Pub Date : 2023-03-30DOI: 10.1142/s0218127423500402
Junhai Ma, Dexia Wang, Xiao Li, Bingnan Zhang
Based on China’s green energy development strategy, this paper constructs a basic model of recycling and a channel expansion model for the circular economy foundation of Zaozhuang, Shandong Province, China. Through numerical simulation, it is found that each member of the supply chain should control the rate of price adjustment, otherwise it will cause market disruption. The model is controlled based on a chaos control method. Then, based on the fuzzy comprehensive evaluation method, an early warning system for the circular economy of Zaozhuang City is constructed. It is found that the economic development of Zaozhuang is a serious warning, resources are moderate warning, and the environment is not in an alarm state. In addition to paying attention to energy conservation and emission reduction of enterprises, the government should pay attention to creating awareness of energy conservation and emission reduction in society, and strengthen the technological investment in reducing pollutant emissions. This paper provides a strategic reference for the circular economy model in Zaozhuang, Shandong, China.
{"title":"Inherent Complexity and Early Warning of Zaozhuang Circular Economy System","authors":"Junhai Ma, Dexia Wang, Xiao Li, Bingnan Zhang","doi":"10.1142/s0218127423500402","DOIUrl":"https://doi.org/10.1142/s0218127423500402","url":null,"abstract":"Based on China’s green energy development strategy, this paper constructs a basic model of recycling and a channel expansion model for the circular economy foundation of Zaozhuang, Shandong Province, China. Through numerical simulation, it is found that each member of the supply chain should control the rate of price adjustment, otherwise it will cause market disruption. The model is controlled based on a chaos control method. Then, based on the fuzzy comprehensive evaluation method, an early warning system for the circular economy of Zaozhuang City is constructed. It is found that the economic development of Zaozhuang is a serious warning, resources are moderate warning, and the environment is not in an alarm state. In addition to paying attention to energy conservation and emission reduction of enterprises, the government should pay attention to creating awareness of energy conservation and emission reduction in society, and strengthen the technological investment in reducing pollutant emissions. This paper provides a strategic reference for the circular economy model in Zaozhuang, Shandong, China.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86278910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.1142/s0218127423300094
Zhigang Zhu, Xiaofeng Zhang, Yisen Wang, Jun Ma
It is important for functional neurons of animals or human beings to adapt to external stimuli, such as sound, pressure, and light. Regarding this aspect, autaptic neuron enables itself to utilize historical information to modulate its instant dynamics, such that it may be able to behave adaptively. In this paper, a FitzHugh–Nagumo based autaptic neuron is employed to investigate the capability of a sound-sensitive neural circuit’s adaptation and filtering to analog acoustic signals. Extensive simulations are performed for excitatory and inhibitory types of autaptic neurons. The results show that the time-delayed feedback of the excitatory chemical autapse can be tuned to play the role of a narrow-band filter in response to a broadband acoustic signal. While the excitatory chemical autaptic neuron cannot saturate its response amplitude due to its positive feedback gain, the inhibitory chemical autapse can drive the neuron’s amplitude to converge as the intensity of external drive increases, which reveals the capability of adaptation. What’s more, the inhibitory chemical autaptic neuron can also exhibit a novel bursting adaptation, in which the number of spikings contained in one bursting changes as the electrical activity evolves. For electrical autaptic neurons, it is also found that both time-delay feedback gains can effectively modulate the response of neuron to acoustic signal. While the variation of time-lags mainly changes the spiking rates of the excitatory electrical autaptic neuron, the feedback gain alters its response amplitude. Lastly, by carefully tuning the time-lags, the expected subthreshold dynamics for larger inhibitory feedback gains can be switched to nearby quasi-periodic firings, which implies a competing relation between the time-delays and the feedback gains in the spiking dynamics of the inhibitory electrical autaptic neurons. The diverse emerging phenomena are expected to facilitate the design of online or interactive learning artificial neural networks with these functional autaptic neurons.
