One-dimensional steady state and evolutionary Ginzburg-Landau equations for superconductivity is disussed. Instability of constant steady state solutions and the existence of limit sets of infinite-dimensional dynamic system are proved. Using the author's definition of chaos for finite-and infinite-dimensional dynamic systems, we conclude that superconductors governed by G-L equations possess chaotic phenomena. Therefore strange phenomena may occur in conducting. The theoretical results indicate that it is advisable to improve the design of experiments or try to find new structures of superconductors in future research to suppress the chaotic behavior.