Due to its rich dynamic behaviors and diversity, the multi-wing chaotic system has garnered widespread research interest, and its emergence and applications are currently under intensive investigation. In this paper, based on the Sprott-A system, a four-dimensional controllable multi-wing chaotic system is proposed by incorporating a cosine function. The system can generate a controllable number of wings by adjusting its parameters, thus realizing the precise control of chaotic behavior. Through numerical simulation, the high randomness and complexity of the system are verified. On account of the superiority of the system, a new color image encryption method is designed based on this new system. The method uses chaotic sequences to implement DNA-level image scrambling operation, and discrete wavelet transform (DWT) to decompose the scrambled image, then scramble these decomposed components. The scrambled matrices are reconstructed with inverse discrete wavelet transform (IDWT), and diffused to produce the ciphertext image. Ultimately, the results of simulations and security evaluations indicate the high-security level of the suggested image encryption technique.