Efficient search and exploration behaviors observed in animals are significant for their survival, encompassing foraging, mate-finding, and predator avoidance. Animal movement patterns underlying the search and exploration behaviors are typically modeled as Lévy flights. Despite their efficacy, current Lévy flight models rely on mathematical formulations and lack connection to biological neural systems. To address this, we propose an adaptive neural system which encodes Lévy distributions for generating and controlling efficient search and exploration behaviors. The neural system consists of three subnetworks: a neural chaotic network, a directional control network, and a step length control network. The neural chaotic network generates chaotic output signals, which are used as exploration seeds, further processed by the directional and step length control networks to control the search and exploration of a flying robot. By simply adjusting the scaling input of the step length control network, our neural control system can generate adaptive exploration behaviors with a combination of global and local search strategies. Our system is implemented on a simulated autonomous flying robot, evaluated across diverse environments, and compared with Gaussian random walk and traditional Lévy flight strategies.