Pub Date : 2024-03-06DOI: 10.1142/s0218127424500342
P. Gokul, S. S. Mohanrasu, A. Kashkynbayev, R. Rakkiyappan
This paper delves into the topics of Finite-Time Stabilization (FTS) and Finite-Time Contractive Stabilization (FTCS) for Fractional-Order Nonlinear Systems (FONSs). To address these issues, we employ a State-Dependent Delayed Impulsive Controller (SDDIC). By leveraging both Lyapunov theory and impulsive control theory, we establish sufficient conditions for achieving stability criteria for fractional-order systems. Initially, we employ the aforementioned sufficient conditions to derive stability criteria for general FONSs within the SDDIC framework, employing Linear Matrix Inequality (LMI) techniques. Furthermore, we apply these theoretical findings to tackle the challenge of finite-time synchronization in fractional-order chaotic systems using the proposed SDDIC. We substantiate the efficacy of these theoretical advancements through numerical simulations that vividly demonstrate their capability to achieve finite-time synchronization in fractional-order cellular neural networks and fractional-order Chua’s circuits. Moreover, we introduce an innovative chaos-based multi-image encryption algorithm, thereby contributing significantly to the field. To ensure the algorithm’s robustness, we subject it to rigorous statistical tests, which confidently affirm its capacity to provide the requisite level of security.
{"title":"Finite-Time Synchronization of Fractional-Order Nonlinear Systems with State-Dependent Delayed Impulse Control","authors":"P. Gokul, S. S. Mohanrasu, A. Kashkynbayev, R. Rakkiyappan","doi":"10.1142/s0218127424500342","DOIUrl":"https://doi.org/10.1142/s0218127424500342","url":null,"abstract":"<p>This paper delves into the topics of Finite-Time Stabilization (FTS) and Finite-Time Contractive Stabilization (FTCS) for Fractional-Order Nonlinear Systems (FONSs). To address these issues, we employ a State-Dependent Delayed Impulsive Controller (SDDIC). By leveraging both Lyapunov theory and impulsive control theory, we establish sufficient conditions for achieving stability criteria for fractional-order systems. Initially, we employ the aforementioned sufficient conditions to derive stability criteria for general FONSs within the SDDIC framework, employing Linear Matrix Inequality (LMI) techniques. Furthermore, we apply these theoretical findings to tackle the challenge of finite-time synchronization in fractional-order chaotic systems using the proposed SDDIC. We substantiate the efficacy of these theoretical advancements through numerical simulations that vividly demonstrate their capability to achieve finite-time synchronization in fractional-order cellular neural networks and fractional-order Chua’s circuits. Moreover, we introduce an innovative chaos-based multi-image encryption algorithm, thereby contributing significantly to the field. To ensure the algorithm’s robustness, we subject it to rigorous statistical tests, which confidently affirm its capacity to provide the requisite level of security.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"44 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124780","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 : 2024-03-06DOI: 10.1142/s021812742450038x
Songan Hou, Haodong Wang, Denggui Fan, Ying Yu, Qingyun Wang
The transiting mechanism of abnormal brain functional activities, such as the epileptic seizures, has not been fully elucidated. In this study, we employ a probabilistic neural network model to investigate the impact of negative regulation, including negative connections and negative inputs, on the dynamical transition behavior of network dynamics. It is observed that negative connections significantly influence the transition behavior of the network, intensifying the oscillation of discharge probability, corresponding to uneven discharge and epileptic states. Negative inputs, within a certain range, exhibited a similar impact on the dynamic state of the network as negative connections, enhancing network oscillations and resulting in higher fragility. However, larger negative inputs can led to the disappearance of oscillations in the discharge probability, indicating a maintenance of lower fragility. We speculate that negative regulation may be an indispensable factor in the occurrence of epileptic seizures, and future research should give it due consideration.
