Understanding the relationship between landscape pattems and ecological processes has been a central yet challenging research theme in landscape ecology. Over the past decades, many landscape metrics have been propose...Understanding the relationship between landscape pattems and ecological processes has been a central yet challenging research theme in landscape ecology. Over the past decades, many landscape metrics have been proposed but few directly incorporated ecological processes. In this paper, we developed a landscape index, namely, location-weighted landscape index （LWLI） to highlight the role of landscape type in ecological processes, such as nutrient losses and soil erosion. Within the framework of the Lorenz curve theory, we develop this index by integrating land- scape pattern and point-based measurements at a watershed scale. The index can be used to characterize the contribution of landscape pattern to ecological processes （e.g. nutrient losses） with respect to a specific monitoring point in a watershed. Through a case study on nutrient losses in an agricultural area in northeastern China, we found that nutrient losses tended to be higher for a watershed with a higher LWLI value, and vice versa. It implied that LWLI can be used to evaluate the potential risk of nutrient losses or soil erosion by comparing their values across watersheds. In addition, this index can be extended to characterize ecological processes, such as the effect of landscape pattern on wildlife inhabitation and urban heat island effect. Finally, we discuss several problems that should be paid attention to when applying this index to a heterogeneous landscape site.展开更多
This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assim...This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme （E4DVAR） benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence： the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.展开更多
Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies ...Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies the complex proofs of the two famous estimations presented by the Russian scholar Leonov. Our uniform formula can derive a series of the new estimations. Employing the idea of intersection in set theory, we extract a new Leonov formula-like estimation from the family of the estimations. With our method and the new estimation, one can confirm that there are no equilibrium, periodic solutions, almost periodic motions, wandering motions or other chaotic attractors outside the global attractive set. The Lorenz butterfly-like singular attractors are located in the global attractive set only. This result is applied to the chaos control and chaos synchronization. Some feedback control laws are obtained to guarantee that all the trajectories of the Lorenz systems track a periodic solution, or globally stabilize an unstable (or locally stable but not globally asymptotically stable) equilibrium. Further, some new global exponential chaos synchronization results are presented. Our new method and the new results are expected to be applied in real secure communication systems.展开更多
We propose a new image encryption algorithm on the basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded...We propose a new image encryption algorithm on the basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded in the proposed algorithm to enhance the security. Such an algorithm is detailed in terms of security analyses, including correlation analysis, information entropy analysis, run statistic analysis, mean-variance gray value analysis, and key sensitivity analysis. The experimental results demonstrate that the proposed image encryption scheme has the advantages of large key space and high security for practical image encryption.展开更多
In this paper, the concept of globally exponentially attractive set is proposed and used to consider the ultimate bounds of the family of Lorenz systems with varying parameters. Explicit estimations of the ultimate bo...In this paper, the concept of globally exponentially attractive set is proposed and used to consider the ultimate bounds of the family of Lorenz systems with varying parameters. Explicit estimations of the ultimate bounds are derived. The results presented in this paper contain all the existing results as special cases. In particular, the critical cases, b→ 1^＋ and a→0^＋, for which the previous methods failed, have been solved using a unified formula.展开更多
In this paper, taking the Lorenz system as an example, we compare the influences of the arithmetic mean and the geometric mean on measuring the global and local average error growth. The results show that the geometri...In this paper, taking the Lorenz system as an example, we compare the influences of the arithmetic mean and the geometric mean on measuring the global and local average error growth. The results show that the geometric mean error （GME） has a smoother growth than the arithmetic mean error （AME） for the global average error growth, and the GME is directly related to the maximal Lyapunov exponent, but the AME is not, as already noted by Krishnamurthy in 1993. Besides these, the GME is shown to be more appropriate than the AME in measuring the mean error growth in terms of the probability distribution of errors. The physical meanings of the saturation levels of the AME and the GME are also shown to be different. However, there is no obvious difference between the local average error growth with the arithmetic mean and the geometric mean, indicating that the choices of the AME or the GME have no influence on the measure of local average predictability.展开更多
