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The study of bifurcations of differential-algebraic equations (DAEs) is the topic of interest for many applied sciences, such as electrical engineering, robotics, etc. While some of them were investigated already, the full classification of such bifurcations has not been done yet. In this paper, we consider bifurcations of quasilinear DAEs with a singularity and provide a full list of all codimension-one bifurcations in lower-dimensional cases. Among others, it includes singularity-induced bifurcations (SIBs), which occur when an equilibrium branch intersects a singular manifold causing certain eigenvalues of the linearized problem to diverge to infinity. For these and other bifurcations, we construct the normal forms, establish the non-degeneracy conditions and give a qualitative description of the dynamics. Also, we study singular homoclinic and heteroclinic bifurcations, which were not considered before., Comment: 35 pages, 17 figures

We consider two mathematical dynamical models of gene regulatory networks (GRNs) and obtain results on robust synchronization on these dynamical models based on the existing theoretical results in the coupled cell network formalism. We also explore the concepts of quotient networks and network lifting in the context of GRNs which are related to the process of gene duplication and the phenomenon of genetic redundancy.

For a regular coupled cell network, synchrony subspaces are the polydiagonal subspaces that are invariant under the network adjacency matrix. The complete lattice of synchrony subspaces of an $n$-cell regular network can be seen as an intersection of the partition lattice of $n$ elements and a lattice of invariant subspaces of the associated adjacency matrix. We assign integer tuples with synchrony subspaces, and use them for identifying equivalent synchrony subspaces to be merged. Based on this equivalence, the initial lattice of synchrony subspaces can be reduced to a lattice of synchrony subspaces which corresponds to a simple eigenvalue case discussed in our previous work. The result is a reduced lattice of synchrony subspaces, which affords a well-defined non-negative integer index that leads to bifurcation analysis in regular coupled cell networks., Comment: 34 pages, 59 figures

Address event representation (AER) cameras have recently attracted more attention due to the advantages of high temporal resolution and low power consumption, compared with traditional frame-based cameras. Since AER cameras record the visual input as asynchronous discrete events, they are inherently suitable to coordinate with the spiking neural network (SNN), which is biologically plausible and energy-efficient on neuromorphic hardware. However, using SNN to perform the AER object classification is still challenging, due to the lack of effective learning algorithms for this new representation. To tackle this issue, we propose an AER object classification model using a novel segmented probability-maximization (SPA) learning algorithm. Technically, 1) the SPA learning algorithm iteratively maximizes the probability of the classes that samples belong to, in order to improve the reliability of neuron responses and effectiveness of learning; 2) a peak detection (PD) mechanism is introduced in SPA to locate informative time points segment by segment, based on which information within the whole event stream can be fully utilized by the learning. Extensive experimental results show that, compared to state-of-the-art methods, not only our model is more effective, but also it requires less information to reach a certain level of accuracy., Comment: AAAI 2020 (Oral)

This paper proposes an unsupervised address event representation (AER) object recognition approach. The proposed approach consists of a novel multiscale spatio-temporal feature (MuST) representation of input AER events and a spiking neural network (SNN) using spike-timing-dependent plasticity (STDP) for object recognition with MuST. MuST extracts the features contained in both the spatial and temporal information of AER event flow, and meanwhile forms an informative and compact feature spike representation. We show not only how MuST exploits spikes to convey information more effectively, but also how it benefits the recognition using SNN. The recognition process is performed in an unsupervised manner, which does not need to specify the desired status of every single neuron of SNN, and thus can be flexibly applied in real-world recognition tasks. The experiments are performed on five AER datasets including a new one named GESTURE-DVS. Extensive experimental results show the effectiveness and advantages of this proposed approach.

A degenerate dynamical system is characterized by a state-dependent multiplier of the time derivative of the state in the time evolution equation. It can give rise to Hamiltonian systems whose symplectic structure possesses a non-constant rank throughout the phase space. Around points where the multiplier becomes singular, flow can experience abrupt and irreversible changes. We introduce a topological index for degenerate dynamical systems around these {\it degeneracy points} and show that it refines and extends the usual topological index in accordance with the Poincar\'e-Hopf Theorem., Comment: 16 pages, 13 figures. The index definition has been polished, Lemma 3.2 and some references have been added