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In this paper, a hyper-chaotic system with a controllable number of scrolls and multiple pairs of coexistence attractors is proposed based on the Sprott-C system. By introducing a multi-segmental family of square functions, the system is verified to have a complex dynamical behavior by analyzing the dissipation, equilibrium point, Lyapunov exponent, dimension and bifurcation, Poincaré cross- section, power spectrum, and 0-1 test of the system. The system is then further characterized in terms of the chaotic bursting oscillations present in the system, the coexistence attractor of the system, the system complexity, and the effect of the parameter change on the system. Verified the implementability of the system through circuit design and simulation. Finally, the hyper-chaotic system is combined with the DNA encryption algorithm and applied to image encryption, and the feasibility and security of the encryption method are verified by comprehensive analysis of the key space and sensitivity, histogram, correlation, information entropy, and robustness of the encrypted image. The system can be used in the field of image encryption in the future.

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