Search

In this paper, we study a predator–prey model assuming that the prey population gathers together in herd and considering feeding satiation for the predator population as well. After analyzing the equilibrium points of the model, their stability and the existence of bifurcations we show the existence of multistability for three different equilibrium points via numerical simulations. This last analysis is performed using the bSTAB software and its extensions. It allows to compute the basin of stability values and to plot bifurcation diagram surfaces with respect to the model parameters. • A predator-prey model is studied assuming that the prey population gathers in herd. • The equilibrium points of the model are analyzed, as well as their stability. • We show the existence of multistability for three different equilibrium points. [ABSTRACT FROM AUTHOR]

This paper describes a new approach to study the nonlinear dynamical behaviors of a belt-driven two-coupled friction oscillator. First, using Buckingham's π theorem, several dimensionless and physically meaningful parameters are derived. Their relationships with the steady state responses of the system are revealed through numerical simulations. Then a new mapping tool named as sticking phase plot is developed to record the transition process of stick–slip motion. It intuitively shows how bifurcation occurs in sticking region and transforms the dynamical responses. The bifurcation diagrams of the system are obtained by sweeping the dimensionless parameters. Some extraordinary bifurcation phenomena are observed in this system. By using sticking phase plot, it is found that the bifurcations can be broadly divided into four main categories: the Border-Collision Bifurcation, the Grazing-Sliding Bifurcation, the Multi-Sliding Bifurcation, and the Fixed-Point Bifurcation. New nonlinear phenomena are found herein, including chaos caused by the Border-Collision Bifurcation and the Sliding Bifurcations. The local and the global changes caused by the Fixed-Point bifurcation are also observed. In numerical simulation, we adopted the LVIM method proposed by the authors, which can be used to integrate non-smooth systems very accurately and efficiently. • A panoramic view of the dynamics of a belt driven two-coupled nonlinear oscillators is demonstrated. • A new event-driven numerical solver named LVIM is described to efficiently solve the non-smooth vibration system. • A so-called sticking phase plot is proposed to illustrate the complex bifurcations of the sticking–slipping motion. • Interesting phenomena, involving chaotic motions caused by new type of bifurcations are demonstrated. [ABSTRACT FROM AUTHOR]

• A Cournot-Bertrand duopoly mixed competition model is established. • The stability conditions of the system are analyzed by numerical calculation and numerical simulation. • The Arnold's tongues are studied. • The coexistence of attractors is analyzed by using the basin of attraction. • The contact bifurcation in the basin of attraction is studied by using the theory of critical curve and noninvertible map. In this paper, we consider the Cournot-Bertrand duopoly mixed competition model, which is characterized by different decision-making variables and different objective functions of the two enterprises. This form of competition is more consistent with the complex actual economic market. The general stability conditions of the four equilibria are given and analyzed with eigenvalues and Jury criterion, so that enterprise decision makers could choose appropriate parameters to determine the development of the enterprise. Under the specific parameter conditions, the stability conditions are analyzed and demonstrated in detail by using two-dimensional bifurcation diagrams, stable region diagrams and bifurcation curves. Many fractal Arnold's tongues are found in two-dimensional bifurcation diagrams. These tongues are associated with the Neimark-Sacker bifurcation of the fixed point and are arranged in accordance with the organization law of the periodic tree of the Stern-Brocot tree, which is helpful for us to analyze the transitions of system between period and chaos. With the help of the basin of attraction, the coexistence of attractors is analyzed, on which the decision maker can choose the initial conditions that are beneficial to the development of the enterprise. We also analyze the topology structure of basin of attraction and the formation mechanism of holes in the basin of attraction by using the theory of critical curve and noninvertible map. [ABSTRACT FROM AUTHOR]