<p>The degree of complexity and chaoticness of a chaotic system are, respectively, measured by the Lyapunov dimension and the largest Lyapunov exponent of the system. An increase in these two quantities makes a chaotic system a worthy candidate for different chaos-based applications. This paper provides a generalized methodology to find a set of parameters and initial conditions of any chaotic system that results in the maximum Lyapunov dimension. The proposed approach is validated by maximizing the Lyapunov dimension of the well-known Lorenz system. Competitive swarm optimization is chosen to realize the above objective. The maximum Lyapunov dimension of 2.169 is found for the Lorenz system. Furthermore, in the process of maximization of the Lyapunov dimension, the highest Lyapunov exponent of the Lorenz system is obtained as 7.9138. Thus, the prediction time of the Lorenz system, with the considered parameters and initial conditions, reduces. To the best of our knowledge, our results are the highest among the existing Lyapunov dimension and positive Lyapunov exponent of the Lorenz system. The significance of the increased Lyapunov dimension is demonstrated by using it in an image encryption process. To validate the proposed generalized approach, two more chaotic systems are considered to get higher Lyapunov dimensions. Extensive simulations and analyses are done and presented to substantiate the claims.</p>