Pub Date : 2016-11-10DOI: 10.4236/IJMNTA.2016.54020
Chengfei Ai, Huixian Zhu, Guoguang Lin
This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.
{"title":"Approximate Inertial Manifold for a Class of the Kirchhoff Wave Equations with Nonlinear Strongly Damped Terms","authors":"Chengfei Ai, Huixian Zhu, Guoguang Lin","doi":"10.4236/IJMNTA.2016.54020","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54020","url":null,"abstract":"This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"218-234"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707411","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 : 2016-11-10DOI: 10.4236/IJMNTA.2016.54015
K. Kirimoto
The tidal data of Kushimoto and Uragami on flow path of Kuroshio from 2004 to 2005 are investigated and discussed by time-frequency methods and nonlinear methods in this paper. These analyzing methods based on mathematical science show us new findings about the tidal motion observed on Kuroshio flow path. On the time-frequency analysis, 12 hours component and 24 hours component swing during the period of 350 hours and 320 hours respectively. However, any remarkable differences or changes depending on Kuroshio flow path weren’t seen on the result of time-frequency analysis. On the nonlinear analysis, a periodical structure has seen on the mutual information of tidal difference data, while Kuroshio flow is stable. In addition, the mutual information showed a characteristic of randomness and irregularity, while Kuroshio flow is unstable. The important results brought us a new finding such as classification of tidal motion regardless of the flow path of Kuroshio.
{"title":"Time-Frequency and Nonlinear Analysis of Tidal Data Observed on the Kuroshio Path","authors":"K. Kirimoto","doi":"10.4236/IJMNTA.2016.54015","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54015","url":null,"abstract":"The tidal data of Kushimoto and Uragami on flow path of Kuroshio from 2004 to 2005 are investigated and discussed by time-frequency methods and nonlinear methods in this paper. These analyzing methods based on mathematical science show us new findings about the tidal motion observed on Kuroshio flow path. On the time-frequency analysis, 12 hours component and 24 hours component swing during the period of 350 hours and 320 hours respectively. However, any remarkable differences or changes depending on Kuroshio flow path weren’t seen on the result of time-frequency analysis. On the nonlinear analysis, a periodical structure has seen on the mutual information of tidal difference data, while Kuroshio flow is stable. In addition, the mutual information showed a characteristic of randomness and irregularity, while Kuroshio flow is unstable. The important results brought us a new finding such as classification of tidal motion regardless of the flow path of Kuroshio.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"147-159"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707208","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 : 2016-11-10DOI: 10.4236/IJMNTA.2016.54017
Mingshan Li, Xiumin Liu, Xiaoliang Zhou
In this paper, a class of discrete vertical and horizontal transmitted disease model under constant vaccination is researched. Under the hypothesis of population being constant size, the model is transformed into a planar map and its equilibrium points and the corresponding eigenvalues are solved out. By discussing the influence of coefficient parameters on the eigenvalues, the hyperbolicity of equilibrium points is determined. By getting the equations of flows on center manifold, the direction and stability of the transcritical bifurcation and flip bifurcation are discussed.
{"title":"The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination","authors":"Mingshan Li, Xiumin Liu, Xiaoliang Zhou","doi":"10.4236/IJMNTA.2016.54017","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54017","url":null,"abstract":"In this paper, a class of discrete vertical and horizontal transmitted disease model under constant vaccination is researched. Under the hypothesis of population being constant size, the model is transformed into a planar map and its equilibrium points and the corresponding eigenvalues are solved out. By discussing the influence of coefficient parameters on the eigenvalues, the hyperbolicity of equilibrium points is determined. By getting the equations of flows on center manifold, the direction and stability of the transcritical bifurcation and flip bifurcation are discussed.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"171-184"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707369","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 : 2016-11-10DOI: 10.4236/IJMNTA.2016.54018
Yunlong Gao, Yuting Sun, Guoguang Lin
In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.
{"title":"The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping","authors":"Yunlong Gao, Yuting Sun, Guoguang Lin","doi":"10.4236/IJMNTA.2016.54018","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54018","url":null,"abstract":"In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"185-202"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707539","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 : 2016-11-10DOI: 10.4236/IJMNTA.2016.54021
T. Nemaura
This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state at some particular concentration. The movement states’ description is found for the primary and secondary level. The primary system is found to be predominantly an unstable system while the secondary system is stable. This is derived from the state of dynamic eigenvalues associated with the system. The saturated solution-particle is projected to be stable both for the primary potential and secondary state. A volume conserving linear system has been suggested to describe the dynamical state of movement of a solution particle.
{"title":"Modelling the Dynamical State of the Projected Primary and Secondary Intra-Solution-Particle Movement System of Efavirenz In-Vivo","authors":"T. Nemaura","doi":"10.4236/IJMNTA.2016.54021","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54021","url":null,"abstract":"This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state at some particular concentration. The movement states’ description is found for the primary and secondary level. The primary system is found to be predominantly an unstable system while the secondary system is stable. This is derived from the state of dynamic eigenvalues associated with the system. The saturated solution-particle is projected to be stable both for the primary potential and secondary state. A volume conserving linear system has been suggested to describe the dynamical state of movement of a solution particle.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"235-247"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707187","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 : 2016-11-10DOI: 10.4236/IJMNTA.2016.54014
Yanan Xue, L. Nie
A delayed mathematical model of Dengue dynamical transmission between vector mosquitoes and human, incorporating a control strategy of perfect pediatric vaccination is proposed in this paper. By some analytical skills, we obtain the existence of disease-free equilibria and endemic equilibrium, the necessary conditions of global asymptotical stability about two disease-free equilibria. Further, by Pontryagin’s maximum principle, we obtain the optimal control of the disease. Finally, numerical simulations are carried out to verify the correctness of the theoretical results and feasibility of the control measure.
