Abstract
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.
Highlights
A limit cycle, which is represented by an isolated closed trajectory in the phase plane, describes a phenomenon of oscillation that is widely observed and studied in various research fields such as electrical circuits and control theory [1], chemistry and medicine [2,3], ecological systems [4], human population [5], etc
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
We propose a new framework for identifying the limit cycles of polynomial systems, inspired by the latest innovation in multivariate polynomial representation and roots finding by formulating the polynomial equations via Macaulay matrices [13,14], thereby turn
Summary
A limit cycle, which is represented by an isolated closed trajectory in the phase plane, describes a phenomenon of oscillation that is widely observed and studied in various research fields such as electrical circuits and control theory [1], chemistry and medicine [2,3], ecological systems [4], human population [5], etc. We propose a new framework for identifying the limit cycles of polynomial systems, inspired by the latest innovation in multivariate polynomial representation and roots finding by formulating the polynomial equations via Macaulay matrices [13,14], thereby turn-. A set of polynomial equations are constructed with respect to the coefficients of the limit cycle, whose roots are found through eigenvalue computation. This framework has a general efficacy for polynomial systems of arbitrary orders and dimensions.
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