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Anti-control of Hopf bifurcation is one of the hot topics in nonlinear dynamics research, which is to make the system generate or strengthen bifurcation at a prespecified location. The Willamowski–Rössler system is taken as the research object, which is a nonlinear dynamic system derived from chemical reaction processes. Using the higher-dimensional Hopf bifurcation theory, the critical value of the Hopf bifurcation of a non-zero equilibrium point is obtained. A state feedback control method is proposed. With this method, anti-control of Hopf bifurcation for the system is accomplished by a hybrid controller, which is composed of linear and nonlinear controllers. The relationship between the bifurcation parameter and the control parameters of the linear controller is obtained. The values of the control parameters of the linear controller are determined by the bifurcation parameter. Although the critical values of the bifurcation parameters are not determined by the control parameters of the nonlinear controller, the parameter can change the amplitude of the limit cycle, which is inversely proportional to the amplitude of the limit cycle. Finally, the theoretical analysis is verified by numerical simulation.

We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can be obtained from the coefficients of the systems without using explicit solutions of their linearizations at the given branch. Our constructions are based on a comparison principle for the spectral flow and a generalization of a bifurcation theorem due to Szulkin.

A dynamic bifurcation analysis on a three-species cooperating model was presented and it was proved that the problem bifurcated an attractor as the parameter λ crossed the critical value λ 0 . The analysis was based on the attractor bifurcation theory together with the central manifold reduction.

Spontaneous pattern formation by a large number of dislocations is commonly observed during the initial stages of metal fatigue under cyclic straining. It was experimentally found that the geometry of the dislocation pattern undergoes a crossover from a 2D spot-scattered pattern to a 1D ladder-shaped pattern as the amplitude of external shear strain increases. However, the physical mechanism that causes the crossover between different dislocation patterns remains unclear. In this study, we theorized a bifurcation diagram that explains the crossover between the two dislocation patterns. The proposed theory is based on a weakly nonlinear stability analysis that considers the mutual interaction of dislocations as a nonlinearity. It was found that the selection rule among the two dislocation patterns, “spotted” and “ladder-shaped”, can be described by inequalities with respect to nonlinearity parameters contained in the governing equations.

This is the second of two volumes dedicated to the centennial of the distinguished mathematician Selim Grigorievich Krein. The companion volume is Contemporary Mathematics, Volume 733.Krein was a major contributor to functional analysis, operator theory, partial differential equations, fluid dynamics, and other areas, and the author of several influential monographs in these areas. He was a prolific teacher, graduating 83 Ph.D. students. Krein also created and ran, for many years, the annual Voronezh Winter Mathematical Schools, which significantly influenced mathematical life in the former Soviet Union.The articles contained in this volume are written by prominent mathematicians, former students and colleagues of Selim Krein, as well as lecturers and participants of Voronezh Winter Schools. They are devoted to a variety of contemporary problems in ordinary and partial differential equations, fluid dynamics, and various applications.

This volume is a tribute to one of the founders of modern theory of dynamical systems, the late Dmitry Victorovich Anosov.It contains both original papers and surveys, written by some distinguished experts in dynamics, which are related to important themes of Anosov's work, as well as broadly interpreted further crucial developments in the theory of dynamical systems that followed Anosov's original work.Also included is an article by A. Katok that presents Anosov's scientific biography and a picture of the early development of hyperbolicity theory in its various incarnations, complete and partial, uniform and nonuniform.

In this paper, we propose a reaction-diffusion genetic regulatory network with Neumann boundary conditions. We incorporate the diffusion factors into the delayed mathematical model of genetic networks, and with the time delays, we demonstrate the local stability and Hopf bifurcation in accordance with the Hopf bifurcation theory. When the sum of the delays exceeds the critical value, a Hopf bifurcation occurs. Finally, we select the appropriate system parameters and present a numerical simulation to illustrate the theoretical analysis.

In this paper, the dynamic bifurcation of the three-species cooperating model is considered. It worth noting that the main theory of this paper is the Center manifold reduction and attractor bifurcation theory, which is developed by Ma [1,2]. The main work of this paper shows that if the algebraic multiplicity of the first eigenvalue is 2, there exists an S1 attractor bifurcation, and the number of its singular points can only be eight. Besides, we show that the simplified governing equations bifurcate to an S1 attractor, when the Control parameter λ crosses a critical value λ0.

In this paper, the codimension-one bifurcations of a monopoly model with memory are explored. Based on bifurcation theory and the center manifold theorem, we rigorously derive the existence of the flip bifurcation and the supercritical Neimark–Sacker bifurcation. With respect to these two bifurcations, we not only derive the bifurcation directions and stability properties, but also provide concise non-degeneracy conditions expressed in terms of the system parameters, and validate the theoretical conclusions through numerical simulations. The results indicate that with the increase in the memory weight w . The system first exhibits quasi-periodicity and Neimark-Sacker bifurcation, and then evolves into a stable state. However, when w exceeds a critical value, the system undergoes a flip bifurcation on a route to chaos. From an economic perspective, if w is too low, a company will be constrained by past successful experiences and sunk costs, trapped in path dependence, and resistant to innovation. Conversely, if w is too high, the company will be unable to accumulate experience or reduce costs, leading to inefficient and volatile development.

Many natural systems exhibit tipping points where slowly changing environmental conditions spark a sudden shift to a new and sometimes very different state. As the tipping point is approached, the dynamics of complex and varied systems simplify down to a limited number of possible “normal forms” that determine qualitative aspects of the new state that lies beyond the tipping point, such as whether it will oscillate or be stable. In several of those forms, indicators like increasing lag-1 autocorrelation and variance provide generic early warning signals (EWS) of the tipping point by detecting how dynamics slow down near the transition. But they do not predict the nature of the new state. Here we develop a deep learning algorithm that provides EWS in systems it was not explicitly trained on, by exploiting information about normal forms and scaling behavior of dynamics near tipping points that are common to many dynamical systems. The algorithm provides EWS in 268 empirical and model time series from ecology, thermoacoustics, climatology, and epidemiology with much greater sensitivity and specificity than generic EWS. It can also predict the normal form that characterizes the oncoming tipping point, thus providing qualitative information on certain aspects of the new state. Such approaches can help humans better prepare for, or avoid, undesirable state transitions. The algorithm also illustrates how a universe of possible models can be mined to recognize naturally occurring tipping points.