In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems: the Lorenz model, which possesses a single characteristic time scale, and the c...In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems: the Lorenz model, which possesses a single characteristic time scale, and the coupled Lorenz model, which possesses two different characteristic time scales. The limit of predictability is defined here as the time at which the error reaches 95% of its saturation level; nonlinear behaviors of the error growth are therefore involved in the definition of the limit of predictability. Our results show that the logarithmic function performs well in describing the relationship between the limit of predictability and initial error in both models, although the coefficients in the logarithmic function were not constant across the examined range of initial errors. Compared with the Lorenz model, in the coupled Lorenz model in which the slow dynamics and the fast dynamics interact with each other--there is a more complex relationship between the limit of predictability and initial error. The limit of predictability of the Lorenz model is unbounded as the initial error becomes infinitesimally small; therefore, the limit of predictability of the Lorenz model may be extended by reducing the amplitude of the initial error. In contrast, if there exists a fixed initial error in the fast dynamics of the coupled Lorenz model, the slow dynamics has an intrinsic finite limit of predictability that cannot be extended by reducing the amplitude of the initial error in the slow dynamics, and vice versa. The findings reported here reveal the possible existence of an intrinsic finite limit of predictability in a coupled system that possesses many scales of time or motion.展开更多
Four-dimensional variational(4D-VAR) data assimilation method is a perfect data assimilation solution in theory, but the computational issue is quite difficult in operational implementation.The incremental 4D-VAR assi...Four-dimensional variational(4D-VAR) data assimilation method is a perfect data assimilation solution in theory, but the computational issue is quite difficult in operational implementation.The incremental 4D-VAR assimilation scheme is set up in order to reduce the computational cost. It is shown that the accuracy of the observations, the length of the assimilation window and the choice of the first guess have an important influence on the assimilation outcome through the contrast experiment. Compared with the standard 4D-VAR assimilation scheme, the incremental 4D-VAR assimilation scheme shows its advantage in the computation speed through an assimilation experiment.展开更多
The theoretical basis and application of an analogue-dynamical model (ADM) in the Lorenz system is studied. The ADM can effectively combine statistical and dynamical methods in which the small disturbance of the cur...The theoretical basis and application of an analogue-dynamical model (ADM) in the Lorenz system is studied. The ADM can effectively combine statistical and dynamical methods in which the small disturbance of the current initial value superimposed on the historical analogue reference state can be regarded as a prediction objective. Primary analyses show that under the condition of appending disturbances in model parameters, the model errors of ADM are much smaller than those of the pure dynamical model (PDM). The characteristics of predictability on the ADM in the Lorenz system are analyzed in phase space by conducting case studies and global experiments. The results show that the ADM can quite effectively reduce prediction errors and prolong the valid time of the prediction in most situations in contrast to the PDM, but when model errors are considerably small, the latter will be superior to the former. To overcome such a problem, the multi-reference-state updating can be applied to introduce the information of multi-analogue and update analogue and can exhibit exciting performance in the ADM.展开更多
The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype...The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined. In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi-Sugeno (TS) fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex (two-wing) butterfly attractors observed from this system in the three dimensional (3D) X-Y-Z space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.展开更多
Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;back...Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is one of the interesting models because of the idea of consolidation of the two models<span style="font-family:Verdana;">:</span><span style="font-family:Verdana;"> Lorenz and <span style="white-space:nowrap;"><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span><span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""></span>ssler. This paper discusses the Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model from the bifurcation phenomena and the optimal control problem (OCP). The bifurcation property at the system equilibrium <img alt="" src="Edit_128925fa-e315-4db4-b9e4-9cd999342cb9.bmp" /> </span><span style="font-family:Verdana;">is studied and it is found that saddle-node and Hopf bifurcations can be holed under some conditions on the parameters. Also, the problem of the optimal control of Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is discussed and </span><span style="font-family:Verdana;">it </span><span style="font-family:Verdana;">u</span><span style="font-family:Verdana;">ses</span><span style="font-family:Verdana;"> the Pontryagin’s Maximum Principle (PMP) to derive the optimal control inputs that achieve the optimal trajectory. Numerical examples and solutions for bifurcation cases and the optimal controlled system are carried out and shown graphically to show the effectiveness of the used procedure.</span>展开更多
The breeding method has been widely used in studies of data assimilation, predictability and instabilities. The bred vectors (BVs), which are the nonlinear difference between the control and perturbed runs, represen...The breeding method has been widely used in studies of data assimilation, predictability and instabilities. The bred vectors (BVs), which are the nonlinear difference between the control and perturbed runs, represent the time-evolving rapidly growing errors in dynamic systems. The Lorenz (1963) model (hereafter Lorenz63 model) has chaotic dynamics similar to weather and climate. This study investigates the features of BVs of the Lorenz63 model and its impact on regime prediction of the Lorenz63 model. The results show that the Lorenz63 model has two different BVs for each breeding cycle, and the two BVs approach being identical when growth rate is high. The duration of the current and next regime is associated with the relative directions between the BV with high growth rate and the model trajectory.展开更多
Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they a...Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.展开更多
基金sprovided jointly by the 973 Program (Grant No.2010CB950400)National Natural Science Foundation of China (Grant Nos. 40805022 and 40821092)
文摘In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems: the Lorenz model, which possesses a single characteristic time scale, and the coupled Lorenz model, which possesses two different characteristic time scales. The limit of predictability is defined here as the time at which the error reaches 95% of its saturation level; nonlinear behaviors of the error growth are therefore involved in the definition of the limit of predictability. Our results show that the logarithmic function performs well in describing the relationship between the limit of predictability and initial error in both models, although the coefficients in the logarithmic function were not constant across the examined range of initial errors. Compared with the Lorenz model, in the coupled Lorenz model in which the slow dynamics and the fast dynamics interact with each other--there is a more complex relationship between the limit of predictability and initial error. The limit of predictability of the Lorenz model is unbounded as the initial error becomes infinitesimally small; therefore, the limit of predictability of the Lorenz model may be extended by reducing the amplitude of the initial error. In contrast, if there exists a fixed initial error in the fast dynamics of the coupled Lorenz model, the slow dynamics has an intrinsic finite limit of predictability that cannot be extended by reducing the amplitude of the initial error in the slow dynamics, and vice versa. The findings reported here reveal the possible existence of an intrinsic finite limit of predictability in a coupled system that possesses many scales of time or motion.
