This article briefly introduces the generalized Lorenz systems family, which includes the classical Lorenz system and the relatively new Chen system as special cases, with infinitely many related but not topologically equivalent chaotic systems in between.
In network science, the non-homogeneity of node degrees has been a concerned issue for study. Yet, with the modern web technologies today, the traditional social communication topologies have evolved from node-central structures to online cycle-based
communities, urgently requiring new network theories and tools. Switching the focus from node degrees to network cycles, it could reveal many interesting properties from the perspective of totally homogeneous networks, or sub-networks in a complex network, especially basic simplexes (cliques) such as links and triangles. Clearly, comparing to node degrees it is much more challenging to deal with network cycles. For studying the latter, a new clique vector space framework is introduced in this paper, where the vector space with a basis consisting of links has the dimension equal to the number of links, that with a basis consisting of triangles has the dimension equal to the number of triangles, and so on. These two vector spaces are related through a boundary operator, e.g., mapping the boundary of a triangle in one space to the sun of three links in the other space. Under the new framework, some important concepts and methodologies from algebraic topology, such as characteristic number, homology group and Betti number, will have a play in network science leading to foreseeable new research directions. As immediate applications, the paper illustrates some important characteristics affecting the collective behaviors of complex networks, some new cycle-dependent importance indexes of nodes, and implications for network synchronization and brain network analysis.
When implemented in the digital domain with time, space and value discretized in the binary form, many good dynamical properties of chaotic systems in continuous domain may be degraded or even diminish. To measure the dynamic complexity of a digital
chaotic system, the dynamics can be transformed to the form of a state-mapping network. Then, the parameters of the network are verified by some typical dynamical metrics of the original chaotic system in infinite precision, such as Lyapunov exponent and entropy. This article reviews some representative works on the network-based analysis of digital chaotic dynamics and presents a general framework for such analysis, unveiling some intrinsic relationships between digital chaos and complex networks. As an example for discussion, the dynamics of a state-mapping network of the Logistic map in a fixed-precision computer is analyzed and discussed.
In this paper, we consider the state controllability of networked systems, where the network topology is directed and weighted and the nodes are higher-dimensional linear time-invariant (LTI) dynamical systems. We investigate how the network topology
, the node-system dynamics, the external control inputs, and the inner interactions affect the controllability of a networked system, and show that for a general networked multi-input/multi-output (MIMO) system: 1) the controllability of the overall network is an integrated result of the aforementioned relevant factors, which cannot be decoupled into the controllability of individual node-systems and the properties solely determined by the network topology, quite different from the familiar notion of consensus or formation controllability; 2) if the network topology is uncontrollable by external inputs, then the networked system with identical nodes will be uncontrollable, even if it is structurally controllable; 3) with a controllable network topology, controllability and observability of the nodes together are necessary for the controllability of the networked systems under some mild conditions, but nevertheless they are not sufficient. For a networked system with single-input/single-output (SISO) LTI nodes, we present precise necessary and sufficient conditions for the controllability of a general network topology.
We study the effects of free will and massive opinion of multi-agents in a majority rule model wherein the competition of the two types of opinions is taken into account. To address this issue, we consider two specific models (model I and model II) i
nvolving different opinion-updating dynamics. During the opinion-updating process, the agents either interact with their neighbors under a majority rule with probability $1-q$, or make their own decisions with free will (model I) or according to the massive opinion (model II) with probability $q$. We investigate the difference of the average numbers of the two opinions as a function of $q$ in the steady state. We find that the location of the order-disorder phase transition point may be shifted according to the involved dynamics, giving rise to either smooth or harsh conditions to achieve an ordered state. For the practical case with a finite population size, we conclude that there always exists a threshold for $q$ below which a full consensus phase emerges. Our analytical estimations are in good agreement with simulation results.
This paper provides a unified method for analyzing chaos synchronization of the generalized Lorenz systems. The considered synchronization scheme consists of identical master and slave generalized Lorenz systems coupled by linear state error variable
s. A sufficient synchronization criterion for a general linear state error feedback controller is rigorously proven by means of linearization and Lyapunovs direct methods. When a simple linear controller is used in the scheme, some easily implemented algebraic synchronization conditions are derived based on the upper and lower bounds of the master chaotic system. These criteria are further optimized to improve their sharpness. The optimized criteria are then applied to four typical generalized Lorenz systems, i.e. the classical Lorenz system, the Chen system, the Lv system and a unified chaotic system, obtaining precise corresponding synchronization conditions. The advantages of the new criteria are revealed by analytically and numerically comparing their sharpness with that of the known criteria existing in the literature.
Recently, an image encryption scheme based on a compound chaotic sequence was proposed. In this paper, the security of the scheme is studied and the following problems are found: (1) a differential chosen-plaintext attack can break the scheme with on
ly three chosen plain-images; (2) there is a number of weak keys and some equivalent keys for encryption; (3) the scheme is not sensitive to the changes of plain-images; and (4) the compound chaotic sequence does not work as a good random number resource.
In this paper, the synchronizability problem of dynamical networks is addressed, where better synchronizability means that the network synchronizes faster with lower-overshoot. The L2 norm of the error vector e is taken as a performance index to meas
ure this kind of synchronizability. For the equilibrium synchronization case, it is shown that there is a close relationship between the L2 norm of the error vector e and the H2 norm of the transfer function G of the linearized network about the equilibrium point. Consequently, the effect of the network coupling topology on the H2 norm of the transfer function G is analyzed. Finally, an optimal controller is designed, according to the so-called LQR problem in modern control theory, which can drive the whole network to its equilibrium point and meanwhile minimize the L2 norm of the output of the linearized network.
In this paper, the problem of pinning control for synchronization of complex dynamical networks is discussed. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is
that lower cost is achieved by the control scheme of pinning nodes with smaller degrees. Some rigorous mathematical analysis is presented for achieving lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barabasi-Albert model and various star-shaped networks are shown for verification and illustration.
In this paper, subgraphs and complementary graphs are used to analyze the network synchronizability. Some sharp and attainable bounds are provided for the eigenratio of the network structural matrix, which characterizes the network synchronizability,
especially when the networks corresponding graph has cycles, chains, bipartite graphs or product graphs as its subgraphs.