The source and sink landscape patterns refer to landscape types or units that can either promote positive evolvement of non-point source(NPS) pollution process, or can prevent/defer the ecological process, respectivel...The source and sink landscape patterns refer to landscape types or units that can either promote positive evolvement of non-point source(NPS) pollution process, or can prevent/defer the ecological process, respectively. Therefore, the role of a catchment landscape pattern in nutrient losses can be identified based on the spatial arrangement of source and sink landscapes. To reveal the relations between landscape spatial characteristics and NPS pollution in small catchment, a case study was carried out in a Wangjiagou small catchment of the Three Gorges Reservoir Region(TGRR), China. Google earth imagery for 2015 were processed and used to differentiate source and sink landscape types, and six subcatchments were selected as sample regions for monitoring nitrogen and phosphorus nutrients.Relative elevation, slope gradient and relative flow length was used to construct the Lorenz curves of different source and sink landscape types in the catchment, in order to assess the source and sink landscape spatial characteristics. By calculating the location-weighted landscape indices of each subcatchment and total catchment, the landscape spatial load characteristics affecting the NPS pollution was identified, with a further Pearson correlation analysis for location-weighted landscape indices and nitrogen-phosphorus monitoring indicators. The analysis of Lorenz curve has revealed that the obtained distribution trend of Lorenz curve and curve area quantified well the spatial characteristics of source and sink landscape pattern related to the relative elevation, slope gradient and relative flow length in small catchment. Results of Pearson correction analysis indicated that location-weighted landscape index(LWLI) combining of terrain and landscape type factor did better in reflecting the status of nitrogen and phosphorus loss than the indices related to relative elevation, slope gradient and relative flow length.展开更多
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
The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point ...The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state;moreover the motion of phase point will not approach infinite at last;system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.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 chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to th...The chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to the circuit unit, an electronic circuit is designed to realize a 3.8-order generalized Lorenz chaotic system. Furthermore, synchronization between two fractional-order systems is achieved by utilizing a single-variable feedback method. Circuit experiment simulation results verify the effectiveness of the proposed scheme.展开更多
基金Under the auspices of Chinese Academy of Sciences (No. KZCX2-YW-421)National Natural Science Foundation of China (No. 40621061, 30570319)
文摘Understanding the relationship between landscape pattems and ecological processes has been a central yet challenging research theme in landscape ecology. Over the past decades, many landscape metrics have been proposed but few directly incorporated ecological processes. In this paper, we developed a landscape index, namely, location-weighted landscape index （LWLI） to highlight the role of landscape type in ecological processes, such as nutrient losses and soil erosion. Within the framework of the Lorenz curve theory, we develop this index by integrating land- scape pattern and point-based measurements at a watershed scale. The index can be used to characterize the contribution of landscape pattern to ecological processes （e.g. nutrient losses） with respect to a specific monitoring point in a watershed. Through a case study on nutrient losses in an agricultural area in northeastern China, we found that nutrient losses tended to be higher for a watershed with a higher LWLI value, and vice versa. It implied that LWLI can be used to evaluate the potential risk of nutrient losses or soil erosion by comparing their values across watersheds. In addition, this index can be extended to characterize ecological processes, such as the effect of landscape pattern on wildlife inhabitation and urban heat island effect. Finally, we discuss several problems that should be paid attention to when applying this index to a heterogeneous landscape site.
基金sponsored by the U.S. National Science Foundation (Grant No.ATM0205599)the U.S. Offce of Navy Research under Grant N000140410471Dr. James A. Hansen was partially supported by US Offce of Naval Research (Grant No. N00014-06-1-0500)
文摘This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme （E4DVAR） benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence： the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.60274007,60474011)the Guangdong Povince Science Foundation for Program of Research Team(Grant No.04205783).
文摘Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies the complex proofs of the two famous estimations presented by the Russian scholar Leonov. Our uniform formula can derive a series of the new estimations. Employing the idea of intersection in set theory, we extract a new Leonov formula-like estimation from the family of the estimations. With our method and the new estimation, one can confirm that there are no equilibrium, periodic solutions, almost periodic motions, wandering motions or other chaotic attractors outside the global attractive set. The Lorenz butterfly-like singular attractors are located in the global attractive set only. This result is applied to the chaos control and chaos synchronization. Some feedback control laws are obtained to guarantee that all the trajectories of the Lorenz systems track a periodic solution, or globally stabilize an unstable (or locally stable but not globally asymptotically stable) equilibrium. Further, some new global exponential chaos synchronization results are presented. Our new method and the new results are expected to be applied in real secure communication systems.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 61004078 and 60971022)the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2009GQ009 and ZR2009GM005)+1 种基金the China Postdoctoral Science Foundation (Grant No. 20100481293)the Special Funds for Postdoctoral Innovative Projects of Shandong Province, China (Grant No. 201003037)
文摘We propose a new image encryption algorithm on the basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded in the proposed algorithm to enhance the security. Such an algorithm is detailed in terms of security analyses, including correlation analysis, information entropy analysis, run statistic analysis, mean-variance gray value analysis, and key sensitivity analysis. The experimental results demonstrate that the proposed image encryption scheme has the advantages of large key space and high security for practical image encryption.
