Dao, Hien Thi LeThe subject of this thesis is deterministic behaviors generated from a microwave time-delayed feedback loop. Time-delayed feedback systems are especially interesting because of the rich variety of dynamical behaviors that they can support. While ordinary differential equations must be of at least third-order to produce chaos, even a simple first-order nonlinear delay differential equation can produce higher-dimensional chaotic dynamics. The system reported in the thesis is governed by a very simple nonlinear delay differential equation. The experimental implementation uses both microwave and digital components to achieve the nonlinearity and time-delayed feedback, respectively. When a sinusoidal nonlinearity is incorporated, the dynamical behaviors range from fixed-point to periodic to chaotic depending on the feedback strength. The microwave frequency modulated chaotic signal generated by this system offers advantages in range and velocity sensing applications. When the sinusoidal nonlinearity is replaced by a binary nonlinearity, the system exhibits a complex periodic attractor with no fixed-point solution. Although there are many classic electronic circuits that produce chaotic behavior, microwave sources of chaos are especially relevant in communication and sensing applications where the signal must be transmitted between locations. The system also can exhibit random walk behavior when being operated in a higher feedback strength regime. Depending on the feedback strength values, the random behaviors can have properties of a regular or fractional Brownian motion. By unidirectional coupling two systems in the baseband, envelope synchronization between two deterministic Brownian motions can be achieved.enDissertationPhysicsChaosDelay differential equationDeterministic Brownian motionFPGAMicrowave chaosTime-delayed