This paper is part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši'lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši'lnikov homoclinic orbit type; (2) chaos of the Ši'lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši'lnikov homoclinic and homoclinic orbits; (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g. one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits.
No abstract is provided for this article.
As regional relationship studies shift from the central place paradigm to a network paradigm, understanding the structural patterns and evolutionary mechanisms of urban network is crucial for network-oriented regional planning. This study leverages firm-level big data, comprising 105,123 headquarters and 253,535 branches, and applies the Temporal Exponential Random Graph Model to analyze the structural evolution and driving mechanisms of the urban network in the Yangtze River Delta (YRD). From 1995 to 2020, the YRD exhibited a clear trend toward polycentric and networked development, accompanied by significant regional disparities. The YRD urban network is characterized as a scale-free network, exhibitting distinct hierarchical patterns and a tendency for preferential attachment. The network comprising both horizontal connections between cities of the same tier and vertical connections between cities of different tiers and higher-tier cities show stronger enterprise connections, while lower-tier cities prioritize linking with higher-tier cities over those of the same or lower tiers. The evolution of the urban network in the YRD is driven by mechanisms such as size-based agglomeration and dispersion effects, temporal dependence, network self-organization, preferential attachment, assortative mechanism, and multi-dimensional proximities. This study enhances our theoretical understanding of the structural patterns and evolutionary mechanisms of urban network.
No abstract is provided for this article.
Ordered bursting synchronization and complex propagation are investigated for a ring neuronal network in which each neuron exhibits chaotic bursting behaviour. The neurons become more and more synchronous in chaotic bursting as the synaptic strength is increased. It is shown that excitatory chemical synapses can effectively tame the chaos, and ordered bursting synchronization can be observed as the synaptic strength is further increased. However, synchronization among neurons is weakened as the number of neurons is increased. More importantly, it is shown that ordered bursting synchronization can be turned into spiking synchronization at certain noise intensity. Complex spatio-temporal patterns propagating towards both sides of pacemaker are found in this network before the emergence of spiking synchronization.
In this paper, we present some conventional feedback controller design principles for chaos control, with mathematical controllability conditions derived via the Lyapunov function methods. The chaotic Chua's circuit and Duffing oscillator are used as examples to illustrate the fundamental concepts and basic methodology employed by this unified Lyapunov approach, in both linear and non-linear controllers design, for the control of chaotic dynamics.
Density evolution is a new method for analyzing the asymptotic performance of network capability approaching error-correcting codes. For irregular LDPC codes with message-passing decoding, the density evolution method can track the messages to find out the threshold, enabling optimization of the degree distribution. In this paper, the principle of density evolution combined with the decoding process is firstly explored. Then, two algorithms for programming the evolution proceeding are discussed: the discretized density evolution and the Gaussian approximation, as well as their application conditions. Finally, simulation results are presented.
As embedded microprocessors are applied widerly to multi-agent systems, control scheduling and time-delay problems arose in the case of limited energy and computational ability. It has been shown that the event-triggered actuation strategy is an effective methodology for designing distributed control of multi-agent systems with limited computational resources. In this paper, a tracking control problem of leader-follower multi-agent systems with/without communication delays is formulated and a distributed dynamic tracking control is designed by employing event-triggered technique. Then, the input-to-state stability of the closed-loop multi-agent system with directed interconnections is analyzed. Finally, a numerical example is given to validate the proposed control.
Cluster synchronization in a network of non-identical dynamic systems is studied in this paper, using two-cluster synchronization for detailed analysis and discussion. The results show that the common intercluster coupling condition is not always needed for the diffusively coupled network. Several sufficient conditions are obtained by using the Schur unitary triangularization theorem, which extends previous results. Some numerical examples are presented for illustration.