基金Supported partly by the National Natural Science Foundation of China (Grant Nos. 60474011 and 60274007)the National Natural Science Foun-dation of China for Excellent Youth (Grant No. 60325310)+2 种基金the Guangdong Province Science Foundation for Program of Research Team (Grant No. 04205783)the Natural Science Fund of Guangdong Province, China (Grant No. 05006508)the Natural Science and Engineering Re-search Council of Canada (Grant No. R2686A02)
文摘In this paper, the concept of globally exponentially attractive set is proposed and used to consider the ultimate bounds of the family of Lorenz systems with varying parameters. Explicit estimations of the ultimate bounds are derived. The results presented in this paper contain all the existing results as special cases. In particular, the critical cases, b→ 1^＋ and a→0^＋, for which the previous methods failed, have been solved using a unified formula.
基金Supported by the National Natural Science Foundation of China （40805022 and 40675046）the National Basic Research and Development （973） Program of China （2010CB950400）
文摘In this paper, taking the Lorenz system as an example, we compare the influences of the arithmetic mean and the geometric mean on measuring the global and local average error growth. The results show that the geometric mean error （GME） has a smoother growth than the arithmetic mean error （AME） for the global average error growth, and the GME is directly related to the maximal Lyapunov exponent, but the AME is not, as already noted by Krishnamurthy in 1993. Besides these, the GME is shown to be more appropriate than the AME in measuring the mean error growth in terms of the probability distribution of errors. The physical meanings of the saturation levels of the AME and the GME are also shown to be different. However, there is no obvious difference between the local average error growth with the arithmetic mean and the geometric mean, indicating that the choices of the AME or the GME have no influence on the measure of local average predictability.
基金funded by the National Natural Science Foundation of China (Grant No.41671291)
文摘The source and sink landscape patterns refer to landscape types or units that can either promote positive evolvement of non-point source(NPS) pollution process, or can prevent/defer the ecological process, respectively. Therefore, the role of a catchment landscape pattern in nutrient losses can be identified based on the spatial arrangement of source and sink landscapes. To reveal the relations between landscape spatial characteristics and NPS pollution in small catchment, a case study was carried out in a Wangjiagou small catchment of the Three Gorges Reservoir Region(TGRR), China. Google earth imagery for 2015 were processed and used to differentiate source and sink landscape types, and six subcatchments were selected as sample regions for monitoring nitrogen and phosphorus nutrients.Relative elevation, slope gradient and relative flow length was used to construct the Lorenz curves of different source and sink landscape types in the catchment, in order to assess the source and sink landscape spatial characteristics. By calculating the location-weighted landscape indices of each subcatchment and total catchment, the landscape spatial load characteristics affecting the NPS pollution was identified, with a further Pearson correlation analysis for location-weighted landscape indices and nitrogen-phosphorus monitoring indicators. The analysis of Lorenz curve has revealed that the obtained distribution trend of Lorenz curve and curve area quantified well the spatial characteristics of source and sink landscape pattern related to the relative elevation, slope gradient and relative flow length in small catchment. Results of Pearson correction analysis indicated that location-weighted landscape index(LWLI) combining of terrain and landscape type factor did better in reflecting the status of nitrogen and phosphorus loss than the indices related to relative elevation, slope gradient and relative flow length.
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.
文摘The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state;moreover the motion of phase point will not approach infinite at last;system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.
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supported by the Natural Science Foundation of Hebei Province,China (Grant Nos A2008000136 and A2006000128)
文摘The chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to the circuit unit, an electronic circuit is designed to realize a 3.8-order generalized Lorenz chaotic system. Furthermore, synchronization between two fractional-order systems is achieved by utilizing a single-variable feedback method. Circuit experiment simulation results verify the effectiveness of the proposed scheme.