{"title":"The Effects of Negative Regulation on the Dynamical Transition in Epileptic Network","authors":"Songan Hou, Haodong Wang, Denggui Fan, Ying Yu, Qingyun Wang","doi":"10.1142/s021812742450038x","DOIUrl":"https://doi.org/10.1142/s021812742450038x","url":null,"abstract":"<p>The transiting mechanism of abnormal brain functional activities, such as the epileptic seizures, has not been fully elucidated. In this study, we employ a probabilistic neural network model to investigate the impact of negative regulation, including negative connections and negative inputs, on the dynamical transition behavior of network dynamics. It is observed that negative connections significantly influence the transition behavior of the network, intensifying the oscillation of discharge probability, corresponding to uneven discharge and epileptic states. Negative inputs, within a certain range, exhibited a similar impact on the dynamic state of the network as negative connections, enhancing network oscillations and resulting in higher fragility. However, larger negative inputs can led to the disappearance of oscillations in the discharge probability, indicating a maintenance of lower fragility. We speculate that negative regulation may be an indispensable factor in the occurrence of epileptic seizures, and future research should give it due consideration.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140129806","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 : 2024-03-06DOI: 10.1142/s0218127424500354
Yousu Huang, Qigui Yang
Global bifurcations with sliding have rarely been studied in three-dimensional piecewise smooth systems. In this paper, codimension-2 bifurcations of nondegenerate sliding heteroclinic cycle are investigated in three-dimensional Filippov systems. Two cases of sliding heteroclinic cycle are discussed: connecting two saddle-foci, connecting one saddle-focus and one saddle. It is proved that at most one sliding homoclinic or one sliding periodic orbit can bifurcate from under certain conditions at the eigenvalues of the equilibria, but they cannot coexist. The asymptotic stability of the sliding periodic orbit and the structural feature of the bifurcation curves of homoclinic orbits are further studied. Finally, two numerical examples corresponding to cases and , respectively, are simulated to verify the theoretical results.
{"title":"Bifurcations of Sliding Heteroclinic Cycles in Three-Dimensional Filippov Systems","authors":"Yousu Huang, Qigui Yang","doi":"10.1142/s0218127424500354","DOIUrl":"https://doi.org/10.1142/s0218127424500354","url":null,"abstract":"<p>Global bifurcations with sliding have rarely been studied in three-dimensional piecewise smooth systems. In this paper, codimension-2 bifurcations of nondegenerate sliding heteroclinic cycle <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"normal\">Γ</mi></math></span><span></span> are investigated in three-dimensional Filippov systems. Two cases of sliding heteroclinic cycle are discussed: <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> connecting two saddle-foci, <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> connecting one saddle-focus and one saddle. It is proved that at most one sliding homoclinic or one sliding periodic orbit can bifurcate from <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"normal\">Γ</mi></math></span><span></span> under certain conditions at the eigenvalues of the equilibria, but they cannot coexist. The asymptotic stability of the sliding periodic orbit and the structural feature of the bifurcation curves of homoclinic orbits are further studied. Finally, two numerical examples corresponding to cases <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span>, respectively, are simulated to verify the theoretical results.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"35 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140129721","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/s0218127423501973
Lujie Ren, Lei Qin, Hadi Jahanshahi, Jun Mou
The neural network activation functions enable neural networks to have stronger fitting abilities and richer dynamical behaviors. In this paper, an improved fractional-order discrete tabu learning neuron (FODTLN) model map with a nonlinear periodic function as the activation function is proposed. The fixed points of the map are discussed. Then, the rich and complex dynamical behaviors of the map under different parameters and order conditions are investigated by using some common nonlinear dynamical analysis methods combined with the fractional-order approximate entropy method. Furthermore, it is found that fractional-order differential operators affect the generation of multiscrolls, and the model has infinitely many coexisting attractors obtained by changing the initial conditions. Interestingly, attractor growth and state transition are found. Finally, the map is implemented on the DSP hardware platforms to verify the realizability. The results show that the map exhibits complex and interesting dynamical behaviors. It provides a fundamental theory for the research of artificial neural networks.