{"title":"A Model of Perfect Pediatric Vaccination of Dengue with Delay and Optimal Control","authors":"Yanan Xue, L. Nie","doi":"10.4236/IJMNTA.2016.54014","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54014","url":null,"abstract":"A delayed mathematical model of Dengue dynamical transmission between vector mosquitoes and human, incorporating a control strategy of perfect pediatric vaccination is proposed in this paper. By some analytical skills, we obtain the existence of disease-free equilibria and endemic equilibrium, the necessary conditions of global asymptotical stability about two disease-free equilibria. Further, by Pontryagin’s maximum principle, we obtain the optimal control of the disease. Finally, numerical simulations are carried out to verify the correctness of the theoretical results and feasibility of the control measure.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"133-146"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707135","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 : 2016-11-10DOI: 10.4236/IJMNTA.2016.54022
M. Al-Mazmumy, S. O. Almuhalbedi
The Adomian decomposition method (ADM) can be used to solve a wide range of problems and usually gets the solution in a series form. In this paper, we propose two-step Adomian Decomposition Method (TSAM) for nonlinear integro-differential equations that will facilitate the calculations. In this modification, compared to the standard Adomian decomposition method, the size of calculations was reduced. This modification also avoids computing Adomian polynomials. Numerical results are given to show the efficiency and performance of this method.
{"title":"Solution of Nonlinear Integro Differential Equations by Two-Step Adomian Decomposition Method (TSAM)","authors":"M. Al-Mazmumy, S. O. Almuhalbedi","doi":"10.4236/IJMNTA.2016.54022","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54022","url":null,"abstract":"The Adomian decomposition method (ADM) can be used to solve a wide range of problems and usually gets the solution in a series form. In this paper, we propose two-step Adomian Decomposition Method (TSAM) for nonlinear integro-differential equations that will facilitate the calculations. In this modification, compared to the standard Adomian decomposition method, the size of calculations was reduced. This modification also avoids computing Adomian polynomials. Numerical results are given to show the efficiency and performance of this method.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"248-255"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707928","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 : 2016-11-10DOI: 10.4236/IJMNTA.2016.54019
Yuting Sun, Yunlong Gao, Guoguang Lin
We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.
{"title":"The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term","authors":"Yuting Sun, Yunlong Gao, Guoguang Lin","doi":"10.4236/IJMNTA.2016.54019","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.54019","url":null,"abstract":"We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear \u0000damping σ and ?, we make assumptions (H1) \u0000- (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"203-217"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707250","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 : 2016-09-16DOI: 10.4236/IJMNTA.2016.53012
Guangqing Zhang
We consider the defocusing mass-critical nonlinear Schr?dinger equation in the exterior domain in (). By analyzing Strichartz estimate and utilizing the inductive hypothesis method, we prove the stability for all initial data in .
{"title":"On the Stability of the Defocusing Mass-Critical Nonlinear Schrödinger Equation","authors":"Guangqing Zhang","doi":"10.4236/IJMNTA.2016.53012","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.53012","url":null,"abstract":"We consider the defocusing mass-critical nonlinear Schr?dinger equation in the exterior domain in (). By analyzing Strichartz estimate and utilizing the inductive hypothesis method, we prove the stability for all initial data in .","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"115-121"},"PeriodicalIF":0.0,"publicationDate":"2016-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70707011","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 : 2016-09-16DOI: 10.4236/IJMNTA.2016.53013
Zhanhong Liang, Jinfeng Gao
This paper has numerically studied the dynamical behaviors of a fractional-order single-machine infinite-bus (FOSMIB) power system. Periodic motions, period- doubling bifurcations and chaotic attractors are observed in the FOSMIB power system. The existence of chaotic behavior is affirmed by the positive largest Lyapunov exponent (LLE). Based on the fractional-order backstepping method, an adaptive controller is proposed to suppress chaos in the FOSMIB power system. Numerical simulation results demonstrate the validity of the proposed controller.
{"title":"Chaos in a Fractional-Order Single-Machine Infinite-Bus Power System and Its Adaptive Backstepping Control","authors":"Zhanhong Liang, Jinfeng Gao","doi":"10.4236/IJMNTA.2016.53013","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.53013","url":null,"abstract":"This \u0000paper has numerically studied the dynamical behaviors of a fractional-order \u0000single-machine infinite-bus (FOSMIB) power system. Periodic motions, period- doubling \u0000bifurcations and chaotic attractors are observed in the FOSMIB power system. \u0000The existence of chaotic behavior is affirmed by the positive largest Lyapunov \u0000exponent (LLE). Based on the fractional-order backstepping method, an adaptive \u0000controller is proposed to suppress chaos in the FOSMIB power system. Numerical \u0000simulation results demonstrate the validity of the proposed controller.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"05 1","pages":"122-131"},"PeriodicalIF":0.0,"publicationDate":"2016-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706671","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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