A Bivariate Chebyshev Polynomials Method for Nonlinear Dynamic Systems With Interval Uncertainties

A bivariate Chebyshev polynomials approach is proposed to estimate the dynamic response bounds of nonlinear systems with interval uncertainties. The existing collocation method directly searches the maximum and minimum values of the surrogate model in the entire interval space by the scanning method (SM). The presence of too many uncertain parameters will lead to expansive computational cost. To overcome this shortcoming, the dynamic response is decomposed by a bivariate function decomposition (BFD), established based on high-order Taylor expansion, into the sum of multiple univariate and bivariate response functions. The above univariate and bivariate functions are fitted using Chebyshev polynomials, and polynomial coefficients are obtained through one-dimensional (1D) and two-dimensional (2D) interpolation points. Thus, the solution of the nonlinear dynamic systems with uncertain parameters can be transformed into that of univariate and bivariate Chebyshev interval functions. The extremum values of the low-dimensional Chebyshev interval functions can be found by SM, and then the bounds of dynamic response are acquired by interval arithmetic. Since SM searches for extreme values only in 1D and 2D uncertain domains, the amount of calculation is reduced compared to searching the whole uncertain space. The efficiency, practicability and effectiveness of the proposed interval uncertainty analysis method are proved by three dynamic examples.

Topological entropy of Markov set-valued functions

We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.

INTERVAL EXTENSION STRUCTURE BASIC TYPES OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FORTH ORDER

In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the fourth order has been developed. Used Method of R-functions for constructed coordinate sequences. Constructing interval extensions of structural formulas, we consider problems (1) on the transverse bending of thin plates and 5 problems on a plate - rigidly clamped plate, loosely supported plate, elastically fixed plates, partially rigidly clamped and partially elastically fixed plates, plates, partially rigidly clamped and partially free . For the problem, the rigidly clamped plate Formula (7) is an interval structure for solving the boundary value problem (4). Here L={▁ω ▁ψ,ω ̅▁ψ,▁ω ψ ̅,ω ̅ψ ̅ },L_1={▁ω D_1 ▁φ,ω ̅D_1 ▁φ,▁ω D_1 φ ̅,ω ̅D_1 φ ̅ } L_2={▁ω^2 ▁Φ,ω ̅^2 ▁Φ,▁ω^2 Φ ̅,ω ̅^2 Φ ̅ }, [ ▁Φ,Φ ̅ ] is an indefinite interval function. For the free-supported plate problem, a solution is obtained for the interval expansion of the structure in the form (15), (17), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ]- indefinite interval function, D_2, T_2 - differential operators of the form (11) and (12). For the problem of elastically fixed plates, a solution was obtained in the interval expansion of the structure in the form (21) - (24), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ]- indefinite interval functions, D_2,T_2,D_1- differential operators of the form (11) and (12), (3). For the problem of partially rigidly clamped and partially elastically fixed plates, a solution was obtained in the interval expansion of the structure in the form of (28), (30), (32), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ]- indefinite interval functions, D_2,T_2,D_1- differential operators of the form (11) and (12), (6). For the plate problem, partially rigidly pinched and partially free, a solution is obtained in the interval extension of the structure (40), (41), (42), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ] - indefinite interval functions, D_2,T_2,D_1,D_3- differential operators of the form (11), (12), (6) and (38).

Interval Enclosures for Reliability Metrics

The computation of reliability metrics involves real numbers. Therefore, numerical problems are generated due to the limitation of representing and operating with real numbers in computers. This paper proposes interval functions for controlling numeric errors in the computation of reliability metrics values of complex systems, based on interval mathematics and high accuracy arithmetic. The interval functions calculate interval enclosures, using Intlab toolbox, for real values of reliability metrics and the SHARPE software was used to validate the results. Analysis of the numerical results obtained with the proposed functions showed that the intervals really enclose the real numbers calculated by the software SHARPE, indicating that these functions, in fact, are an alternative for auto-validating representation of these reliability values of complex systems.