In this paper, a new memristor-based fourth-order hyperchaotic circuit that can generate multiple vortex volumes is constructed with the aid of a cosine memristor proposed by scholars. Thanks to the properties of cosine memristors, we find new hyperchaotic systems with full even symmetry based on parameters. The new system can produce multi-vortex properties with infinite equilibrium points and exhibits extreme multi-stability phenomena with the coexistence of an infinite number of attractors. After further study of phase trajectory diagrams, Lyapunov exponential spectrum, bifurcation diagrams, intermittent chaos and transient phenomena, we find that there are complex state transfer phenomena such as period 1 windows, chaos, multiperiodic bifurcations, multi-periodic windows and hyperchaotics, which are constantly alternating. In terms of circuit implementation, Multisim was used to build the circuit and to realise the characteristics of the memristor and the simulation of the hyperchaotic circuit. Finally, a DNA encryption algorithm is designed based on memristor hyperchaotic system, with emphasis on adjacent pixel correlation, encryption histogram, key sensitivity, anti-cropping and anti-noise performance. The results show that the new system has a very large key space, strong key sensitivity, outstanding anti-cropping, and anti-noise performance. The proposed hyperchaotic system thus has high security performance when applied to image encryption.