{"title":"Functional Responses of Autaptic Neural Circuits to Acoustic Signals","authors":"Zhigang Zhu, Xiaofeng Zhang, Yisen Wang, Jun Ma","doi":"10.1142/s0218127423300094","DOIUrl":"https://doi.org/10.1142/s0218127423300094","url":null,"abstract":"It is important for functional neurons of animals or human beings to adapt to external stimuli, such as sound, pressure, and light. Regarding this aspect, autaptic neuron enables itself to utilize historical information to modulate its instant dynamics, such that it may be able to behave adaptively. In this paper, a FitzHugh–Nagumo based autaptic neuron is employed to investigate the capability of a sound-sensitive neural circuit’s adaptation and filtering to analog acoustic signals. Extensive simulations are performed for excitatory and inhibitory types of autaptic neurons. The results show that the time-delayed feedback of the excitatory chemical autapse can be tuned to play the role of a narrow-band filter in response to a broadband acoustic signal. While the excitatory chemical autaptic neuron cannot saturate its response amplitude due to its positive feedback gain, the inhibitory chemical autapse can drive the neuron’s amplitude to converge as the intensity of external drive increases, which reveals the capability of adaptation. What’s more, the inhibitory chemical autaptic neuron can also exhibit a novel bursting adaptation, in which the number of spikings contained in one bursting changes as the electrical activity evolves. For electrical autaptic neurons, it is also found that both time-delay feedback gains can effectively modulate the response of neuron to acoustic signal. While the variation of time-lags mainly changes the spiking rates of the excitatory electrical autaptic neuron, the feedback gain alters its response amplitude. Lastly, by carefully tuning the time-lags, the expected subthreshold dynamics for larger inhibitory feedback gains can be switched to nearby quasi-periodic firings, which implies a competing relation between the time-delays and the feedback gains in the spiking dynamics of the inhibitory electrical autaptic neurons. The diverse emerging phenomena are expected to facilitate the design of online or interactive learning artificial neural networks with these functional autaptic neurons.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74632294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.1142/s0218127423500426
Ziqiang Zhu, Pingping Chen, Zhijian Lin, Haoyu Chen, Yi Fang, Yong Li
This paper proposes a novel [Formula: see text]-ary differential permutation index (DPI) differential chaos shift keying (DCSK) by using the Bahl–Cocke–Jeline–Raviv (BCJR) decoding (DPI-DCSK-BCJR) for chaos-based communications. By exploiting the characteristics of differential modulation and quasi-orthogonality of chaotic signals, the proposed DPI-DCSK-BCJR can provide two more calculated energies to decide the transmitted bits. Theoretical analysis of bit-error-rate (BER) is carried out over multipath Rayleigh fading channel and then validated via simulations. The proposed scheme is demonstrated to outperform the conventional DPI-DCSK by 2[Formula: see text]dB for different [Formula: see text]. This performance gain is especially increased over high-delay channels. Furthermore, the advantage of the proposed frequency-modulated DPI-DCSK-BCJR is verified over practical ultra-wideband (UWB) wireless channels. In addition, we investigate the proposed DPI-DCSK-BCJR for more generalized multiuser communications. The simulation result shows that DPI-DCSK-BCJR can achieve performance gains up to 5[Formula: see text]dB as compared to the conventional PI-DCSK and DPI-DCSK for both two and three user systems. Consequently, the proposed DPI-DCSK-BCJR can be considered as a low-cost alternative for low-complexity and high-rate chaotic-based wireless communication.