{"title":"Infinitely Many Coexisting Attractors and Scrolls in a Fractional-Order Discrete Neuron Map","authors":"Lujie Ren, Lei Qin, Hadi Jahanshahi, Jun Mou","doi":"10.1142/s0218127423501973","DOIUrl":"https://doi.org/10.1142/s0218127423501973","url":null,"abstract":"The neural network activation functions enable neural networks to have stronger fitting abilities and richer dynamical behaviors. In this paper, an improved fractional-order discrete tabu learning neuron (FODTLN) model map with a nonlinear periodic function as the activation function is proposed. The fixed points of the map are discussed. Then, the rich and complex dynamical behaviors of the map under different parameters and order conditions are investigated by using some common nonlinear dynamical analysis methods combined with the fractional-order approximate entropy method. Furthermore, it is found that fractional-order differential operators affect the generation of multiscrolls, and the model has infinitely many coexisting attractors obtained by changing the initial conditions. Interestingly, attractor growth and state transition are found. Finally, the map is implemented on the DSP hardware platforms to verify the realizability. The results show that the map exhibits complex and interesting dynamical behaviors. It provides a fundamental theory for the research of artificial neural networks.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139140125","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/s0218127423501961
Jaume Llibre, C. Valls
We study discontinuous piecewise linear differential systems formed by linear centers and/or linear Hamiltonian saddles and separated by a nonregular straight line. There are two classes of limit cycles: the ones that intersect the separation line at two points and the ones that intersect the separation line in four points, named limit cycles of type [Formula: see text] and limit cycles of type [Formula: see text], respectively. We prove that the maximum numbers of limit cycles of types [Formula: see text] and [Formula: see text] are two and one, respectively. We show that all these upper bounds are reached providing explicit examples.
{"title":"The Extended 16th Hilbert Problem for Discontinuous Piecewise Systems Formed by Linear Centers and Linear Hamiltonian Saddles Separated by a Nonregular Line","authors":"Jaume Llibre, C. Valls","doi":"10.1142/s0218127423501961","DOIUrl":"https://doi.org/10.1142/s0218127423501961","url":null,"abstract":"We study discontinuous piecewise linear differential systems formed by linear centers and/or linear Hamiltonian saddles and separated by a nonregular straight line. There are two classes of limit cycles: the ones that intersect the separation line at two points and the ones that intersect the separation line in four points, named limit cycles of type [Formula: see text] and limit cycles of type [Formula: see text], respectively. We prove that the maximum numbers of limit cycles of types [Formula: see text] and [Formula: see text] are two and one, respectively. We show that all these upper bounds are reached providing explicit examples.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 14","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139141762","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/s0218127423501948
Sattwika Acharya, R. K. Upadhyay, Bapin Mondal
In the current study, we have formulated an [Formula: see text] epidemic model incorporating the influence of Allee effect on the dynamics of the model system in space and time. During the spread of the disease, inhibition among the susceptible and infected individuals is seen, which is measured using the Crowley–Martin type incidence function. Extensive analysis explores the system’s stability and different bifurcation scenarios, including Hopf, saddle-node, transcritical, and Bogdanov–Takens. We also examine how these bifurcations react to parameter variations within the proposed model. The temporal model has further been extended to a spatial one to investigate the impact of the Allee effect on the formation of patterns for different values of the cross-diffusion coefficient. The well-posedness of the stationary solution and the global stability of the spatial system are derived. Also, the Turing instability, Hopf, and Turing bifurcation are calculated considering the transmission rate as critical parameter. In a heterogeneous environment, the spatial distribution of the susceptible population shows a complex structure with flat tabular surface and a few almost circular holes in surface plots for both strong and weak Allee effects. To further explore the role of Allee effect on pattern development for various cross-diffusion coefficient values, the temporal model has been spatially extended. Additionally, the transmission rate is taken into account while calculating the Turing instability and Hopf and Turing bifurcations. We ran numerical simulations to validate our analytical results for both spatial and nonspatial models. Our current model system can explain the recent occurrence of respiratory distress syndrome in endangered penguin species.
