurs, the stability of the periodic point discontinuously changes in some cases but maintains in other cases. In addition, we demonstrate that a border-collision bifurcation set successively branches at distinct points. This behavior results in a tree-like structure being observed in the border-collision bifurcation diagram; we refer to this structure as a border-collision bifurcation tree. We observe that a periodic point disappears at the border-collision bifurcation in the discrete-time dynamical system and is simultaneously replaced by another periodic point; this phenomenon corresponds to a change in the firing order in the continuous-time dynamical system.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 29","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139141837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-30DOI: 10.1142/s0218127423501936
Chenyuan Tian, Shangjiang Guo
In this paper, we consider a single-species reaction–diffusion–advection population model with nonlinear boundary condition in heterogenous space. We not only investigate the existence, nonexistence and stability of positive steady-state solutions through a linear elliptic eigenvalue problem by means of variational approach, but also verify the existence of steady-state bifurcations at zero solution through Crandall and Robinowitz bifurcation theory and discuss the stability of bifurcations, which can lead to Allee effect when the bifurcation is subcritical.
{"title":"Dynamics of a Reaction–Diffusion–Advection System with Nonlinear Boundary Conditions","authors":"Chenyuan Tian, Shangjiang Guo","doi":"10.1142/s0218127423501936","DOIUrl":"https://doi.org/10.1142/s0218127423501936","url":null,"abstract":"In this paper, we consider a single-species reaction–diffusion–advection population model with nonlinear boundary condition in heterogenous space. We not only investigate the existence, nonexistence and stability of positive steady-state solutions through a linear elliptic eigenvalue problem by means of variational approach, but also verify the existence of steady-state bifurcations at zero solution through Crandall and Robinowitz bifurcation theory and discuss the stability of bifurcations, which can lead to Allee effect when the bifurcation is subcritical.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 3","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139137401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-30DOI: 10.1142/s0218127423501882
Yingying Zhang, Ruijuan Niu
In this study, we extend an SIS epidemic model by introducing a piecewise smooth incidence rate. By assuming that the demographic parameters are much smaller than the disease-related ones, the proposed model is converted to a slow–fast system. Utilizing the geometrical singular perturbation theory and entry-exit function, we prove the coexistence of two relaxation oscillations surrounding the unique positive equilibrium of the model. Numerical simulations are performed to verify our theoretical results. The phenomenon presented in this study can be a potential explanation for that several infectious diseases can re-emerge many years after being almost extinct.
{"title":"Relaxation Oscillations in an SIS Epidemic Model with a Nonsmooth Incidence","authors":"Yingying Zhang, Ruijuan Niu","doi":"10.1142/s0218127423501882","DOIUrl":"https://doi.org/10.1142/s0218127423501882","url":null,"abstract":"In this study, we extend an SIS epidemic model by introducing a piecewise smooth incidence rate. By assuming that the demographic parameters are much smaller than the disease-related ones, the proposed model is converted to a slow–fast system. Utilizing the geometrical singular perturbation theory and entry-exit function, we prove the coexistence of two relaxation oscillations surrounding the unique positive equilibrium of the model. Numerical simulations are performed to verify our theoretical results. The phenomenon presented in this study can be a potential explanation for that several infectious diseases can re-emerge many years after being almost extinct.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 28","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139139371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-30DOI: 10.1142/s0218127423501985
Xiao-Juan Liu, Song-Mei Huan
In this paper, we investigate the existence of one type of homoclinic double loops (i.e. figure-eight loops) in a family of planar sector-wise linear systems with saddle–saddle dynamics. We obtain necessary and sufficient conditions for the existence of a figure-eight loop. Moreover, we prove that such systems can have simultaneously three types of invariant sets: a figure-eight loop, a homoclinic loop and three different types of periodic orbits. We also provide an example to show that a crossing limit cycle can bifurcate from this figure-eight loop.
