Steady-state Bifurcation Research Articles

Bifurcation and Chaos in Couplings with Clearance

The paper is focused on analysis of dynamic properties of drive systems. It describes the possible ways of stability analysis and possible ways of analysis of bifurcation of steady states and possible occurrence of chaotic behavior.

Bifurcation analysis of a diffusive ratio-dependent predator–prey model

In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.

Mixed quadratic-cubic autocatalytic reaction–diffusion equations: Semi-analytical solutions

Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a one-dimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction–diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations.

Bifurcation and stability analysis of steady states to a Brusselator model

In this paper we consider the structure of the nonnegative steady state solutions of a Brusselator model. We prove the existence and boundedness of continua of the steady state solutions by applying the theorems of global bifurcation theory. The concentration of one intermediary reactant is treated as a bifurcation parameter. Through the asymptotic bifurcation analysis, we derive an explicit formula for the non-constant steady states (i.e., stationary patterns). Using it as a base, we establish the stability criteria and find a selection mechanism of the wave modes for the stable patterns by computing the leading term of the principal eigenvalue. Our results demonstrate that all of the bifurcations other than the one at the first bifurcation location are unstable, and if the pattern is stable, then its mode must be a positive integer which minimizes the bifurcation parameter. A specific example is presented to illustrate our analytical results.

Nonlinear Vibrations of Viscoelastic Plane Truss Under Harmonic Excitation

This paper is concerned with the steady state bifurcations of a harmonically excited two-member plane truss system. A two-degree-of-freedom Duffing system having nonlinear fractional derivatives is derived to govern the dynamic behaviors of the truss system. Viscoelastic properties are described by the fractional Kelvin–Voigt model based on the Caputo definition. The combined method of harmonic balance and polynomial homotopy continuation is adopted to obtain steady state solutions analytically. A parametric study is conducted with the help of amplitude-response curves. Despite its seeming simplicity, the mechanical system exhibits a wide variety of structural responses. The primary and sub-harmonic resonances and chaos are found in specific regions of system parameters. The dynamic snap-through phenomena are observed when the forcing amplitude exceeds some critical values. Moreover, it has been shown that, suppression of undesirable responses can be achieved via changing of viscosity of the system.

Bifurcation Analysis of a Generic Reaction–Diffusion Turing Model

A generic Turing type reaction–diffusion system derived from the Taylor expansion near a constant equilibrium is analyzed. The existence of Hopf bifurcations and steady state bifurcations is obtained. The bifurcation direction and the stability of the bifurcating periodic obits are calculated. Numerical simulations are included to show the rich spatiotemporal dynamics.

Reducibility of steady-state bifurcations in coupled cell systems

A general theory for coupled cell systems was formulated recently by I. Stewart, M. Golubitsky and their collaborators. In their theory, a coupled cell system is a network of interacting dynamical systems whose coupling architecture is expressed by a directed graph called a coupled cell network. An equivalence relation on cells in a regular network (a coupled cell network with identical nodes and identical edges) determines a new network called quotient network by identifying cells in the same equivalence class and determines a quotient system as well. In this paper we develop an idea of reducibility of bifurcations in coupled cell systems associated with regular networks. A bifurcation of equilibria from subspace where states of all cells are equal is called a synchrony-breaking bifurcation. We say that a synchrony-breaking steady-state bifurcation is reducible in a coupled cell system if any bifurcation branch for the system is lifted from those for some quotient system. First, we give the complete classification of codimension-one synchrony-breaking steady-state bifurcations in 1-input regular networks (where each cell receives only one edge). Second, we show that under a mild condition on the multiplicity of critical eigenvalues, codimension-one synchrony-breaking steady-state bifurcations in generic coupled cell systems associated with an n-cell coupled cell network with Dn symmetry, a regular network, is reducible for n>2.

Bifurcations of Tumor-Immune Competition Systems with Delay

A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.

Bifurcations in time-delay fully-connected networks with symmetry

In this work a brief method for finding steady-state and Hopf bifurcations in a (R + 1)-th order N-node time-delay fully-connected network with symmetry is explored. A self-sustained Phase-Locked Loop is used as node. The irreducible representations found due to the network symmetry are used to find regions of time-delay independent stability/instability in the parameter space. Symmetry-preserving and symmetry-breaking bifurcations can be computed numerically using the S n map proposed in (1). The analytic results show the existence of symmetry-breaking bifurcations with multiplicity N − 1. A second-order N-node network is used as application example. This work is a generalization of some results presented in (2).

Stability and Dynamical Analysis of a Biological System

This study considers the spatiotemporal dynamics of a reaction-diffusion phytoplankton-zooplankton system with a double Allee effect on prey under a homogeneous boundary condition. The qualitative properties are analyzed, including the local stability of all equilibria and the global asymptotic property of the unique positive equilibrium. We also discuss the Hopf bifurcation and the steady state bifurcation of the system. These results are expected to help understand the complexity of the Allee effect and the interaction between phytoplankton and zooplankton.

Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling

We investigate the dynamics of a delayed neural network model consisting ofnidentical neurons. We first analyze stability of the zero solution and then study the effect of time delay on the dynamics of the system. We also investigate the steady state bifurcations and their stability. The direction and stability of the Hopf bifurcation and the pitchfork bifurcation are analyzed by using the derived normal forms on center manifolds. Then, the spatiotemporal patterns of bifurcating periodic solutions are investigated by using the symmetric bifurcation theory, Lie group theory andS1-equivariant degree theory. Finally, two neural network models with four or seven neurons are used to verify our theoretical results.

