Cognitive dynamics, which refer to the evolution in human cognitive processes, are pivotal to advance human understanding of the world. Recent advancements in large language models (LLMs) highlight their potential for cognitive simulation. However, these LLM-based cognitive studies primarily focus on replicating human cognition in specific contexts, overlooking the inherently dynamic nature of cognition. To bridge this gap, we explore the cognitive dynamics of LLMs and present a corresponding task inspired by longitudinal studies. Toward the task, we develop CogBench, a novel benchmark to assess the cognitive dynamics of LLMs and validate it through participant surveys. We also design two evaluation metrics for CogBench, including Authenticity and Rationality. Recognizing the inherent static nature of LLMs, we further introduce CogGPT for the task, which features an innovative iterative cognitive mechanism to develop lifelong cognitive dynamics. Empirical results demonstrate the superiority of CogGPT over several existing methods, particularly in its ability to facilitate role-specific cognitive dynamics under continuous information flows. We will release the code and data to enable further research.