{"title":"Existence and Number of Figure-Eight Loops in Planar Sector-Wise Linear Systems with Saddle–Saddle Dynamics","authors":"Xiao-Juan Liu, Song-Mei Huan","doi":"10.1142/s0218127423501985","DOIUrl":"https://doi.org/10.1142/s0218127423501985","url":null,"abstract":"In this paper, we investigate the existence of one type of homoclinic double loops (i.e. figure-eight loops) in a family of planar sector-wise linear systems with saddle–saddle dynamics. We obtain necessary and sufficient conditions for the existence of a figure-eight loop. Moreover, we prove that such systems can have simultaneously three types of invariant sets: a figure-eight loop, a homoclinic loop and three different types of periodic orbits. We also provide an example to show that a crossing limit cycle can bifurcate from this figure-eight loop.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 36","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139137267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-30DOI: 10.1142/s021812742350195x
Yanqiu Li, Lei Zhang
The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.
{"title":"Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System","authors":"Yanqiu Li, Lei Zhang","doi":"10.1142/s021812742350195x","DOIUrl":"https://doi.org/10.1142/s021812742350195x","url":null,"abstract":"The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 27","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139141839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-11DOI: 10.1142/s0218127423501791
Xinhao Huang, Lijuan Chen, Yue Xia, Fengde Chen
In this paper, a predator–prey model in which the prey has the additive Allee effect and the predator has artificially controlled migration is proposed. When the system introduces additive Allee effect and artificially controlled migration, more complicated dynamical behavior is obtained. The system can undergo saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Two limit cycles are found and discussed. The influence of the additive Allee effect and artificially controlled migration on the dynamics of the system is also presented. In detail, when the Allee effect is large, the prey will become extinct. When the artificially controlled migration rate is larger, the intensity of the prey (pest) will be smaller and the intensity of the predator will be larger. This indicates that artificially controlled migration can be effectively used to control the pest.
{"title":"Dynamical Analysis of a Predator–Prey Model with Additive Allee Effect and Migration","authors":"Xinhao Huang, Lijuan Chen, Yue Xia, Fengde Chen","doi":"10.1142/s0218127423501791","DOIUrl":"https://doi.org/10.1142/s0218127423501791","url":null,"abstract":"In this paper, a predator–prey model in which the prey has the additive Allee effect and the predator has artificially controlled migration is proposed. When the system introduces additive Allee effect and artificially controlled migration, more complicated dynamical behavior is obtained. The system can undergo saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Two limit cycles are found and discussed. The influence of the additive Allee effect and artificially controlled migration on the dynamics of the system is also presented. In detail, when the Allee effect is large, the prey will become extinct. When the artificially controlled migration rate is larger, the intensity of the prey (pest) will be smaller and the intensity of the predator will be larger. This indicates that artificially controlled migration can be effectively used to control the pest.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"68 6","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138979108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-11DOI: 10.1142/s021812742350178x
Ning Xiao, Kuilin Wu
This paper is concerned with the number of limit cycles for a planar piecewise linear (PWL) system with two zones separated by a straight line. Assume that one of the subsystems of the PWL system has an improper node. The number of limit cycles for saddle-improper node type, focus-improper node type and center-improper node type (the focus or the center is a virtual or boundary equilibrium) are studied. First, we introduce displacement functions and study the number of zeros of displacement functions for different types. Then, we give the parameter regions where the exact number of limit cycles is one or two (at least two) for different types.
{"title":"Limit Cycles of a Planar Piecewise Linear System with an Improper Node","authors":"Ning Xiao, Kuilin Wu","doi":"10.1142/s021812742350178x","DOIUrl":"https://doi.org/10.1142/s021812742350178x","url":null,"abstract":"This paper is concerned with the number of limit cycles for a planar piecewise linear (PWL) system with two zones separated by a straight line. Assume that one of the subsystems of the PWL system has an improper node. The number of limit cycles for saddle-improper node type, focus-improper node type and center-improper node type (the focus or the center is a virtual or boundary equilibrium) are studied. First, we introduce displacement functions and study the number of zeros of displacement functions for different types. Then, we give the parameter regions where the exact number of limit cycles is one or two (at least two) for different types.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"17 6","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138980534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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