We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for m...We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for more accurate calculation of the mean exit time by computing large deviation prefactors with the aid of machine learning.More specifically,we design a neural network framework to compute quasipotential,most probable paths and prefactors based on the orthogonal decomposition of a vector field.We corroborate the higher effectiveness and accuracy of our algorithm with two toy models.Numerical experiments demonstrate its powerful functionality in exploring the internal mechanism of rare events triggered by weak random fluctuations.展开更多
Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the v...Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.展开更多
The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems (RDS), which can be generated by solutions of some stochastic partial differential equations on...The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems (RDS), which can be generated by solutions of some stochastic partial differential equations on unbounded domains. The conceptual analysis for the long time behavior of RDS will be done through some examples. An application of those analysis will be demonstrated through the proof of the existence of random attractors for asymptotically compact dissipative RDS.展开更多
We introduce a new dynamical evolutionary algorithm(DEA) based on the theory of statistical mechanics and investigate the reconstruction problem for the nonlinear dynamical systems using observation data. The conver...We introduce a new dynamical evolutionary algorithm(DEA) based on the theory of statistical mechanics and investigate the reconstruction problem for the nonlinear dynamical systems using observation data. The convergence of the algorithm is discussed. We make the numerical experiments and test our model using the two famous chaotic systems (mainly the Lorenz and Chen systems). The results show the relatively accurate reconstruction of these chaotic systems based on observational data can be obtained. Therefore we may conclude that there are broad prospects using our method to model the nonlinear dynamical systems.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
A control method is presented for the problem of decentralized stabilizationof large scale nonlinear systems by designing robust controllers, in the sense of L2-gaincontrol, for each subsystem. An uncertainty toleranc...A control method is presented for the problem of decentralized stabilizationof large scale nonlinear systems by designing robust controllers, in the sense of L2-gaincontrol, for each subsystem. An uncertainty tolerance matrix is defined to characterize thedesired robustness leve1 of the overall system. It is then identified that, for a given uncer-tainty tolerance matrix, the design problem is related to the existence of a smooth Positivedefinite solution to a modified Ham ilton -Jacobi - Bellman (H-J-B ) equa tion. The solution,if exists, is exactly the payoff function in terms of the game theory. A decentralized statefeedback law is duly designed, which, under the weak assumption of the zero-state ob-servability on the system, renders the overall closed-loop system aspoptotically stable withan explicitly expressed stability region. Finally, relation between the payoff function andthe uncertainty tolerance matrix is provided, highlighting the 'knowing less and payingmore' philosophy.展开更多
This paper uses Poincaré formalism to extend the Levi-Civita theorem to cope with nonholonomic sys- tems admitting certain invariant relations whose equations of motion involve constraint multipliers.Sufficient c...This paper uses Poincaré formalism to extend the Levi-Civita theorem to cope with nonholonomic sys- tems admitting certain invariant relations whose equations of motion involve constraint multipliers.Sufficient condi- tions allowing such extension are obtained and,as an application of the theory a generalization of Routh's motion is presented.展开更多
We construct new unidirectional coupling schemes for autonomous and nonautonomous drive systems, respectively. Each of these schemes makes the state of the response system asymptotically approach the first-order deriv...We construct new unidirectional coupling schemes for autonomous and nonautonomous drive systems, respectively. Each of these schemes makes the state of the response system asymptotically approach the first-order derivative of the state of the driver. From the point of view of geometry, the first-order derivative of the state of the driver can be viewed as a tangent vector of the trajectory of the driver, so the proposed schemes are named tangent response schemes. Numerical simulations of the Lorenz system and the forced Duffing oscillator verify the validity of the tangent response schemes. We further point out that the tangent response can be interpreted as a special kind of generalised synchronisation, thereby explaining why the response system can exhibit rich geometrical structures in its state space.展开更多
We examine through the lens of dynamical systems a “one dimensional” time mapping of emergent VEV from Pre-Planckian space time conditions. As it is, we will from first principles examine what adding acceleration do...We examine through the lens of dynamical systems a “one dimensional” time mapping of emergent VEV from Pre-Planckian space time conditions. As it is, we will from first principles examine what adding acceleration does as to the HUP previously derived. In doing so, we will be trying it in our discussion with the earlier work done on the HUP. not equal to zero, constant, but large would frequently imply which would have three dissimilar real valued roots. And the situation with not equal to zero yields more tractable result for which will have implications for the HUP inequality in Pre-Planckian space-time, and buttresses an analysis of a 1 dimensional “time” mapping for emergent VEV (vacuum expectation values).展开更多
In this paper we introduce the concept of tensor sum semigroups. Also we have given the examples of tensor sum operators which induce dynamical system on weighted locally convex function spaces.
