Three Examples in the Dynamical Systems Theory

Abstract

. We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms R , S of a closed two-dimensional annulus that possess the intersection property but their composition RS does not ( R being just the rotation by π/ 2). The second example is that of a non-Lagrangian n -torus L 0 in the cotangent bundle T ∗ T n of T n ( n ≥ 2) such that L 0 intersects neither its images under almost all the rotations of T ∗ T n nor the zero section of T ∗ T n . The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form ˙ x = f ( x, y ), ˙ y = µg ( x, y ) in the closed upper half-plane { y ≥ 0 } such that the corresponding phase portraits for 0 < µ < 1 and for µ > 1 are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.