Chaotic behaviors can be observed in all kinds of natural and non-natural phenomena, such as weather forecasting in meteorology and population growth in sociology. A chaotic system demonstrating chaotic behavior has properties of ergodicity, unpredictability and initial state sensitivity. Thanks for these significant properties, chaotic theory has wide applications in the fields of science and engineering, especially in cryptography. However, when a chaotic system is implemented in the digital devices with finite precision, its extremely close states may overlap and its chaotic behavior may degrade to periodic behavior. Besides, the fast development of discerning chaos technologies makes some chaotic systems easily being attacked. Chaotic systems with more complex behaviors have less chance to degrade and are more difficult to be attacked. Thus, developing chaotic systems with complex behaviors becomes attractive. The main contributions of this thesis involve two parts. The first part is to develop two one-dimensional (1D) chaotic frameworks: the cascade chaotic system (CCS) and dynamic parameter-control chaotic system (DPCCS). Using existing 1D chaotic maps as the seed maps, they both can generate a large number of new chaotic maps. The second part is to propose two two-dimensional (2D) chaotic maps: the 2D Sine Logistic modulation map (2D-SLMM) and 2D Logistic-adjusted-Sine map (2D-LASM). They are used to design novel image encryption schemes. The detail achievements of this thesis are listed as follows. 1. CCS is proposed for generating 1D chaotic maps. It connects two 1D chaotic maps (seed maps) in series. The output of the first seed map is linked to the input of the second one. The output of the second one is fed back into the input of the first one for recursive iterations, and it is also the output of CCS. Three examples of new chaotic map are provided. Evaluations and analysis results show that these new chaotic maps have better performance than their corresponding seed maps. Using one chaotic map generated by CCS as an example, we further propose a pseudo-random number generator (PRNG) and a data encryption system. Simulations and security analysis demonstrate that the PRNG can generate iii pseudo-random numbers with high randomness and the data encryption system has a high security level. 2. DPCCS is proposed as a general framework for enhancing dynamics of 1D chaotic map. In DPCCS, a chaotic map (control map) is used to dynamically control the parameter of another chaotic map (seed map) to form a new chaotic map. Thus, the seed map generates outputs using a dynamically changed or fixed parameter provided by the control map in each unit time. A comprehensive analysis of the DPCCS’s properties and chaotic behavior is performed theoretically and experimentally. Using three existing chaotic maps as the control and seed maps, DPCCS produces nine new chaotic maps. Analysis and comparison results of these maps are provided to show their complex chaotic behaviors. We introduce an FPGA design of DPCCS to show its simplicity in hardware implementation, and we also propose a new PRNG using a chaotic map of DPCCS, and evaluate its randomness using different test standards. 3. 2D-SLMM is designed for image encryption. It is derived from the Sine and Logistic maps. Performance analysis is provided to show its complex chaotic behavior. A chaotic magic transform (CMT) for image encryption is introduced to show the effectiveness of 2D-SLMM. Integrating 2D-SLMM and CMT, a new CMT-based image encryption algorithm (CMT-IEA) is also proposed. Experimental results and security analysis are given to show that CMT-IEA has a high level of security and a low time complexity. 4. 2D-LASM is introduced for image encryption. It uses the Logistic map to adjust the input of the Sine map and then extends its phase plane from 1D to 2D. Performance evaluations prove its excellent performance. Using 2D-LASM, a 2D-LASM-based image encryption scheme (LAS-IES) is further proposed. An additional mechanism of adding random values to the plain-image is designed to ensure that each encrypted result is different. Simulation results and security analysis show that LAS-IES has a strong capability against various attacks.