Narrow Results

A calculation of site-bond percolation thresholds in many lattices in two to five dimensions is presented. The line of threshold values has been parametrized in the literature, but we show here that there are strong deviations from the known approximate equations. We propose an alternative parametrization that lies much closer to the numerical values.

We study a delayed system with feedback modulation of the nonlinear parameter. Study of the system as a function of nonlinearity and modulation parameters reveals complex dynamical phenomena: different types of coexisting attractors, local or global bifurcations and transitions. Bistability and dynamical attractors can be distinguished in some parameter-space regions, which may be useful to drive chaotic dynamics, unstable attractors or bistability towards regular dynamics. At the bifurcation to bistability, two striking features are that they lead to fundamentally unpredictable behavior of orbits and crisis of attractors as system parameters are varied slowly through the critical curve. Unstable attractors are also investigated in bistable regions, which are easily mistaken for true multi-periodic orbits judging merely from zero measure local basins. Lyapunov exponents and basins of attraction are also used to characterize the phenomenon observed.

Many applications give rise to systems that can be described by maps that do not have a unique inverse. We consider here the case of a planar noninvertible map. Such a map folds the phase plane, so that there are regions with different numbers of preimages. The locus, where the number of preimages changes, is made up of so-called critical curves, that are defined as the images of the locus where the Jacobian is singular. A typical critical curve corresponds to a fold under the map, so that the number of preimages changes by two.

We consider the question of how the stable set of a hyperbolic saddle of a planar noninvertible map changes when a parameter is varied. The stable set is the generalization of the stable manifold for the case of an invertible map. Owing to the changing number of preimages, the stable set of a noninvertible map may consist of finitely or even infinitely many disjoint branches. It is now possible to compute stable sets with the Search Circle algorithm that we developed recently.

We take a bifurcation theory point of view and consider the two basic codimension-one interactions of the stable set with a critical curve, which we call the outer-fold and the inner-fold bifurcations. By taking into account how the stable set is organized globally, these two bifurcations allow one to classify the different possible changes to the structure of a basin of attraction that are reported in the literature. The fundamental difference between the stable set and the unstable manifold is discussed. The results are motivated and illustrated with a single example of a two-parameter family of planar noninvertible maps.

A dynamic oligopoly game model with nonlinear cost and strategic delegation is built on the basis of isoelastic demand in this paper. And the dynamic characteristics of this game model are investigated. The local stability of the boundary equilibrium points is analyzed by means of the stability theory and Jacobian matrix, and the stability region of the Nash equilibrium point is obtained by Jury criterion. It is concluded that the system may lose stability through Flip bifurcation and Neimark–Sacker bifurcation. And the effects of speed of adjustment, price elasticity, profit weight coefficient and marginal cost on the system stability are discussed through numerical simulation. After that, the coexistence of attractors is analyzed through the basin of attraction, where multiple stability always means path dependence, implying that the long-term behavior of enterprises is strongly affected by historical contingency. In other words, a small perturbation of the initial conditions will have a significant impact on the system. In addition, the global dynamical behavior of the system is analyzed by using the critical curves, the basin of attraction, absorbing areas and a noninvertible map, revealing that three global bifurcations, the first two of which are caused by the interconversion of simply-connected and multiply-connected regions in the basin of attraction, and the third global bifurcation, that is, the final bifurcation is caused by the contact between attractors and the boundary of the basin of attraction.

Taking into account the nonlinear demand function, we have developed a multi-agent fishery economic model, where a multitude of agents are bounded by rationality. The fishing decisions of these agents are driven by a profit gradient mechanism. To assess the local stability of the system, stability analysis is performed with the Jury criterion. The investigation has revealed the presence of two conventional paths to chaos, namely, the flip bifurcation and the Neimark–Sacker bifurcation. This was achieved by mapping the stability regions and stability curves of the Nash equilibrium. The multistability of the system is further explored on two-dimensional planes on which the influence of joint parameters on the system’s stability is demonstrated. The existence of Arnold’s tongue has demonstrated unparalleled complexity and intricate interactions across different scales of the system. Both critical curves and basins of attraction are illustrated to gain insight into global bifurcations. The chaotic attractor is found to be confined within specific boundaries. The findings clearly show higher maximum instantaneous demand, relatively slower adjustment speed, and lower price sensitivity. Arguably, a controlled cost would lead to sustainable fishing resources. Moreover, the results also suggest that the agents would benefit more from confined conditions.

Many applications give rise to systems that can be described by maps that do not have a unique inverse. We consider here the case of a planar noninvertible map. Such a map folds the phase plane, so that there are regions with different numbers of preimages. The locus, where the number of preimages changes, is made up of so-called critical curves, that are defined as the images of the locus where the Jacobian is singular. A typical critical curve corresponds to a fold under the map, so that the number of preimages changes by two.

We consider the question of how the stable set of a hyperbolic saddle of a planar noninvertible map changes when a parameter is varied. The stable set is the generalization of the stable manifold for the case of an invertible map. Owing to the changing number of preimages, the stable set of a noninvertible map may consist of finitely or even infinitely many disjoint branches. It is now possible to compute stable sets with the Search Circle algorithm that we developed recently.

We take a bifurcation theory point of view and consider the two basic codimension-one interactions of the stable set with a critical curve, which we call the outer-fold and the inner-fold bifurcations. By taking into account how the stable set is organized globally, these two bifurcations allow one to classify the different possible changes to the structure of a basin of attraction that are reported in the literature. The fundamental difference between the stable set and the unstable manifold is discussed. The results are motivated and illustrated with a single example of a two-parameter family of planar noninvertible maps.