Affiliation:
1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain
Abstract
Planar differential systems whose angular velocity is constant are called rigid or uniform differential systems. The first rigid system goes back to the pendulum clock of Christiaan Huygens in 1656; since then, the interest for the rigid systems has been growing. Thus, at this moment, in MathSciNet there are 108 articles with the words rigid systems or uniform systems in their titles. Here, we study the dynamics of the planar rigid polynomial differential systems with homogeneous nonlinearities of arbitrary degree. More precisely, we characterize the existence and non-existence of limit cycles in this class of rigid systems, and we determine the local phase portraits of their finite and infinite equilibrium points in the Poincaré disc. Finally, we classify the global phase portraits in the Poincaré disc of the rigid polynomial differential systems of degree two, and of one class of rigid polynomial differential systems with cubic homogeneous nonlinearities that can exhibit one limit cycle.
Funder
Agencia Estatal de Investigación of Spain
AGAUR
Reial Acadèmia de Ciències i Arts de Barcelona
Reference14 articles.
1. Centers with degenerate infinity and their commutators;Algaba;J. Math. Anal. Appl.,2003
2. Computing center conditions for vector fields with constant angular speed;Algaba;J. Comput. Appl. Math.,2003
3. On the center conditions of certain cubic systems;Alwash;Proc. Am. Math. Soc.,1998
4. Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree;Collins;J. Math. Anal. Appl.,1995
5. Uniformly isochronous centers of polynomial systems in R2;Conti;Differential Equations, Dynamical Systems, and Control Science,1994