We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condi...We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.展开更多
In this note,we will generalize some results about the Cohen-Macaulaymess and Goronsteinness of Rees rings and associated graded rings of ideals having higher analytic deviation.
文摘托马斯·巴宾顿·麦考利(Thomas Babington Macaulay)(1800—1859)是英国著名的散文家,是十九世纪瞩目的文学家之一。他年幼好学,富有天资。五岁时说话滔滔不绝。十岁时韵文、史诗、宗教论文、史纲等无不属于他习作之列。他手不释卷,读了不少的书。他聪明伶俐,记忆非凡,不但能大段大段地背诵名家杰作,就连他四十年前在咖啡馆里一略而过的报上小诗,四十年后,不需重温,仍然记忆犹新,脱口而出,虽然他有如此天才,但是仍刻苦努力,一丝不苟。他撰写的《英国历史》(History of England)便是一例。
文摘We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.
文摘In this note,we will generalize some results about the Cohen-Macaulaymess and Goronsteinness of Rees rings and associated graded rings of ideals having higher analytic deviation.