{"title":"Impact of Allee Effect on the Spatio-Temporal Behavior of a Diffusive Epidemic Model in Heterogenous Environment","authors":"Sattwika Acharya, R. K. Upadhyay, Bapin Mondal","doi":"10.1142/s0218127423501948","DOIUrl":"https://doi.org/10.1142/s0218127423501948","url":null,"abstract":"In the current study, we have formulated an [Formula: see text] epidemic model incorporating the influence of Allee effect on the dynamics of the model system in space and time. During the spread of the disease, inhibition among the susceptible and infected individuals is seen, which is measured using the Crowley–Martin type incidence function. Extensive analysis explores the system’s stability and different bifurcation scenarios, including Hopf, saddle-node, transcritical, and Bogdanov–Takens. We also examine how these bifurcations react to parameter variations within the proposed model. The temporal model has further been extended to a spatial one to investigate the impact of the Allee effect on the formation of patterns for different values of the cross-diffusion coefficient. The well-posedness of the stationary solution and the global stability of the spatial system are derived. Also, the Turing instability, Hopf, and Turing bifurcation are calculated considering the transmission rate as critical parameter. In a heterogeneous environment, the spatial distribution of the susceptible population shows a complex structure with flat tabular surface and a few almost circular holes in surface plots for both strong and weak Allee effects. To further explore the role of Allee effect on pattern development for various cross-diffusion coefficient values, the temporal model has been spatially extended. Additionally, the transmission rate is taken into account while calculating the Turing instability and Hopf and Turing bifurcations. We ran numerical simulations to validate our analytical results for both spatial and nonspatial models. Our current model system can explain the recent occurrence of respiratory distress syndrome in endangered penguin species.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139138261","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/s0218127423501900
Xiaoyang Chen, Jun Mou, Yinghong Cao, Santo Banerjee
In this paper, a chaotic multiple-image encryption algorithm using block scrambling and dynamic DNA coding is designed. The algorithm can achieve simultaneous encryption of many different types of images. Firstly, a new block scrambling algorithm is proposed, which is divided into two scrambling modes according to the size of the initial block. And it combines with the cyclic shift operation to achieve scrambling. Secondly, dynamic DNA coding is utilized to diffuse images, which enhances the complexity and security of the proposed multiple-image encryption algorithm. Through the analysis of key space, correlation, information entropy, histogram, differential attacks and robustness, it is verified that the algorithm is safe and effective. Experimental results show that the new multiple-image encryption algorithm is suitable for multiple color and gray images encryption. The algorithm has excellent encryption performance, and can be applied in secure communication.
本文设计了一种使用块加扰和动态 DNA 编码的混沌多图像加密算法。该算法可实现多种不同类型图像的同时加密。首先，提出了一种新的块加扰算法，根据初始块的大小分为两种加扰模式。并结合循环移位操作实现加扰。其次，利用动态 DNA 编码对图像进行扩散，增强了所提多图像加密算法的复杂性和安全性。通过对密钥空间、相关性、信息熵、直方图、差分攻击和鲁棒性的分析，验证了算法的安全性和有效性。实验结果表明，新的多图像加密算法适用于多彩色和灰色图像加密。该算法具有优异的加密性能，可应用于安全通信领域。
{"title":"Chaotic Multiple-Image Encryption Algorithm Based on Block Scrambling and Dynamic DNA Coding","authors":"Xiaoyang Chen, Jun Mou, Yinghong Cao, Santo Banerjee","doi":"10.1142/s0218127423501900","DOIUrl":"https://doi.org/10.1142/s0218127423501900","url":null,"abstract":"In this paper, a chaotic multiple-image encryption algorithm using block scrambling and dynamic DNA coding is designed. The algorithm can achieve simultaneous encryption of many different types of images. Firstly, a new block scrambling algorithm is proposed, which is divided into two scrambling modes according to the size of the initial block. And it combines with the cyclic shift operation to achieve scrambling. Secondly, dynamic DNA coding is utilized to diffuse images, which enhances the complexity and security of the proposed multiple-image encryption algorithm. Through the analysis of key space, correlation, information entropy, histogram, differential attacks and robustness, it is verified that the algorithm is safe and effective. Experimental results show that the new multiple-image encryption algorithm is suitable for multiple color and gray images encryption. The algorithm has excellent encryption performance, and can be applied in secure communication.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 17","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139141333","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/s0218127423501894
Monireh Maleki, F. N. Rahatabad, Majid Pouladian
