Hidden Attractor in a Dynamical System Without Fixed Points and Perpetual Points

This study examines the emergence of chaotic dynamics in Three-Dimensional (3D) quadratic systems that lack both fixed points and perpetual points. Although systems without fixed points are already uncommon, the simultaneous absence of fixed and perpetual points imposes severe constraints that have not been explored. Starting from previously reported systems with no fixed points, the existence of perpetual points was first examined, revealing that all known cases retain at least one perpetual point. Motivated by this observation, a class of parametric quadratic systems was constructed under explicit analytical constraints that rigorously exclude both fixed points and perpetual points. An extensive and systematic exploration of the admissible parameter space was then performed to identify simple, bounded, and potentially chaotic dynamics. This search yielded only a single parameter set, whose behavior is periodic. However, a detailed bifurcation analysis of this system revealed a very narrow parameter interval near the transition between bounded and unbounded motion in which chaotic behavior arises. Lyapunov exponent analysis and bifurcation diagrams confirm that chaos is confined to this critical boundary regime. These results demonstrate that chaotic dynamics in 3D systems without fixed points and perpetual points are extremely rare and structurally fragile, emerging only under highly restricted transitional conditions, thereby clarifying the strong suppressive role of such phase-space constraints while showing that chaos is not entirely forbidden.