A mathematical framework for critical transitions: normal forms, variance and applications

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio

New Results in Discrete-Time Nonlinear Filtering

We consider a discrete-time linear system with correlated Gaussian plant and observation noises and non-Gaussian initial condition independent of the plant and observation noises. We firstly find a solution of the filtering problem; we find a representation for the conditional distribution of the state at time t given the observations up to time t - 1. This representation is in terms of a finite collection of easily- computable statistics. With this solution to the filtering problem, we then find representations for the MMSE and LLSE estimates of the state given the previous observations, and the mean-square error between the two. (Of course the MMSE estimate will in general be a nonlinear function of the observations, whereas the LLSE estimate is by definition linear and is given by the Kalman filtering equations.) We then consider the asymptotic behavior of the mean-square error between the MMSE and LLSE estimates as time tends to infinity. We find conditions on the system dynamics under which the effects of the initial condition die out; under these conditions the non-Gaussian nature of the initial condition becomes unimportant as t becomes large. The practical value of this result is clear - under these conditions, the LLSE estimate, which is usally less costly to generate than the MMSE estimate, is asymptotically as good as the MMSE estimate (i.e., asymptotically optimal) in the mean-square sense

Discrete-Time Filtering for Linear Systems in Correlated Noise with Non-Gaussian Initial Conditions.

We consider the one-step prediction problem for discrete-time linear systems in correlated plant and observation noises, and non-gaussian initial conditions. Explicit representations are obtained for the MMSE and LMSE ( or Kalman) estimates of the state given past observations. These formulae are obtained with the help of the Girsanov transformation for Gaussian white noise sequences, and display explicitly the dependence of the quantities of interest on the initial distribution. Applications of these results can be found in [5] and [6]

Discrete-Time Filtering for Linear Syskms in Correlated Noise with Non-Gaussian Initial Conditions: Asymptotic Behavior of the Difference between the MMSE and LMSE Estimates.

We consider the one-step prediction problem for discrete-time linear systems in correlated plant and observation noises, and non-gaussian initial conditions. We investigate the asymptotic behavior of the expected square {GREEK LETTER SUB t} of the difference between the MMSE and LMSE (or Kalman) estimates of the state given past observations. We characterize the limit of the error sequence ( {GREEK LETTER SUB t}, t = 0,1, ...} and obtain some related rates of convergence, with complete analysis being provided for the scalar case. The discussion is based on the explicit representations which were obtained by the authors in [,] for the MMSE and LMSE estimates, and which explicitly display the dependence of these quantities on the initial distribution

On the Effects of the Initial Condition in State Estimation for Discrete-Time Linear Systems

We consider the one-step prediction problem for discrete-time linear systems in correlated Gaussian white plant and observation noises, and non-Gaussian initial conditions. Explicit representations are obtained for the MMSE and LLSE (or Kalman) estimates square of their difference. These formulae are obtained with the help of the Girsanov transformation for Gaussian white noise sequences, and explicitly display the effects of the distribution of the initial condition. With the help of these formulae, we investigate the large-time asymptotics of et , the expected squared difference between the MMSE and LLSE estimates at time t. We characterize the limit of the error sequence {et, t = 1,2,...} and obtain some related rates of convergence. A complete large-time analysis is provided for the scalar case

From ballistic to diffusive behavior in periodic potentials

The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator

Influência do preparo inicial sobre a estrutura do solo quando da adoção do sistema plantio direto, avaliada por meio da pressão de preconsolidação Influence of initial tillage operations on the soil structure appraised through the preconsolidation pressure when adopting the no till system

Os efeitos do tráfego e do tipo de preparo sobre a estrutura dos solos agrícolas, quando da adoção do sistema plantio direto na região dos Cerrados, têm sido pouco pesquisados. Os estudos desenvolvidos são apenas qualitativos e utilizam-se, geralmente, de propriedades, tais como: a densidade do solo e a resistência do solo à penetração, as quais não possibilitam predizer quanto de pressão o solo pode receber de forma que, em manejos futuros, a compactação possa ser evitada. Este trabalho teve como objetivo avaliar a influência do preparo inicial do solo quando da adoção do sistema plantio direto sobre a estrutura de um Latossolo Vermelho distrófico por meio da pressão de preconsolidação (sigmap). Os valores de sigmap foram obtidos a partir da elaboração de modelos de compressibilidade, os quais consideraram a influência dos seguintes fatores: (1) preparo inicial do solo: arado de aivecas (AA), arado de discos (AD), grade aradora (GA) e vibrosubsolador (VS); (2) manejo: sistema plantio convencional (PC), o qual foi utilizado como testemunha para avaliar a influência dos preparos e (3) profundidade: superficial (SP - 0,00 a 0,05 m) e profundidade média de trabalho dos implementos (PMT- 0,24 a 0,27 m). Os resultados evidenciaram que a sigmap mostrou-se eficiente na avaliação da influência do preparo inicial sobre a estrutura do solo quando da instalação do sistema plantio direto no Latossolo Vermelho distrófico, tendo o seu valor variado entre preparos e profundidades estudados. A sigmap evidenciou maior resistência mecânica e, portanto, maior consolidação da estrutura do solo na profundidade SP dos tratamentos da área sob plantio direto. Verificou-se, também, que os preparos iniciais avaliados reduziram a resistência mecânica do solo na profundidade PMT, quando comparados à do sistema plantio convencional. Todavia, são o vibrosubsolador e o arado de discos os implementos recomendados para alívio dessa resistência, isto é, para a melhoria da estrutura do solo nas profundidades SP e PMT, respectivamente, quando da adoção do sistema plantio direto.<br>Little research has yet been done into the effects of traffic and tillage operations on agricultural soil structures of the Cerrado region before and during the implantation of no till systems. However, the developed studies are only qualitative and usually use properties such as soil bulk density and penetration resistance. These variables cannot predict the pressure the soil can withstand so that soil compaction through future soil management could be avoided. The objective of this work was to evaluate the influence of the initial tillage operations at the adoption of the no till system on the soil structure of a dystrophic Red Latosol through the preconsolidation pressure (sigmap). The sigmap values were obtained from soil compressibility models, which take the influence of the following factors into consideration: (1) initial tillage operations: moldboard plow (AA), disk plow (AD) disk harrow (GA) and vibrosubsoiler (VS); (2) soil management: conventional till (PC) which was used as a reference to evaluate the influence of the tillage operations; and (3) depth: Surface layer (SP - 0.00 to 0.05 m) and medium work depth of the tillage implement (PMT - 0.24 to 0.27 m). Results evidenced that the sigmap was efficient at the evaluation of the initial tillage operations when implanting the no till system in the dystrophic Red Latosol, with its varying values according to tillage operation and depth. The sigmap evidenced a larger mechanical resistance and, therefore, larger soil consolidation of the soil structure at SP depth of the area under no till. It was also verified that the appraised initial tillage operations reduced the soil mechanical resistance at the PMT depth, compared to the conventional tillage system. However, the vibrosubsolador and the disk plow are the recommended tillage tools to relieve this soil resistance, that is, to improve the soil structure in the SP and PMT, respectively, when adopting the no till system