The aim of modeling musculoskeletal systems is to understand the mechanisms of locomotion control in terms of neurophysiology and neuroanatomy. The complexity and unique nature of neuromuscular systems, however, make control problems in these systems very challenging due to several characteristics including speed and precision. Thus, their investigation requires the use of simple and analyzable methods. Consequently, taking into account the central pattern generator’s (CPG) function, we attempted to create a structured chaotic model of how human joints and muscles interact for the purpose of facilitating gait and rehabilitation in patients with incomplete spinal cord injury. The four muscle groups used in this model are gluteus, and hip flexor groups for flexion and extension of the hip joints as well as hamstring muscles and vasti muscles for flexion and extension of the knee joint. The results indicate that the output of the chaotic model of muscle and joint interactions in a healthy state would be chaotic, while in the incomplete spinal cord injury state, it would remain a fixed point. For model rehabilitation, afferent nerve stimulation is used in a CPG model; based on the modeling results, by applying coefficients of 1.98, 2.21, and 3.1 to the values of Ia, II, and Ib afferent nerves, the incomplete spinal cord injury model state is changed from a fix-point to periodic in a permanent fashion, suggesting locomotion with rehabilitation in our model. Based on the results obtained from the chaotic model of muscle and joint interactions as well as the comparisons made with reference papers, it can be stated that this model has acceptable output while enjoying simple computations and can predict different norms.
{"title":"Chaotic Model of Muscle and Joint Interactions Based on CPG for Rehabilitation of Incomplete Spinal Cord Injury Patients","authors":"Monireh Maleki, F. N. Rahatabad, Majid Pouladian","doi":"10.1142/s0218127423501894","DOIUrl":"https://doi.org/10.1142/s0218127423501894","url":null,"abstract":"The aim of modeling musculoskeletal systems is to understand the mechanisms of locomotion control in terms of neurophysiology and neuroanatomy. The complexity and unique nature of neuromuscular systems, however, make control problems in these systems very challenging due to several characteristics including speed and precision. Thus, their investigation requires the use of simple and analyzable methods. Consequently, taking into account the central pattern generator’s (CPG) function, we attempted to create a structured chaotic model of how human joints and muscles interact for the purpose of facilitating gait and rehabilitation in patients with incomplete spinal cord injury. The four muscle groups used in this model are gluteus, and hip flexor groups for flexion and extension of the hip joints as well as hamstring muscles and vasti muscles for flexion and extension of the knee joint. The results indicate that the output of the chaotic model of muscle and joint interactions in a healthy state would be chaotic, while in the incomplete spinal cord injury state, it would remain a fixed point. For model rehabilitation, afferent nerve stimulation is used in a CPG model; based on the modeling results, by applying coefficients of 1.98, 2.21, and 3.1 to the values of Ia, II, and Ib afferent nerves, the incomplete spinal cord injury model state is changed from a fix-point to periodic in a permanent fashion, suggesting locomotion with rehabilitation in our model. Based on the results obtained from the chaotic model of muscle and joint interactions as well as the comparisons made with reference papers, it can be stated that this model has acceptable output while enjoying simple computations and can predict different norms.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 13","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139137739","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/s0218127423501924
Suvankar Majee, T. K. Kar, Soovoojeet Jana, D. K. Das, J. J. Nieto
In this study, we have developed a novel SIR epidemic model by incorporating fractional-order differential equations and utilizing saturated-type functions to describe both disease incidence and treatment. The intricate dynamical characteristics of the proposed model, encompassing the determination of the conditions for the existence of all possible feasible equilibria with their local and global stability criteria, are investigated thoroughly. The model undergoes backward bifurcation with respect to the parameter representing the side effects due to treatment. This phenomenon emphasizes the critical role of treatment control parameters in shaping epidemic outcomes. In addition, to understand the optimal role of the treatment in mitigating the disease prevalence and minimizing the associated cost, we investigated a fractional-order optimal control problem. To further visualize the analytical results, we have conducted simulation works considering feasible parameter values for the model. Finally, we have employed local and global sensitivity analysis techniques to identify the factors that have the greatest potential to reduce the impact of the disease.