In this paper, we present an innovative non–linear, discrete, dynamical system trying to model the historic battle of Salamis between Greeks and Persians. September 2020 marks the anniversary of the 2500 years that h...In this paper, we present an innovative non–linear, discrete, dynamical system trying to model the historic battle of Salamis between Greeks and Persians. September 2020 marks the anniversary of the 2500 years that have passed since this famous naval battle which took place in late September 480 B.C. The suggested model describes very well the most effective strategic behavior between two participants during a battle (or in a war). Moreover, we compare the results of the Dynamical Systems analysis to Game Theory, considering this conflict as a “war game”.展开更多
The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan...The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan canonical form of involved matrices. This improves the computational complexity of the algorithms used in literature.展开更多
Existence of traveling wave solutions for some lattice differential equations is investigated. We prove that there exists c<sub>*</sub>>0 such that for each c≥c*</sub>, the systems und...Existence of traveling wave solutions for some lattice differential equations is investigated. We prove that there exists c<sub>*</sub>>0 such that for each c≥c*</sub>, the systems under consideration admit monotonic nondecreasing traveling waves.展开更多
The objective of dynamical system learning tasks is to forecast the future behavior of a system by leveraging observed data.However,such systems can sometimes exhibit rigidity due to significant variations in componen...The objective of dynamical system learning tasks is to forecast the future behavior of a system by leveraging observed data.However,such systems can sometimes exhibit rigidity due to significant variations in component parameters or the presence of slow and fast variables,leading to challenges in learning.To overcome this limitation,we propose a multiscale differential-algebraic neural network(MDANN)method that utilizes Lagrangian mechanics and incorporates multiscale information for dynamical system learning.The MDANN method consists of two main components:the Lagrangian mechanics module and the multiscale module.The Lagrangian mechanics module embeds the system in Cartesian coordinates,adopts a differential-algebraic equation format,and uses Lagrange multipliers to impose constraints explicitly,simplifying the learning problem.The multiscale module converts high-frequency components into low-frequency components using radial scaling to learn subprocesses with large differences in velocity.Experimental results demonstrate that the proposed MDANN method effectively improves the learning of dynamical systems under rigid conditions.展开更多
Wediscuss the idea of using continuous dynamicalsystemstomodel generalhigh-dimensional nonlinear functions used in machine learning.We also discuss theconnection with deep learning.
The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissi...The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissipative dynamical systems, and to extend the existing relevant results of Runge-Kutta methods by Humphries and Stuart(1994). [ABSTRACT FROM AUTHOR]展开更多
In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existenc...In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, a-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.展开更多
Distributed state estimation is of paramount importance in many applications involving the large-scale complex systems over spatially deployed networked sensors.This paper provides an overview for analysis of distribu...Distributed state estimation is of paramount importance in many applications involving the large-scale complex systems over spatially deployed networked sensors.This paper provides an overview for analysis of distributed state estimation algorithms for linear time invariant systems.A number of previous works are reviewed and a clear classification of the main approaches in this field are presented,i.e.,Kalman-filter-type methods and Luenberger-observer-type methods.The design and the stability analysis of these methods are discussed.Moreover,a comprehensive comparison of the existing results is provided in terms of some standard metrics including the graph connectivity,system observability,optimality,time scale and so on.Finally,several important and challenging future research directions are discussed.展开更多
The present paper is devoted to the existence of the random attractor for partly dissipative stochastic lattice dynamical systems with multiplicative white noises.
