In this paper, by introducing the concept of topological equivalence on measure chain, we investigate the relationship between the linear system $x^{\Delta }=A(t)x$ and the nonlinear system $x^{\Delta }=A(t)x+f(t,x)$. Some sufficient conditions are obtained to guarantee the existence of a equivalent function $H(t,x)$ sending the $(c,d)$-quasibounded solutions of nonlinear system $x^{\Delta }=A(t)x+f(t,x)$ onto those of linear system $x^{\Delta }=A(t)x$. Our results generalize the Palmer's linearization theorem in [K.J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl. 41 (1973) 753–758] to dynamic equation measure chains. In the present paper, we give a new analytical method to study the topological equivalence problem on measure chains. As we will see, due to the completely different method to investigate the topological equivalence problem, we have a considerably different result from that in the pioneering work of Hilger [S. Hilger, Generalized theorem of Hartman–Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157–191]. Moreover, we prove that equivalent function $H(t,x)$ is also ω-periodic when the systems are ω-periodic. Hilger [S. Hilger, Generalized theorem of Hartman–Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157–191] never considered this important property of the equivalent function $H(t,x)$.

In this paper, we report the existence of twelve small limit cycles in a planar system with 3rd-degree polynomial functions. The system has $Z_2$-symmetry, with a saddle point, or a node, or a focus point at the origin, and two focus points which are symmetric about the origin. It is shown that such a $Z_2$-equivariant vector field can have twelve small limit cycles. Fourteen or sixteen small limit cycles, as expected before, cannot not exist.