Pub Date : 2021-07-14DOI: 10.4236/ijmnta.2021.103008
Suqi Ma
A van der Pol equation underlying state feedback control is investigated and the triple-zero bifurcation arises at the bifurcation point which is of codimension three singularity. By applying Schmidt-Lyapunov reduction method combined with center manifold analytical technique, the near approximating formal norm is derived at the triple-zero point. Hence after, as varying parameters continuously, the numerical simulation produces homoclinic bifurcation solutions appearing in system. In addition, the numerical simulation also exhibits the produced double-period limit cycle with chosen bifurcation parameters and the routes to chaos via period-doubling bifurcation are also verified.
研究了一种基于状态反馈控制的van der Pol方程，在具有余维三奇点的分岔点上产生了三零分岔。应用Schmidt-Lyapunov约简方法结合中心流形解析技术，导出了三零点处的近似形式范数。因此，随着参数的不断变化，数值模拟得到了系统中出现的同斜分岔解。此外，数值模拟还展示了在选择分岔参数的情况下产生的双周期极限环，并验证了通过双周期分岔到达混沌的路径。
{"title":"Triple-Zero Bifurcation of Van Der Pol Oscillator with Delay Feedback","authors":"Suqi Ma","doi":"10.4236/ijmnta.2021.103008","DOIUrl":"https://doi.org/10.4236/ijmnta.2021.103008","url":null,"abstract":"A van der Pol equation underlying state feedback control is investigated and the triple-zero bifurcation arises at the bifurcation point which is of codimension three singularity. By applying Schmidt-Lyapunov reduction method combined with center manifold analytical technique, the near approximating formal norm is derived at the triple-zero point. Hence after, as varying parameters continuously, the numerical simulation produces homoclinic bifurcation solutions appearing in system. In addition, the numerical simulation also exhibits the produced double-period limit cycle with chosen bifurcation parameters and the routes to chaos via period-doubling bifurcation are also verified.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47827428","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 : 2021-05-17DOI: 10.4236/IJMNTA.2021.102004
S. Ma
The HIV problem is studied by version of delay mathematical models which consider the apoptosis of uninfected CD4+ T cells which cultured with infected T cells in big volume. The opportunistic infection and the apoptosis of uninfected CD4+ T cells are caused directly or indirectly by a toxic substance produced from HIV genes. Ubiquitously, the nonlinear incidence rate brings forth the increasing number of infected CD4+ T cells with introduction of small time delay, and in addition, there also exists a natural time delay factor during the process of virus replication. With state feedback control of time delay, the bifurcating periodical oscillating phenomena is induced via Hopf bifurcation. Mathematically, with the geometrical criterion applied in the stability analysis of delay model, the critical threshold of Hopf bifurcation in multiple delay differential equations which satisfy the transversal condition is derived. By applying reduction dimensional method combined with the center manifold theory, the stability of the bifurcating periodical solution is analyzed by the perturbation near Hopf point.
{"title":"Bifurcation and Stability Analysis of HIV Infectious Model with Two Time Delays","authors":"S. Ma","doi":"10.4236/IJMNTA.2021.102004","DOIUrl":"https://doi.org/10.4236/IJMNTA.2021.102004","url":null,"abstract":"The HIV problem is studied by version of delay mathematical models which consider the apoptosis of uninfected CD4+ T cells which cultured with infected T cells in big volume. The opportunistic infection and the apoptosis of uninfected CD4+ T cells are caused directly or indirectly by a toxic substance produced from HIV genes. Ubiquitously, the nonlinear incidence rate brings forth the increasing number of infected CD4+ T cells with introduction of small time delay, and in addition, there also exists a natural time delay factor during the process of virus replication. With state feedback control of time delay, the bifurcating periodical oscillating phenomena is induced via Hopf bifurcation. Mathematically, with the geometrical criterion applied in the stability analysis of delay model, the critical threshold of Hopf bifurcation in multiple delay differential equations which satisfy the transversal condition is derived. By applying reduction dimensional method combined with the center manifold theory, the stability of the bifurcating periodical solution is analyzed by the perturbation near Hopf point.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44162514","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 : 2021-05-17DOI: 10.4236/IJMNTA.2021.102003
S. Ma
G0/G1 “gaps” joint the S phase and M phase to form the cell cycle. The dynamics of enzyme reaction to drive the target protein production in M phase is analyzed mathematically. Time delay is introduced since the signal transmission need time in G0/G1 “gaps” phase. Hopf bifurcation of DDEs model is analyzed by applying geometrical analytical method. The instability oscillating periodic solutions arise as subcritical Hopf bifurcation occurs. The Hysteresis phenomena of the limit cycle are also observed underlying the saddle-node bifurcation of the limit cycle. Due to stability switching, interestingly, the bifurcating periodical solution dies out near the vicinity of Hopf lines. By Lyapunov-Schmidt reduction scheme, the normal form is computed on the center manifold. Finally, it is verified that the theory analytical results are in coincidence with the numerical simulation.