基金the National Basic Research Program of China under Natural contract Nos 2007CB816001 and 2006CB400603Natinal Natural Science Foundation of China under contract Nos 40346027 and 40676008the China"908"-Project under Grant No.908-02-01-03 and 908-IC-I-13
文摘Four-dimensional variational(4D-VAR) data assimilation method is a perfect data assimilation solution in theory, but the computational issue is quite difficult in operational implementation.The incremental 4D-VAR assimilation scheme is set up in order to reduce the computational cost. It is shown that the accuracy of the observations, the length of the assimilation window and the choice of the first guess have an important influence on the assimilation outcome through the contrast experiment. Compared with the standard 4D-VAR assimilation scheme, the incremental 4D-VAR assimilation scheme shows its advantage in the computation speed through an assimilation experiment.
基金jointly supported by the National Natural Science Foundation of China (Grant Nos. 40805028, 40675039 and 40575036)the Meteorological Special Project (GYHY200806005)the National Science and Technology Support Program of China (2006BAC02B04 and 2007BAC29B03)
文摘The theoretical basis and application of an analogue-dynamical model (ADM) in the Lorenz system is studied. The ADM can effectively combine statistical and dynamical methods in which the small disturbance of the current initial value superimposed on the historical analogue reference state can be regarded as a prediction objective. Primary analyses show that under the condition of appending disturbances in model parameters, the model errors of ADM are much smaller than those of the pure dynamical model (PDM). The characteristics of predictability on the ADM in the Lorenz system are analyzed in phase space by conducting case studies and global experiments. The results show that the ADM can quite effectively reduce prediction errors and prolong the valid time of the prediction in most situations in contrast to the PDM, but when model errors are considerably small, the latter will be superior to the former. To overcome such a problem, the multi-reference-state updating can be applied to introduce the information of multi-analogue and update analogue and can exhibit exciting performance in the ADM.
基金Project partially supported by the Natural Science Foundation of Educational Committee of Anhui Province, China (Grant No 2006kj250B).
文摘The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined. In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi-Sugeno (TS) fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex (two-wing) butterfly attractors observed from this system in the three dimensional (3D) X-Y-Z space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.
文摘Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is one of the interesting models because of the idea of consolidation of the two models<span style="font-family:Verdana;">:</span><span style="font-family:Verdana;"> Lorenz and <span style="white-space:nowrap;"><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span><span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""></span>ssler. This paper discusses the Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model from the bifurcation phenomena and the optimal control problem (OCP). The bifurcation property at the system equilibrium <img alt="" src="Edit_128925fa-e315-4db4-b9e4-9cd999342cb9.bmp" /> </span><span style="font-family:Verdana;">is studied and it is found that saddle-node and Hopf bifurcations can be holed under some conditions on the parameters. Also, the problem of the optimal control of Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is discussed and </span><span style="font-family:Verdana;">it </span><span style="font-family:Verdana;">u</span><span style="font-family:Verdana;">ses</span><span style="font-family:Verdana;"> the Pontryagin’s Maximum Principle (PMP) to derive the optimal control inputs that achieve the optimal trajectory. Numerical examples and solutions for bifurcation cases and the optimal controlled system are carried out and shown graphically to show the effectiveness of the used procedure.</span>
基金supported by the ONR(Office of Naval Research)(Grant No.N00014-10-1-0557)the Civil,Mechanical and Manufacturing Innovation Division of the NSF(National Science Foundation)(Grant No.CMMI112585)NASA(National Aeronautic and Space Administration)(Grant No.5069UMNASAMI3G)
文摘The breeding method has been widely used in studies of data assimilation, predictability and instabilities. The bred vectors (BVs), which are the nonlinear difference between the control and perturbed runs, represent the time-evolving rapidly growing errors in dynamic systems. The Lorenz (1963) model (hereafter Lorenz63 model) has chaotic dynamics similar to weather and climate. This study investigates the features of BVs of the Lorenz63 model and its impact on regime prediction of the Lorenz63 model. The results show that the Lorenz63 model has two different BVs for each breeding cycle, and the two BVs approach being identical when growth rate is high. The duration of the current and next regime is associated with the relative directions between the BV with high growth rate and the model trajectory.
文摘Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.