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In this paper, we use new methods to investigate different bifurcations of fixed points in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect. The nonstandard discretization scheme produces a discrete analog of the continuous-time toxic-phytoplankton–zooplankton model with Allee effect. The local stability for proposed system around all of its fixed points is derived. We obtain the codimension-1 conditions of various bifurcations such as period doubling and Neimark–Sacker. Moreover, the system produces codimension-2 bifurcations such as resonance 1:1, 1:2, 1:3, and 1:4. Furthermore, the system can produce very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexisting periodic orbits, and chaotic behavior. Theoretical analysis is validated by numerical methods. [ABSTRACT FROM AUTHOR]

A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.

A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations. [ABSTRACT FROM AUTHOR]

In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher m -fold symmetry with m ≥ 3 the bifurcation does not occur. This answers to a problem left open in [10] . Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in [10] . The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation. [ABSTRACT FROM AUTHOR]

In this study, a mathematical model of a tri-stable energy harvester is developed and used to investigate its formation mechanisms for multi-stability states, nonlinear dynamic behaviors, and power generation performance. Bifurcation analyses for the equilibrium solution of the derived model system are performed. It is shown that the present energy harvester system can exhibit multi-stable (mono-, bi-, and tri-stable) behaviors depending on the two geometric parameters associated with the locations of the tip and external magnets. It is also found that the tri-stability is initiated by a new pitchfork bifurcation or a degenerate pitchfork bifurcation that leads to a pair of saddle-node bifurcations. Bifurcation set diagram is obtained in the parametric space of these two geometric parameters, which can be used to design the potential wells of the multi-stable energy harvesters. Potential energy diagrams are also obtained and they show that the distance between the outer potential energy wells for the tri-stable state is formed in a way that it is larger than that for the equivalent bi-stable state. A series of numerical simulations performed on the present system well illustrates the high output power generation characteristics of the tri-stable energy harvester over a broad operating frequency band. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical behavior of a discrete-time neural network system from both analytical and numerical points of view. The conditions as well as the critical coefficients for the pitchfork, flip (period-doubling), Neimark-Sacker, and strong resonances are computed analytically. Using critical coefficients, the bifurcation scenarios were determined for each bifurcation point. By changing one or two parameters, bifurcation curves of fixed points and cycles with periods up to four iterates, were obtained. Numerical analysis validates our analytical results and reveals more complex dynamical behaviors. [ABSTRACT FROM AUTHOR]