In this paper, we study the chaotic behaviors in a fractional order logistic delay system. We find that chaos exists in the fractional order logistic delay system with an order being less than 1. In addition, we numer...In this paper, we study the chaotic behaviors in a fractional order logistic delay system. We find that chaos exists in the fractional order logistic delay system with an order being less than 1. In addition, we numerically simulate the continuances of the chaotic behaviors in the logistic delay system with orders from 0.1 to 0.9. The lowest order we find to have chaos in this system is 0.1. Then we further investigate two methods in controlling the fractional order chaotic logistic delay system based on feedback. Finally, we investigate a lag synchronization scheme in this system. Numerical simulations show the effectiveness and feasibility of our approach.展开更多
This paper proposes a mechanism theory on regional development by using a modified Logistic model. It reveals regional evolution is an integration of fluctuation in temporal dimension and disparity in spatial dimensio...This paper proposes a mechanism theory on regional development by using a modified Logistic model. It reveals regional evolution is an integration of fluctuation in temporal dimension and disparity in spatial dimension. T = S model is established by using Logistic model to simulate the growth of per capita GDP in China from 1990 to 1999. The result shows that T=S model accurately simulates the tracks of economic growth.展开更多
Different types of the Logistic model are constructed based on a simple assumption that the microbial populations are all composed of homogeneous members and consequently, the condition of design for the initial value...Different types of the Logistic model are constructed based on a simple assumption that the microbial populations are all composed of homogeneous members and consequently, the condition of design for the initial value of these models has to be rather limited in the case of N(t_0)=N_0. Therefore, these models cannot distinguish the dynamic behavior of the populations possessing the same N0 from heteroge-neous phases. In fact, only a certain ratio of the cells in a population is dividing at any moment during growth progress, termed as θ, and thus, ddNt not only depends on N, but also on θ. So θ is a necessary element for the condition design of the initial value. Unfortunately, this idea has long been neglected in widely used growth models. However, combining together the two factors (N0 and θ ) into the initial value often leads to the complexity in the mathematical solution. This difficulty can be overcome by using instantaneous rates (Vinst) to express growth progress. Previous studies in our laboratory sug-gested that the Vinst curve of the bacterial populations all showed a Guassian function shape and thus, the different growth phases can be reasonably distinguished. In the present study, the Gaussian dis-tribution function was transformed approximately into an analytical form (0.5x ibxYi αe=20) that can be conveniently used to evaluate the growth parameters and in this way the intrinsic growth behavior of a bacterial species growing in heterogeneous phases can be estimated. In addition, a new method has been proposed, in this case, the lag period and the double time for a bacterial population can also be reasonably evaluated. This approach proposed could thus be expected to reveal important insight of bacterial population growth. Some aspects in modeling population growth are also discussed.展开更多
文摘In this paper, we study the chaotic behaviors in a fractional order logistic delay system. We find that chaos exists in the fractional order logistic delay system with an order being less than 1. In addition, we numerically simulate the continuances of the chaotic behaviors in the logistic delay system with orders from 0.1 to 0.9. The lowest order we find to have chaos in this system is 0.1. Then we further investigate two methods in controlling the fractional order chaotic logistic delay system based on feedback. Finally, we investigate a lag synchronization scheme in this system. Numerical simulations show the effectiveness and feasibility of our approach.
基金Supported by the Knowledge Innovation Program of Chinese Academy of Sciences and National Key Technologies R &D Program in the 10th Five-Ycar Plan of china(2001BA901A40)
文摘This paper proposes a mechanism theory on regional development by using a modified Logistic model. It reveals regional evolution is an integration of fluctuation in temporal dimension and disparity in spatial dimension. T = S model is established by using Logistic model to simulate the growth of per capita GDP in China from 1990 to 1999. The result shows that T=S model accurately simulates the tracks of economic growth.
基金Supported by the National Natural Science Foundation of China (Grant No. 30370013)National Basic Research Program of China (Grant No. 2004CB719702)
文摘Different types of the Logistic model are constructed based on a simple assumption that the microbial populations are all composed of homogeneous members and consequently, the condition of design for the initial value of these models has to be rather limited in the case of N(t_0)=N_0. Therefore, these models cannot distinguish the dynamic behavior of the populations possessing the same N0 from heteroge-neous phases. In fact, only a certain ratio of the cells in a population is dividing at any moment during growth progress, termed as θ, and thus, ddNt not only depends on N, but also on θ. So θ is a necessary element for the condition design of the initial value. Unfortunately, this idea has long been neglected in widely used growth models. However, combining together the two factors (N0 and θ ) into the initial value often leads to the complexity in the mathematical solution. This difficulty can be overcome by using instantaneous rates (Vinst) to express growth progress. Previous studies in our laboratory sug-gested that the Vinst curve of the bacterial populations all showed a Guassian function shape and thus, the different growth phases can be reasonably distinguished. In the present study, the Gaussian dis-tribution function was transformed approximately into an analytical form (0.5x ibxYi αe=20) that can be conveniently used to evaluate the growth parameters and in this way the intrinsic growth behavior of a bacterial species growing in heterogeneous phases can be estimated. In addition, a new method has been proposed, in this case, the lag period and the double time for a bacterial population can also be reasonably evaluated. This approach proposed could thus be expected to reveal important insight of bacterial population growth. Some aspects in modeling population growth are also discussed.
