Introduction

Albert C. J. Luo ()
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Albert C. J. Luo: Southern Illinois University Edwardsville, Department of Mechanical and Mechatronics Engineering

Chapter Chapter 1 in Limit Cycles and Homoclinic Networks in Two-Dimensional Polynomial Systems, 2025, pp 1-6 from Springer

Abstract: Abstract Consider a dynamical system with a differential equation as $$\dot{x}_{1} \equiv \frac{{dx_{1} }}{dt} = P(x_{1} ,x_{2} ), \, \dot{x}_{2} \equiv \frac{{dx_{2} }}{dt} = Q(x_{1} ,x_{2} )$$ x ˙ 1 ≡ d x 1 dt = P ( x 1 , x 2 ) , x ˙ 2 ≡ d x 2 dt = Q ( x 1 , x 2 ) where $$P(x_{1} ,x_{2} )$$ P ( x 1 , x 2 ) and $$Q(x_{1} ,x_{2} )$$ Q ( x 1 , x 2 ) are real polynomials of degree $$n$$ n . The second part of Hilbert's 16th problem is to decide an upper bound for the number of limit cycles in polynomial vector fields of degree $$n$$ n and, similar to the first part, investigate their relative positions. The original problem can be found.

Date: 2025
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DOI: 10.1007/978-981-97-2617-2_1

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