A backdoor attack in deep learning inserts a hidden backdoor in the model to trigger malicious behavior upon specific input patterns. Existing detection approaches assume a metric space (for either the original inputs or their latent representations) in which normal samples and malicious samples are separable. We show that this assumption has a severe limitation by introducing a novel SSDT (Source-Specific and Dynamic-Triggers) backdoor, which obscures the difference between normal samples and malicious samples. To overcome this limitation, we move beyond looking for a perfect metric space that would work for different deep-learning models, and instead resort to more robust topological constructs. We propose TED (Topological Evolution Dynamics) as a model-agnostic basis for robust backdoor detection. The main idea of TED is to view a deep-learning model as a dynamical system that evolves inputs to outputs. In such a dynamical system, a benign input follows a natural evolution trajectory similar to other benign inputs. In contrast, a malicious sample displays a distinct trajectory, since it starts close to benign samples but eventually shifts towards the neighborhood of attacker-specified target samples to activate the backdoor. Extensive evaluations are conducted on vision and natural language datasets across different network architectures. The results demonstrate that TED not only achieves a high detection rate, but also significantly outperforms existing state-of-the-art detection approaches, particularly in addressing the sophisticated SSDT attack. The code to reproduce the results is made public on GitHub.