{"title":"Bifurcation Analysis of the Regulatory Modules of the Mammalian M/G 1 Phase with Time Delay","authors":"S. Ma","doi":"10.4236/IJMNTA.2021.102003","DOIUrl":"https://doi.org/10.4236/IJMNTA.2021.102003","url":null,"abstract":"G0/G1 “gaps” joint the S phase and M phase to form the cell cycle. The dynamics of enzyme reaction to drive the target protein production in M phase is analyzed mathematically. Time delay is introduced since the signal transmission need time in G0/G1 “gaps” phase. Hopf bifurcation of DDEs model is analyzed by applying geometrical analytical method. The instability oscillating periodic solutions arise as subcritical Hopf bifurcation occurs. The Hysteresis phenomena of the limit cycle are also observed underlying the saddle-node bifurcation of the limit cycle. Due to stability switching, interestingly, the bifurcating periodical solution dies out near the vicinity of Hopf lines. By Lyapunov-Schmidt reduction scheme, the normal form is computed on the center manifold. Finally, it is verified that the theory analytical results are in coincidence with the numerical simulation.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43533204","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 : 2021-03-05DOI: 10.4236/IJMNTA.2021.101002
Suqi Ma
The observed dynamical property illustrates that state feedback control may stabilize invariant attractor to stable state in a simple version of hematopoietic stem cell model. The stability character of the positive steady state is analyzed by the computation of the rightmost characteristic roots in complex plane. Hopf bifurcation points are tracked as the roots curve crossing imaginary axis from the left half plane to the right half plane continuously. The bifurcation direction and stability of the bifurcating periodical solution are discussed by norm form computation combined with the center manifold theory. Furthermore, the numerical simulation verifies that instead of chaos, system is stabilized to period-1, 2, 3, 4 and period-7 periodical solutions in some delay windows, and the continuous of periodical solutions is also numerical simulated with varying free parameters continuously.
{"title":"Stability and Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model","authors":"Suqi Ma","doi":"10.4236/IJMNTA.2021.101002","DOIUrl":"https://doi.org/10.4236/IJMNTA.2021.101002","url":null,"abstract":"The observed dynamical property illustrates that state feedback control may stabilize invariant attractor to stable state in a simple version of hematopoietic stem cell model. The stability character of the positive steady state is analyzed by the computation of the rightmost characteristic roots in complex plane. Hopf bifurcation points are tracked as the roots curve crossing imaginary axis from the left half plane to the right half plane continuously. The bifurcation direction and stability of the bifurcating periodical solution are discussed by norm form computation combined with the center manifold theory. Furthermore, the numerical simulation verifies that instead of chaos, system is stabilized to period-1, 2, 3, 4 and period-7 periodical solutions in some delay windows, and the continuous of periodical solutions is also numerical simulated with varying free parameters continuously.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46307923","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 : 2021-03-05DOI: 10.4236/IJMNTA.2021.101001
A. E. M. Ahmed, M. Zohdy
In order to deal with unmodeled dynamics in large vehicle systems, which have an ill condition of the state matrix, the use of model order reduction methods is a good approach. This article presents a new construction of the sliding mode controller for singularly perturbed systems. The controller design is based on a linear diagonal transformation of the singularly perturbed model. Furthermore, the use of a single sliding mode controller designed for the slow component of the diagonalized system is investigated. Simulation results indicate the performance improvement of the proposed controllers.
{"title":"Sliding Mode Control for Singularly Perturbed Systems Using Accurate Reduced Model","authors":"A. E. M. Ahmed, M. Zohdy","doi":"10.4236/IJMNTA.2021.101001","DOIUrl":"https://doi.org/10.4236/IJMNTA.2021.101001","url":null,"abstract":"In order to deal with unmodeled dynamics in large vehicle systems, which have an ill condition of the state matrix, the use of model order reduction methods is a good approach. This article presents a new construction of the sliding mode controller for singularly perturbed systems. The controller design is based on a linear diagonal transformation of the singularly perturbed model. Furthermore, the use of a single sliding mode controller designed for the slow component of the diagonalized system is investigated. Simulation results indicate the performance improvement of the proposed controllers.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46399919","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 : 2021-01-01DOI: 10.4236/IJMNTA.2021.102005
Y. Aguilera-Carvajal, Yonathan Tapia Robledo, Sebastian Alexander Pérez Cortés
The current study is based on the DEM computer simulation of three experimental test devices with different dimensions to determine the difference in the results of the formation of shear and repose angles that the particles experience when grouped under the action of the gravitational force. In this respect, the experimental test devices with different height, width, and depth were geometrically modeled with iron pellet particles using morphology and a granulometric variation from 6 mm to 9 mm of equivalent diameter in its spherical shape. Depending on the results obtained, a reliable size of the experimental test device will be available to obtain the necessary data for a cor-rect adjustment of the calibration parameters for the DEM simulation of min-ing-metallurgical processes that use granulated material of iron pellet.