在本研究中，我们结合分数阶微分方程，利用饱和型函数来描述疾病的发生和治疗，从而建立了一种新型 SIR 流行病模型。我们深入研究了所提出模型的复杂动态特性，包括确定所有可能的可行均衡的存在条件及其局部和全局稳定性标准。该模型在代表治疗副作用的参数方面发生了向后分叉。这一现象强调了治疗控制参数在形成流行病结果中的关键作用。此外，为了理解治疗在缓解疾病流行和最小化相关成本方面的最佳作用，我们研究了一个分数阶优化控制问题。为了使分析结果更加直观，我们考虑了模型的可行参数值，进行了模拟工作。最后，我们采用了局部和全局敏感性分析技术，以确定最有可能减少疾病影响的因素。
{"title":"Complex Dynamics and Fractional-Order Optimal Control of an Epidemic Model with Saturated Treatment and Incidence","authors":"Suvankar Majee, T. K. Kar, Soovoojeet Jana, D. K. Das, J. J. Nieto","doi":"10.1142/s0218127423501924","DOIUrl":"https://doi.org/10.1142/s0218127423501924","url":null,"abstract":"In this study, we have developed a novel SIR epidemic model by incorporating fractional-order differential equations and utilizing saturated-type functions to describe both disease incidence and treatment. The intricate dynamical characteristics of the proposed model, encompassing the determination of the conditions for the existence of all possible feasible equilibria with their local and global stability criteria, are investigated thoroughly. The model undergoes backward bifurcation with respect to the parameter representing the side effects due to treatment. This phenomenon emphasizes the critical role of treatment control parameters in shaping epidemic outcomes. In addition, to understand the optimal role of the treatment in mitigating the disease prevalence and minimizing the associated cost, we investigated a fractional-order optimal control problem. To further visualize the analytical results, we have conducted simulation works considering feasible parameter values for the model. Finally, we have employed local and global sensitivity analysis techniques to identify the factors that have the greatest potential to reduce the impact of the disease.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 17","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139138198","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/s0218127423501912
Yu Yang, Shijie Qin, Shijun Liao
A liquid bath vibrating vertically can lead to the emergence of a self-propelled walking droplet on its free surface, which can exhibit chaotic motion. It is well-known that trajectories of a chaotic system are sensitive to its initial condition, known as the “butterfly-effect”, while its statistics normally remain stable to small disturbances: this type of chaos is called “normal-chaos”. However, a concept called “ultra-chaos” has been recently introduced, whose statistical features are unstable, i.e. extremely sensitive to small disturbances. Up to now, a few examples of ultra-chaos have been reported. In this paper, the influence of tiny disturbances on the motion of walking droplet is investigated. It is found that both normal-chaos and ultra-chaos exist in the motion of the walking droplet. Different from the normal-chaotic motion, even the statistical properties of the droplet’s ultra-chaotic motion are sensitive to tiny disturbances. Therefore, this illustrates once again that ultra-chaos indeed exists widely and represents a higher disorder compared with normal-chaos. The ultra-chaos as a new concept can widen our knowledge about chaos and provide us with a new point of view to study chaotic properties. It should be emphasized that, for an ultra-chaos, it is impossible to repeat any results of its physical experiments or numerical simulations even in the meaning of statistics! Unfortunately, reproducibility is a corner stone of modern science. Thus, the paradigm of modern scientific research might be invalid for an ultra-chaotic system.
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