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A quasi-three-dimensional mathematical model has been developed to study the morphological processes based on equilibrium sediment transport method. The flow velocities are computed by a two-dimensional horizontal depth-averaged flow model (H2D) in combination with logarithmic velocity profiles. The transport of sediment particles by a flow water has been considered in the form of bed load and suspended load. The bed load transport rate is defined as the transport of particles by rolling and saltating along the bed surface and is given by the Van Rijn relationship (1987). The equilibrium suspended load transport is described in terms of an equilibrium sediment concentration profile (ce) and a logarithmic velocity (u). Based on the equilibrium transport, the bed change rate is given by integration of the sediment mass-balance equation. The model results have been compared with a Van Rijn results (equilibrium approach) and good agreement has been found.

In the study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In this paper, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for the study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.

Results of our recent studies concerning possible effects of Λ > 0 for equilibrium positions of spinning test particles and stationary configurations of perfect-fluid tori are presented.

We examine the long-term behavior of nonintegrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasi-equilibrium. Here, we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. This phase is described by equilibrium statistical mechanics, opening up the possibility that the machinery of nonequilibrium statistical mechanics may be used to understand the behavior of these systems away from equilibrium.

Shortest path plays an important role in the study of complex networks. But in real transportation systems, choosing the shortest path may not be the best way for the drivers. Based on the traffic equilibrium theory, we generalize the concept of shortest path. Flux distribution is also investigated by using the generalized concept on various types of complex networks. We find that the flux differs little in all the edges of lattice while in small-world and scale-free networks, the flux distribution follows a power law, and in the random network, the flux distribution has an exponential tail. We consider lattice may be the optimal topology in design a transportation network.

In this paper, we present two epidemic models with a nonlinear incidence and transfer from infectious to recovery. For epidemic models, the basic reproductive number is calculated. A dynamic system based on threshold, using LaSalle’s invariance principle and Lyapunov function, is structured completely by the basic reproductive number. By studying the SIR and SIRS models under the nonlinear condition, the general validity of the method is verified.

In this paper, the problem of the global asymptotic stability (GAS) of a class of delayed neural network is investigated. Under the generalization of dropping the boundedness and differentiability hypotheses for activation functions, using some existing results for the existence and uniqueness of the equilibrium point, we obtain a couple of general results concerning GAS by means of Lyapunov functional method without the assumption of symmetry of interconnection matrix. Our results improve and extend some previous works of other researchers. Moreover, our conditions are presented in terms of system parameters, which have leading significance in designs and applications of GAS for Hopfield neural network (HNNs) and delayed cellular neural network (DCNNs).

We consider the following system of delay differential equations

This paper addresses bifurcation properties of equilibria in lumped electrical circuits. The goal is to tackle these properties in circuit-theoretic terms, characterizing the bifurcation conditions in terms of the underlying network digraph and the electrical features of the circuit devices. The attention is mainly focused on so-called singular bifurcations, resulting from the semistate (differential-algebraic) nature of circuit models, but the scope of our approach seems to extend to other types of bifurcations. The bifurcation analysis combines different tools coming from graph theory (such as proper trees in circuit digraphs, Maxwell's determinantal expansions or the colored branch theorem) with several results from linear algebra (matrix pencils, the Cauchy–Binet formula, Schur complements). Several examples illustrate the results.

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincaré map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.

In this letter we investigate the role of complex fixed-points in finding hidden attractors in chaotic flows with no equilibria. If these attractors could be found by starting the trajectory in the neighborhood of complex fixed-points, maybe it would be better not to call them hidden.

Discovering unknown features of no-equilibrium systems with hidden strange attractors is an attractive research topic. This paper presents a novel no-equilibrium chaotic system that is constructed by using a state feedback controller. Interestingly, the new system can exhibit multiwing butterfly attractors. Moreover, a new chaotic system with an infinite number of equilibrium points, which can generate multiscroll attractors, is also proposed by applying the introduced methodology.

Chua's circuit, an archetypal example of nonsmooth dynamical systems, exhibits mostly discontinuous bifurcations. More complex dynamical phenomena of Chua's circuit are presented here due to discontinuity-induced bifurcations. Some new kinds of classical bifurcations are revealed and analyzed, including the coexistence of two classical bifurcations and bifurcations of equilibrium manifolds. The local dynamical behavior of the boundary equilibrium points located on switch boundaries is found to be determined jointly by the Jacobian matrices evaluated before and after switching. Some new discontinuous bifurcations are also observed, such as the coexistence of two discontinuous and one classical bifurcation.

The nonlinear dynamical behavior of an atmospheric circulation in a beta-plane channel is examined on a five-spectral mode model, truncated from the Charney and DeVore quasi-geostrophic equation. Bifurcation and chaos are observed when subjected to a topographic driving disturbance and a thermally driving zonal source. An equilibrium state undergoes supercritical Hopf bifurcation and becomes a stable periodic state with respect to the magnitude of the thermally driving source, whereas the periodic state undergoes a subcritical Hopf bifurcation and transforms into a low-index equilibrium state with respect to the increasing topographic driving disturbance. The stable periodic state further develops into a pair of stable periodic states when increasing the thermally driving source. The first one with the period of 4.3 days exhibits an oscillation of strong and weak zonal flow patterns, whereas the second one with the period of 6.8 days demonstrates a fluctuation amongst weak zonal disturbance flow patterns. Moreover, the two periodic states transform respectively into chaos through separate period-doubling cascades with the further development of the thermally driving source.

Although many chaotic systems have been introduced in the literature, a few of them possess uncountably infinite equilibrium points. The aim of our short work is to widen the current knowledge of the chaotic systems with an infinite number of equilibria. A three-dimensional system with special properties, for example, exhibiting chaotic attractor with circular equilibrium, chaotic attractor with ellipse equilibrium, chaotic attractor with square-shaped equilibrium, and chaotic attractor with rectangle-shaped equilibrium, is proposed.

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

The chaotic systems with hidden attractors, such as chaotic systems with a stable equilibrium, chaotic systems with infinite equilibria or chaotic systems with no equilibrium have been investigated recently. However, the relationships between them still need to be discovered. This work explains how to transform a system with one stable equilibrium into a new system with an infinite number of equilibrium points by using a memristive device. Furthermore, some other new systems with infinite equilibria are also constructed to illustrate the introduced methodology.

Applying some transformation to an autonomous system, we obtain a new system, which might keep the dynamical behavior of the original system or generate different dynamics. But this is often accompanied by the appearance of discontinuous points, where the vector field for the new system is not continuous at these points. We discuss the effects of the discontinuous points, and provide two methods to construct systems with any preassigned number of chaotic attractors via some transformation. The first one does not change the geometric structure of the attractors, since the discontinuous points are out of the basin of attraction. The second one might make the new systems have different dynamics, like multiscroll chaotic attractors, or infinitely many chaotic attractors. These results illustrate that both the equilibria and the discontinuous points affect the global dynamics.

We present the development of a new theory of the pitchfork bifurcation, which removes the perspective of the third derivative and a requirement of symmetry.

A new chaotic system having variable equilibrium is introduced in this paper. The presence of an infinite number of equilibrium points, a stable equilibrium, and no-equilibrium is observed in the system. Interestingly, this system is classified as a rare system with hidden attractors from the view point of computation. Complex dynamical behavior and a circuital implementation of the new system have been investigated in our work.