Manifold studies in the topic of nonlinear dynamics in memristive systems are presented in this work. The author uses mathematical approaches in analysing various memristive systems. Several new concepts are first proposed for the general mathematical structures, including the categorization of homogeneous and heterogeneous systems, memtronic systems, coupling relation and memristive telegraph equations.
A general analytical solution is given for the titanium dioxide memristor based on a nonlinear ionic drift model within the frame of Abel dynamics, which is expressed in the form of Gaussian hypergeometric functions. For characterizing such systems, a method using the characteristic curve of state (CCOS) is applied to determine the orbital shapes and saturation levels. A novel spin-valve memristive system is constructed on a feedback loop employing giant magnetoresistance and classical Hall effect. The spintronic system demonstrates a non-origin-crossing property, due to a heterogeneous voltage-current relation, which can be categorized as a heterogeneous memristive system. A sufficient condition for asymptotic stability is further established by calculating the Floquet index of the linearized Poincar map. In the domain of neuromorphic systems, a memristive spike-timing-dependent-plasticity (STDP) model compatible with the triplet rule is proposed. The model uses adaptive thresholds to realize the principle of suppression for the triplet-based STDP rule. Comparison is made between simulated synaptic modifications and experimental recordings. Representative applications of memristive systems are demonstrated as an extensional scope in this topic.