The exponential stability of a class of switched systems containing stableand unstable subsystems with impulsive effect is analyzed by using the matrix measure concept andthe average dwell-time approach. It is shown t...The exponential stability of a class of switched systems containing stableand unstable subsystems with impulsive effect is analyzed by using the matrix measure concept andthe average dwell-time approach. It is shown that if appropriately a large amount of the averagedwell-time and the ratio of the total activation time of the subsystems with negative matrix measureto the total activation time of the subsystems with nonnegative matrix measure is chosen, theexponential stability of a desired degree is guaranteed.Using the proposed switching scheme, westudied the robust exponential stability for a class of switched systems with impulsive effect andstructure perturbations.Simulations validate the main results.展开更多
Based on the total time derivative along the trajectory of the system, for noneonservative dynamical system, the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is studied. Fi...Based on the total time derivative along the trajectory of the system, for noneonservative dynamical system, the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is studied. Firstly, the Lie symmetry of the system is given. Then, the necessary and sumeient condition under which the Lie symmetry is a Mei symmetry is presented and the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is obtained. Lastly, an example is given to illustrate the application of the result.展开更多
In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53（1-2） （2015） 613-645] and [J. C. Larsen, The bistability theorem in a mo...In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53（1-2） （2015） 613-645] and [J. C. Larsen, The bistability theorem in a model of metastatic cancer, to appear in Appl. Math.]. I shall prove the bistability theorem for the ODE model from [Larsen, 2015]. It is a mass action kinetic system in the variables C cancer, GF growth factor and GI growth inhibitor. This theorem says that for some values of the parameters, there exist two positive singular points c＊＋ = （C＊＋, GF＊., GI＊＋）, c＊2- = （C＊-, GF＊, GI＊-） of the vector field. Here C＊- 〈 C＊＋ and e. is stable and c＊＋ is unstable, see Sec. 2. There is also a discrete model in [Larsen, 2015], it is a linear map （T） on three-dimensional Euclidean vector space with variables （C, GF, GI）, where these variables have the same meaning as in the ODE model above. In [Larsen, 2015], I showed that one can sometimes find attine vector fields on three-dimensional Euclidean vector space whose time one map is T. I shall also show this in the present paper in a more general setting than in [Larsen, 2015]. This enables me to find an expression for the rate of change of cancer growth on the coordinate hyperplane C = 0 in Euclidean vector space. I also present an ODE model of cancer metastasis with variables C, CM, CF,GI, where C is primary cancer and CM is metastatic cancer and GF, GI are growth factors and growth inhibitors, respectively.展开更多
文摘The exponential stability of a class of switched systems containing stableand unstable subsystems with impulsive effect is analyzed by using the matrix measure concept andthe average dwell-time approach. It is shown that if appropriately a large amount of the averagedwell-time and the ratio of the total activation time of the subsystems with negative matrix measureto the total activation time of the subsystems with nonnegative matrix measure is chosen, theexponential stability of a desired degree is guaranteed.Using the proposed switching scheme, westudied the robust exponential stability for a class of switched systems with impulsive effect andstructure perturbations.Simulations validate the main results.
文摘Based on the total time derivative along the trajectory of the system, for noneonservative dynamical system, the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is studied. Firstly, the Lie symmetry of the system is given. Then, the necessary and sumeient condition under which the Lie symmetry is a Mei symmetry is presented and the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is obtained. Lastly, an example is given to illustrate the application of the result.
文摘In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53（1-2） （2015） 613-645] and [J. C. Larsen, The bistability theorem in a model of metastatic cancer, to appear in Appl. Math.]. I shall prove the bistability theorem for the ODE model from [Larsen, 2015]. It is a mass action kinetic system in the variables C cancer, GF growth factor and GI growth inhibitor. This theorem says that for some values of the parameters, there exist two positive singular points c＊＋ = （C＊＋, GF＊., GI＊＋）, c＊2- = （C＊-, GF＊, GI＊-） of the vector field. Here C＊- 〈 C＊＋ and e. is stable and c＊＋ is unstable, see Sec. 2. There is also a discrete model in [Larsen, 2015], it is a linear map （T） on three-dimensional Euclidean vector space with variables （C, GF, GI）, where these variables have the same meaning as in the ODE model above. In [Larsen, 2015], I showed that one can sometimes find attine vector fields on three-dimensional Euclidean vector space whose time one map is T. I shall also show this in the present paper in a more general setting than in [Larsen, 2015]. This enables me to find an expression for the rate of change of cancer growth on the coordinate hyperplane C = 0 in Euclidean vector space. I also present an ODE model of cancer metastasis with variables C, CM, CF,GI, where C is primary cancer and CM is metastatic cancer and GF, GI are growth factors and growth inhibitors, respectively.