{"title":"DEM Simulation to Determine the Influence on the Experimental Results of Tests of Iron Pellets When the Dimensions of the Test Device Are Varied","authors":"Y. Aguilera-Carvajal, Yonathan Tapia Robledo, Sebastian Alexander Pérez Cortés","doi":"10.4236/IJMNTA.2021.102005","DOIUrl":"https://doi.org/10.4236/IJMNTA.2021.102005","url":null,"abstract":"The current study is based on the DEM computer simulation of three experimental test devices with different dimensions to determine the difference in the results of the formation of shear and repose angles that the particles experience when grouped under the action of the gravitational force. In this respect, the experimental test devices with different height, width, and depth were geometrically modeled with iron pellet particles using morphology and a granulometric variation from 6 mm to 9 mm of equivalent diameter in its spherical shape. Depending on the results obtained, a reliable size of the experimental test device will be available to obtain the necessary data for a cor-rect adjustment of the calibration parameters for the DEM simulation of min-ing-metallurgical processes that use granulated material of iron pellet.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707842","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 : 2021-01-01DOI: 10.4236/IJMNTA.2021.102006
N. Bedelova, A. Asanov, Zhypar Orozmamatova, Zhypargul D. Abdullaeva
The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.
{"title":"Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions","authors":"N. Bedelova, A. Asanov, Zhypar Orozmamatova, Zhypargul D. Abdullaeva","doi":"10.4236/IJMNTA.2021.102006","DOIUrl":"https://doi.org/10.4236/IJMNTA.2021.102006","url":null,"abstract":"The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707898","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 : 2020-11-04DOI: 10.4236/ijmnta.2020.94005
Guoguang Lin, Yalan Yang
The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.
{"title":"The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping","authors":"Guoguang Lin, Yalan Yang","doi":"10.4236/ijmnta.2020.94005","DOIUrl":"https://doi.org/10.4236/ijmnta.2020.94005","url":null,"abstract":"The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47259647","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 : 2020-11-04DOI: 10.4236/ijmnta.2020.94006
Wen-bing Wu, Rihua Liu
Although mechanical vibration is extremely complicated, each fault signal produced by it has its own inherent features, The distinction may be most prominent between the certain components hidden in those features and the same components of normal signals. Three-order cumulant can reduce the Gaussian background noise automatically and its complex formal includes different coupling information of its signal. In the experiment, through these different coupling modes, the same coupling components are fetched from specific fault signal and normal signal, then these components are used to diagnose that certain fault. The results show that the method can fetch the most prominent distinction between normal signal and the specific fault signal, so the specific fault diagnosis by this method is satisfactory.
{"title":"The Application of Three Order Cumulants Coupling Features in Fault Diagnosis","authors":"Wen-bing Wu, Rihua Liu","doi":"10.4236/ijmnta.2020.94006","DOIUrl":"https://doi.org/10.4236/ijmnta.2020.94006","url":null,"abstract":"Although mechanical vibration is extremely complicated, each fault signal produced by it has its own inherent features, The distinction may be most prominent between the certain components hidden in those features and the same components of normal signals. Three-order cumulant can reduce the Gaussian background noise automatically and its complex formal includes different coupling information of its signal. In the experiment, through these different coupling modes, the same coupling components are fetched from specific fault signal and normal signal, then these components are used to diagnose that certain fault. The results show that the method can fetch the most prominent distinction between normal signal and the specific fault signal, so the specific fault diagnosis by this method is satisfactory.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44900896","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 : 2020-09-18DOI: 10.4236/IJMNTA.2020.93004
Guoguang Lin, Jie Liu
In this paper, we studied a class of damped high order Beam equation stochas-tic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of ran-dom attractors.
{"title":"A Random Attractor Family of the High Order Beam Equations with White Noise","authors":"Guoguang Lin, Jie Liu","doi":"10.4236/IJMNTA.2020.93004","DOIUrl":"https://doi.org/10.4236/IJMNTA.2020.93004","url":null,"abstract":"In this paper, we studied a class of damped high order Beam equation stochas-tic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of ran-dom attractors.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42525711","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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