We propose a new semi-implicit level set approach to a class of curvature dependent flows.The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving t...We propose a new semi-implicit level set approach to a class of curvature dependent flows.The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi(ROF)model for image regularization.Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions.Since the derivation is based on a semi-implicit time discretization,this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method.As an interesting application of the numerical approach,we propose a new variational approach for extracting limit cycles in dynamical systems.The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles.Further,we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.展开更多
基金Project supported by the Natural Science Foundation of Jiangsu Province (Grant No.BK20220917)the National Natural Science Foundation of China (Grant Nos.12001213 and 12302035)。
文摘We present a large deviation theory that characterizes the exponential estimate for rare events in stochastic dynamical systems in the limit of weak noise.We aim to consider a next-to-leading-order approximation for more accurate calculation of the mean exit time by computing large deviation prefactors with the aid of machine learning.More specifically,we design a neural network framework to compute quasipotential,most probable paths and prefactors based on the orthogonal decomposition of a vector field.We corroborate the higher effectiveness and accuracy of our algorithm with two toy models.Numerical experiments demonstrate its powerful functionality in exploring the internal mechanism of rare events triggered by weak random fluctuations.
基金the National Natural Science Foundation of China (11871188, 12031019)。
文摘Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.
基金the National NSFC under grant No.50579022the Foundation of Pre-973 Program of China under grant No.2004CCA02500+1 种基金the SRF for the ROCS,SEMthe Talent Recruitment Foundation of HUST
文摘The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems (RDS), which can be generated by solutions of some stochastic partial differential equations on unbounded domains. The conceptual analysis for the long time behavior of RDS will be done through some examples. An application of those analysis will be demonstrated through the proof of the existence of random attractors for asymptotically compact dissipative RDS.
基金Supported by the National Natural Science Foun-dation of China (60133010) the Natural Science Foundation ofHubei Province (2004ABA011)
文摘We introduce a new dynamical evolutionary algorithm(DEA) based on the theory of statistical mechanics and investigate the reconstruction problem for the nonlinear dynamical systems using observation data. The convergence of the algorithm is discussed. We make the numerical experiments and test our model using the two famous chaotic systems (mainly the Lorenz and Chen systems). The results show the relatively accurate reconstruction of these chaotic systems based on observational data can be obtained. Therefore we may conclude that there are broad prospects using our method to model the nonlinear dynamical systems.
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.
文摘A control method is presented for the problem of decentralized stabilizationof large scale nonlinear systems by designing robust controllers, in the sense of L2-gaincontrol, for each subsystem. An uncertainty tolerance matrix is defined to characterize thedesired robustness leve1 of the overall system. It is then identified that, for a given uncer-tainty tolerance matrix, the design problem is related to the existence of a smooth Positivedefinite solution to a modified Ham ilton -Jacobi - Bellman (H-J-B ) equa tion. The solution,if exists, is exactly the payoff function in terms of the game theory. A decentralized statefeedback law is duly designed, which, under the weak assumption of the zero-state ob-servability on the system, renders the overall closed-loop system aspoptotically stable withan explicitly expressed stability region. Finally, relation between the payoff function andthe uncertainty tolerance matrix is provided, highlighting the 'knowing less and payingmore' philosophy.
文摘This paper uses Poincaré formalism to extend the Levi-Civita theorem to cope with nonholonomic sys- tems admitting certain invariant relations whose equations of motion involve constraint multipliers.Sufficient condi- tions allowing such extension are obtained and,as an application of the theory a generalization of Routh's motion is presented.
文摘We construct new unidirectional coupling schemes for autonomous and nonautonomous drive systems, respectively. Each of these schemes makes the state of the response system asymptotically approach the first-order derivative of the state of the driver. From the point of view of geometry, the first-order derivative of the state of the driver can be viewed as a tangent vector of the trajectory of the driver, so the proposed schemes are named tangent response schemes. Numerical simulations of the Lorenz system and the forced Duffing oscillator verify the validity of the tangent response schemes. We further point out that the tangent response can be interpreted as a special kind of generalised synchronisation, thereby explaining why the response system can exhibit rich geometrical structures in its state space.
文摘We examine through the lens of dynamical systems a “one dimensional” time mapping of emergent VEV from Pre-Planckian space time conditions. As it is, we will from first principles examine what adding acceleration does as to the HUP previously derived. In doing so, we will be trying it in our discussion with the earlier work done on the HUP. not equal to zero, constant, but large would frequently imply which would have three dissimilar real valued roots. And the situation with not equal to zero yields more tractable result for which will have implications for the HUP inequality in Pre-Planckian space-time, and buttresses an analysis of a 1 dimensional “time” mapping for emergent VEV (vacuum expectation values).
文摘In this paper we introduce the concept of tensor sum semigroups. Also we have given the examples of tensor sum operators which induce dynamical system on weighted locally convex function spaces.