{"title":"DPI DCSK Modulation with BCJR Decoding","authors":"Ziqiang Zhu, Pingping Chen, Zhijian Lin, Haoyu Chen, Yi Fang, Yong Li","doi":"10.1142/s0218127423500426","DOIUrl":"https://doi.org/10.1142/s0218127423500426","url":null,"abstract":"This paper proposes a novel [Formula: see text]-ary differential permutation index (DPI) differential chaos shift keying (DCSK) by using the Bahl–Cocke–Jeline–Raviv (BCJR) decoding (DPI-DCSK-BCJR) for chaos-based communications. By exploiting the characteristics of differential modulation and quasi-orthogonality of chaotic signals, the proposed DPI-DCSK-BCJR can provide two more calculated energies to decide the transmitted bits. Theoretical analysis of bit-error-rate (BER) is carried out over multipath Rayleigh fading channel and then validated via simulations. The proposed scheme is demonstrated to outperform the conventional DPI-DCSK by 2[Formula: see text]dB for different [Formula: see text]. This performance gain is especially increased over high-delay channels. Furthermore, the advantage of the proposed frequency-modulated DPI-DCSK-BCJR is verified over practical ultra-wideband (UWB) wireless channels. In addition, we investigate the proposed DPI-DCSK-BCJR for more generalized multiuser communications. The simulation result shows that DPI-DCSK-BCJR can achieve performance gains up to 5[Formula: see text]dB as compared to the conventional PI-DCSK and DPI-DCSK for both two and three user systems. Consequently, the proposed DPI-DCSK-BCJR can be considered as a low-cost alternative for low-complexity and high-rate chaotic-based wireless communication.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76684928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On the basis of global and BA scale-free coupled Stuart–Landau models, dynamic survivability has been investigated in detail with new definition and measure function, which is different from the previous survivability studies which only focused on static analysis. The effects on dynamic survivability of attractive–repulsive interaction and attack strategies are detected. Our results suggest that the coupling strength and presence of the repulsive interaction reduce the dynamic survivability in globally coupled systems. Furthermore, the dynamic survivability of the BA systems remains stable in the case of random attack with invariable critical attack cost [Formula: see text]. While they have the same features with globally coupled systems when being deliberately attacked; attacking high-degree oscillators show a tendency to spoil the dynamic survivability more effectively. Finally, it is found that the attractive coupling plays a more important role in the dynamic survivability. These findings may help us to prevent systems from being attacked and design survivable systems.
{"title":"Dynamic Survivability in Nonlinear Oscillation Systems with Attractive-Repulsive Interaction","authors":"Yuexin Wang, Zhongkui Sun, Shutong Liu, Yining Zhou, Wei Xu","doi":"10.1142/s0218127423500499","DOIUrl":"https://doi.org/10.1142/s0218127423500499","url":null,"abstract":"On the basis of global and BA scale-free coupled Stuart–Landau models, dynamic survivability has been investigated in detail with new definition and measure function, which is different from the previous survivability studies which only focused on static analysis. The effects on dynamic survivability of attractive–repulsive interaction and attack strategies are detected. Our results suggest that the coupling strength and presence of the repulsive interaction reduce the dynamic survivability in globally coupled systems. Furthermore, the dynamic survivability of the BA systems remains stable in the case of random attack with invariable critical attack cost [Formula: see text]. While they have the same features with globally coupled systems when being deliberately attacked; attacking high-degree oscillators show a tendency to spoil the dynamic survivability more effectively. Finally, it is found that the attractive coupling plays a more important role in the dynamic survivability. These findings may help us to prevent systems from being attacked and design survivable systems.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89377938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we show that any switching hypersurface of [Formula: see text]-dimensional continuous piecewise linear systems is an [Formula: see text]-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.