Coupled Cell Networks and Their Hidden Symmetries

Dynamical systems with a coupled cell network structure can display synchronous solutions, spectral degeneracies, and anomalous bifurcation behavior. We explain these phenomena here for homogeneous networks by showing that every homogeneous network dynamical system admits a semigroup of hidden symmetries. The synchronous solutions lie in the symmetry spaces of this semigroup and the spectral degeneracies of the network are determined by its indecomposable representations. Under a condition on the semigroup representation, we prove that a one-parameter synchrony breaking steady state bifurcation in a coupled cell network must generically occur along an absolutely indecomposable subrepresentation. We conclude with a classification of generic one-parameter bifurcations in monoid networks with two or three cells.

Stability and Sensitivity Analysis of a Plant Disease Model with Continuous Cultural Control Strategy

In this paper, a plant disease model with continuous cultural control strategy and time delay is formulated. Then, how the time delay affects the overall disease progression and, mathematically, how the delay affects the dynamics of the model are investigated. By analyzing the transendental characteristic equation, stability conditions related to the time delay are derived for the disease-free equilibrium. Specially, whenR0=1, the Jacobi matrix of the model at the disease-free equilibrium always has a simple zero eigenvalue for allτ≥0. The center manifold reduction and the normal form theory are used to discuss the stability and the steady-state bifurcations of the model near the nonhyperbolic disease-free equilibrium. Then, the sensitivity analysis of the threshold parameterR0and the positive equilibriumE*is carried out in order to determine the relative importance of different factors responsible for disease transmission. Finally, numerical simulations are employed to support the qualitative results.

Spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law

In this paper, a reaction–diffusion bimolecular model with autocatalysis and saturation law is investigated. Firstly, we provide some conditions for the stability/Turing instability of the constant positive solution. Then we mainly consider Hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system. Our results suggest the existence of spatially non-homogeneous periodic orbit and non-constant positive steady state solutions, which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model. Numerical examples are presented to support our theoretical analysis. Furthermore, non-existence of non-constant steady state is investigated in terms of parameters.

Bifurcation from double eigenvalue for nonlinear equation with third-order nondegenerate singularity

In this paper, the steady state bifurcation from a double eigenvalue for nonlinear equation with singularity being second-order fully degenerate and third-order nondegenerate is investigated. By the normalized Lyapunov–Schmidt reduction method, the precise criteria for the existence and nonexistence of bifurcation and the topological properties of regular bifurcated branches are obtained.

Bifurcation analysis of the Gierer–Meinhardt system with a saturation in the activator production

The reaction–diffusion Gierer–Meinhardt system with a saturation in the activator production is considered. Stability of the unique positive constant steady state solution is analysed, and associated Hopf bifurcations and steady state bifurcations are obtained. A global bifurcation diagram of non-trivial periodic orbits and steady state solutions with respect to key system parameters is obtained, which improves the understanding of dynamics of Gierer–Meinhardt system with a saturation in different parameter regimes.

Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis

In this paper, a homogeneous reaction-diffusion model describing the control growth of mammalian hair is investigated. We provide some global analyses of the model depending upon some parametric thresholds/constraints. We find that when one of the dimensionless parameter is less than one, then the unique positive equilibrium is globally asymptotically stable. On the contrary, when this threshold is greater than one, the existence of both steady-state and Hopf bifurcations can be observed under further parametric constraints. In addition, we find that both spatially homogeneous and heterogeneous oscillatory solutions can be seen for some spatially independent parameters provided that some conditions are met. Under these conditions, the direction and stability of these oscillatory behaviors, global stability of the unique constant steady state and the local orbital asymptotic stability of the spatially homogeneous periodic orbits are also investigated.

Dynamics and Stability Issues of a Discretized Sliding-Mode Controlled DC-DC Buck Converter Governed by Fixed-Event-Time Switching

This paper finds and investigates the application of fixed-event-time based discretized sliding mode (DSM) controller in dc-dc buck converter, to achieve fast transient response and high robustness under wide parameters variation. We show that how these can be achieved by integrating the concept of Utkin's equivalent control law and discontinuous border-collision bifurcation (DBCB) theory developed for 2-D discontinuous piecewise smooth (PWS) maps. Moreover, based on derived 2-D discontinuous maps of DSM-controlled converter, we investigate its inherent steady-state dynamical properties or bifurcation behaviors under different parameters variation. Numerically as well as experimentally obtained bifurcation diagrams are then presented to show the domains of existence of different oscillatory modes and their sequence of occurrence. Such phenomena are not only useful to study the robustness of the system but may also facilitates to design the input filter with fast transient response. The performance of DSM controller is experimentally verified and compared with hysteresis and classical peak current-mode controller.

Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady-state bifurcation analysis are carried out in detail. First, the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated by analysing the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the centre manifold reduction for partial functional differential equations and then steady-state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System

The dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey system subject to Neumann boundary conditions are considered. By choosing the ratio of intrinsic growth rates of predators to preys as a bifurcation parameter, the existence and stability of spatially homogeneous and nonhomogeneous Hopf bifurcations and steady state bifurcation are investigated in detail. Meanwhile, we show that Turing instability takes place at a certain critical value; that is, the stationary solution becomes unstable induced by diffusion. Particularly, the sufficient conditions of the global stability of the positive constant coexistence are given by the upper-lower solutions method.