文摘In this paper, we present an innovative non–linear, discrete, dynamical system trying to model the historic battle of Salamis between Greeks and Persians. September 2020 marks the anniversary of the 2500 years that have passed since this famous naval battle which took place in late September 480 B.C. The suggested model describes very well the most effective strategic behavior between two participants during a battle (or in a war). Moreover, we compare the results of the Dynamical Systems analysis to Game Theory, considering this conflict as a “war game”.
文摘The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan canonical form of involved matrices. This improves the computational complexity of the algorithms used in literature.
文摘Existence of traveling wave solutions for some lattice differential equations is investigated. We prove that there exists c<sub>*</sub>>0 such that for each c≥c*</sub>, the systems under consideration admit monotonic nondecreasing traveling waves.
基金supported by the National Natural Science Foundations of China(Nos.12172186 and 11772166).
文摘The objective of dynamical system learning tasks is to forecast the future behavior of a system by leveraging observed data.However,such systems can sometimes exhibit rigidity due to significant variations in component parameters or the presence of slow and fast variables,leading to challenges in learning.To overcome this limitation,we propose a multiscale differential-algebraic neural network(MDANN)method that utilizes Lagrangian mechanics and incorporates multiscale information for dynamical system learning.The MDANN method consists of two main components:the Lagrangian mechanics module and the multiscale module.The Lagrangian mechanics module embeds the system in Cartesian coordinates,adopts a differential-algebraic equation format,and uses Lagrange multipliers to impose constraints explicitly,simplifying the learning problem.The multiscale module converts high-frequency components into low-frequency components using radial scaling to learn subprocesses with large differences in velocity.Experimental results demonstrate that the proposed MDANN method effectively improves the learning of dynamical systems under rigid conditions.
基金with several collaborators,including Jiequn Han,Qianxiao Li,Jianfeng Lu and Cheng Tai.The author benefitted a great deal from discussions with them,particularly Jiequn Han.This work is supported in part by the Major Program of NNSFC under Grant91130005,ONR NO0014-13-1-0338 and DOE DE-SCo009248.
文摘Wediscuss the idea of using continuous dynamicalsystemstomodel generalhigh-dimensional nonlinear functions used in machine learning.We also discuss theconnection with deep learning.
基金a grant !(No. 19871070) from NSF of China a grant!(No. A757D9I0) from Academy of Mathematics and System Sciences, Academy o
文摘The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissipative dynamical systems, and to extend the existing relevant results of Runge-Kutta methods by Humphries and Stuart(1994). [ABSTRACT FROM AUTHOR]
基金supported by Hunan Provincial Natural Science Foundation of China(No.2015JJ2144)National Natural Science Foundation of China(No.11671343 and No.11171280)+1 种基金the General Project of The Education Department of Hunan Province(No.12C0408)Zhejiang Natural Science Foundation(No.LY14A010012)
文摘In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, a-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.
基金supported by the National Natural Science Foundation of China (No. 61790573)supported by the National Natural Science Foundation of China (Nos. 61890924, 61991404)Liao Ning Revitalization Talents Program (No. XLYC1907087)
文摘Distributed state estimation is of paramount importance in many applications involving the large-scale complex systems over spatially deployed networked sensors.This paper provides an overview for analysis of distributed state estimation algorithms for linear time invariant systems.A number of previous works are reviewed and a clear classification of the main approaches in this field are presented,i.e.,Kalman-filter-type methods and Luenberger-observer-type methods.The design and the stability analysis of these methods are discussed.Moreover,a comprehensive comparison of the existing results is provided in terms of some standard metrics including the graph connectivity,system observability,optimality,time scale and so on.Finally,several important and challenging future research directions are discussed.
基金supported by the National Natural Science Foundations of China(No.11071165,No.11071199)Natural Science Foundation of Guangxi(No.2013GXNSFBA019008)Department of Research Project of Guangxi Provincial(No.2013YB102)
文摘The present paper is devoted to the existence of the random attractor for partly dissipative stochastic lattice dynamical systems with multiplicative white noises.
基金The work of Leung was supported in part by the RGC under Grant 605612。
文摘We propose a new semi-implicit level set approach to a class of curvature dependent flows.The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi(ROF)model for image regularization.Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions.Since the derivation is based on a semi-implicit time discretization,this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method.As an interesting application of the numerical approach,we propose a new variational approach for extracting limit cycles in dynamical systems.The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles.Further,we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.