{"title":"Classification on Boundary-Equilibria and Singular Continuums of Continuous Piecewise Linear Systems","authors":"Hebai Chen, Zhaosheng Feng, Hao Yang, Linfeng zhou","doi":"10.1142/s0218127423500517","DOIUrl":"https://doi.org/10.1142/s0218127423500517","url":null,"abstract":"In this paper, we show that any switching hypersurface of [Formula: see text]-dimensional continuous piecewise linear systems is an [Formula: see text]-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88214383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.1142/s0218127423500505
Wujiu Pan, Liangyu Ling, Haoyong Qu, Minghai Wang
This paper considers the discontinuous characteristics of a real aero-engine rotor system, that is, the existence of bolted connection characteristics, and establishes a new bolted connection rotor system model. Taking into account the bending stiffness and the nonlinear Hertzian contact force of the rolling bearing, the Newmark-[Formula: see text] numerical method is used to solve the system response, and the influence of the bending stiffness on the system is studied. Moreover, the effects of bending stiffness and eccentricity on the system dynamics are analyzed. The results show that the nonlinear phenomena of the system are more abundant and the critical speed of the system is higher when the bending stiffness is involved. With the increase of bending stiffness, the critical speed of the system increases, and the frequency component of the system becomes more complex. Then, the influence of eccentricity on the system is studied based on the bending stiffness. It is found that the greater the eccentricity, the greater the critical speed of the rotor and the greater the amplitude of the main frequency. In the case of the same eccentricity, the main frequency increases as the rotational speed increases, and the frequency doubling component appears in the 2-period motion. This paper provides a basis for predicting the nonlinear response of bolted rotor-bearing system.
{"title":"Nonlinear Vibration of Bolted Rotor Bearing System Accounting for the Bending Stiffness Characteristics of the Connection Interface","authors":"Wujiu Pan, Liangyu Ling, Haoyong Qu, Minghai Wang","doi":"10.1142/s0218127423500505","DOIUrl":"https://doi.org/10.1142/s0218127423500505","url":null,"abstract":"This paper considers the discontinuous characteristics of a real aero-engine rotor system, that is, the existence of bolted connection characteristics, and establishes a new bolted connection rotor system model. Taking into account the bending stiffness and the nonlinear Hertzian contact force of the rolling bearing, the Newmark-[Formula: see text] numerical method is used to solve the system response, and the influence of the bending stiffness on the system is studied. Moreover, the effects of bending stiffness and eccentricity on the system dynamics are analyzed. The results show that the nonlinear phenomena of the system are more abundant and the critical speed of the system is higher when the bending stiffness is involved. With the increase of bending stiffness, the critical speed of the system increases, and the frequency component of the system becomes more complex. Then, the influence of eccentricity on the system is studied based on the bending stiffness. It is found that the greater the eccentricity, the greater the critical speed of the rotor and the greater the amplitude of the main frequency. In the case of the same eccentricity, the main frequency increases as the rotational speed increases, and the frequency doubling component appears in the 2-period motion. This paper provides a basis for predicting the nonlinear response of bolted rotor-bearing system.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76204730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.1142/s0218127423300082
Bhaben Kalita, S. K. Dwivedy
In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle subjected to time-varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales. The time and frequency response, phase portraits, and basin of attraction are plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters are evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters is illustrated.
{"title":"Parametrically Excited Nonlinear Pneumatic Artificial Muscle Under Hard Excitation: A Theoretical and Experimental Investigation","authors":"Bhaben Kalita, S. K. Dwivedy","doi":"10.1142/s0218127423300082","DOIUrl":"https://doi.org/10.1142/s0218127423300082","url":null,"abstract":"In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle subjected to time-varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales. The time and frequency response, phase portraits, and basin of attraction are plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters are evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters is illustrated.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73629301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 10.1142/s0218127423500487
D. Vinko, K. Miličević, Ivan Vidovic, Bruno Zoric
Chaotic systems are often considered to be a basis for various cryptographic methods due to their properties, which correspond to cryptographic properties like confusion, diffusion and algorithm (attack) complexity. In these kinds of applications, chaos robustness is desired. It can be defined by the absence of periodic windows and coexisting attractors in some neighborhoods of the parameter space. On the other hand, when used as a basis for neuromorphic modeling, chaos robustness is to be avoided, and the edge-of-chaos regime is needed. This paper analyses the robustness and edge-of-chaos for Chua’s systems, comprising either a piecewise linear or a smooth function nonlinearity, using a novel figure of merit based on correlation coefficient and Lyapunov exponent. Calculation complexity, which is important when a chaotic system is implemented, is evaluated for double and decimal data types, where needed calculation time varies by a factor of about 1500, depending on the nonlinearity function and the data type. On the other hand, different data types result in different number precision, which has some practical advantages and drawbacks presented in the paper.
{"title":"Chaos Robustness and Computation Complexity of Piecewise Linear and Smooth Chaotic Chua's System","authors":"D. Vinko, K. Miličević, Ivan Vidovic, Bruno Zoric","doi":"10.1142/s0218127423500487","DOIUrl":"https://doi.org/10.1142/s0218127423500487","url":null,"abstract":"Chaotic systems are often considered to be a basis for various cryptographic methods due to their properties, which correspond to cryptographic properties like confusion, diffusion and algorithm (attack) complexity. In these kinds of applications, chaos robustness is desired. It can be defined by the absence of periodic windows and coexisting attractors in some neighborhoods of the parameter space. On the other hand, when used as a basis for neuromorphic modeling, chaos robustness is to be avoided, and the edge-of-chaos regime is needed. This paper analyses the robustness and edge-of-chaos for Chua’s systems, comprising either a piecewise linear or a smooth function nonlinearity, using a novel figure of merit based on correlation coefficient and Lyapunov exponent. Calculation complexity, which is important when a chaotic system is implemented, is evaluated for double and decimal data types, where needed calculation time varies by a factor of about 1500, depending on the nonlinearity function and the data type. On the other hand, different data types result in different number precision, which has some practical advantages and drawbacks presented in the paper.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88983364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-15DOI: 10.1142/s0218127423500293
Yusen Wu, Feng Li
With the help of algebraic manipulator-Mathematica, we identify the order of weak centers at [Formula: see text] and the origin as well as the number of local critical periods in a [Formula: see text]-equivariant vector field of degree 5. We show that [Formula: see text] and the origin can be weak centers of infinite order (i.e. isochronous center) and at most fourth-order weak centers of finite order. Furthermore, we prove that at most four local critical periods bifurcate from the bicenter and the origin, respectively. Our approach is a combination of computational algebraic techniques.
{"title":"Weak Centers and Local Bifurcation of Critical Periods in a Z2-Equivariant Vector Field of Degree 5","authors":"Yusen Wu, Feng Li","doi":"10.1142/s0218127423500293","DOIUrl":"https://doi.org/10.1142/s0218127423500293","url":null,"abstract":"With the help of algebraic manipulator-Mathematica, we identify the order of weak centers at [Formula: see text] and the origin as well as the number of local critical periods in a [Formula: see text]-equivariant vector field of degree 5. We show that [Formula: see text] and the origin can be weak centers of infinite order (i.e. isochronous center) and at most fourth-order weak centers of finite order. Furthermore, we prove that at most four local critical periods bifurcate from the bicenter and the origin, respectively. Our approach is a combination of computational algebraic techniques.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87901084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-15DOI: 10.1142/s0218127423500359
M. Demina
We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.
给出了非退化近无穷多项式Levinson-Smith微分方程Liouvillian可积性的充分必要条件。这些方程推广了lisamadard方程，并用于描述自持续振荡。我们的结果对方程中出现的多项式的任意次都是有效的。我们发现了一些新的Liouvillian可积亚族。在非简并或代数简并的近无穷多项式Levinson-Smith方程中，我们推导了不可约达布多项式阶上的一个变量的上界。对非简并或代数简并的近无穷Rayleigh-Duffing-van der Pol方程，即三次Levinson-Smith方程，进行了Liouvillian第一积分的完全分类。
{"title":"The Darboux Polynomials and Integrability of Polynomial Levinson-Smith Differential Equations","authors":"M. Demina","doi":"10.1142/s0218127423500359","DOIUrl":"https://doi.org/10.1142/s0218127423500359","url":null,"abstract":"We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87094186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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