Bifurcation Of Periodic Solutions Research Articles

Global Hopf bifurcation and dynamics of a stage-structured model with delays for tick population

In this paper, we study a three-stage tick population model with three development delays. Using the total delay τ as the bifurcation parameter, we conduct local and global Hopf bifurcation analysis. Especially, we examine the onset and termination of Hopf bifurcations of periodic solutions from the unique positive equilibrium. We locate all of the stability switches for the equilibrium and demonstrate that the global Hopf bifurcation branches are bounded. This result implies the system undergoes oscillatory behavior only with bounded delays. The key step in the proof is the exclusion of 2τ-periodic solutions due to the negative Lozinskiĭ measure of the corresponding compound matrix. Numerical simulations are provided to show the complicated patterns of the tick population when the development delays are altered by the temperature and other environmental factors.

Bifurcation and Number of Periodic Solutions of Some 2n-Dimensional Systems and Its Application

In high dimension, the bifurcation theory of periodic orbits of nonlinear dynamics systems are difficult to establish in general. In this paper, by performing the curvilinear coordinate frame and constructing a Poincare map, we obtain some sufficient conditions of the bifurcation of periodic solutions of some 2n-dimensional systems for the unperturbed system in two cases: one is a decoupled n-degree-of-freedom nonlinear Hamiltonian system and the other has an isolated invariant torus. We use a new method and study new types of systems compared with the existing results. As an application we study the bifurcation and number of periodic solutions of an ice covered suspension system. Under a certain parametrical condition, the number of periodic solutions of this system can be 2 or 1 with the variation of parameter $$p_{2}$$ .

Stability and bifurcation analysis of $ SIQR $ for the COVID-19 epidemic model with time delay

<abstract> Based on the $ SIQR $ model, we consider the influence of time delay from infection to isolation and present a delayed differential equation (DDE) according to the characteristics of the COVID-19 epidemic phenomenon. First, we consider the existence and stability of equilibria in the above delayed $ SIQR $ model. Second, we analyze the existence of Hopf bifurcations associated with two equilibria, and we verify that Hopf bifurcations occur as delays crossing some critical values. Then, we derive the normal form for Hopf bifurcation by using the multiple time scales method for determining the stability and direction of bifurcation periodic solutions. Finally, numerical simulations are carried out to verify the analytic results. </abstract>

Dynamics of a Coupled Chua’s Circuit with Lossless Transmission Line

This paper proposes a coupled-circuit system composed of two Chua’s circuits with lossless transmission lines. By applying the Kirchhoff’s voltage and current laws, the equations that describe the coupled-circuit system are reduced to two coupled neutral-type differential equations with a time delay. Subsequently, the conditions for global stability are established using the inequality technology, and those for local stability and Hopf bifurcation are obtained by selecting the length of the transmission line as the bifurcation parameter. By using the normal-form theory and central manifold theorem, the formulas for the Hopf bifurcation direction and bifurcation periodic solution are obtained. Finally, the numerical simulations not only verify the theoretical analysis but also show that chaos exists near the Hopf bifurcation point.

Suspension bridges with non-constant stiffness: bifurcation of periodic solutions

We consider a modified version of a suspension bridge model with a spatially variable stiffness parameter to reflect the discrete nature of the placement of the bridge hangers. We study the qualitative and quantitative properties of this model and compare the cases of constant and non-constant coefficients. In particular, we show that for certain values of the stiffness parameter, the bifurcation occurs. Moreover, we can expect also the appearance of blowups, whose existence is closely connected with the so-called Fucik spectrum of the corresponding linear operator.

Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation

In this paper, the periodic solutions of a certain one-dimensional differential equation are investigated for the first order cubic non-autonomous equation. The method used here is the bifurcation of periodic solutions from a fine focus $z=0$. We aimed to find the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For classes $C_{3,8},C_{4,3},C_{7,5},C_{7,6}$ eight periodic multiplicity have been found. To investigate the multiplicity greater than 9, the formula for focal value was not available in the literature. We also succeeded to construct the formula for $\eta _{10}$, by implementing our newly developed formula we are able to get multiplicity ten for classes $ C_{7,3},C_{9,1}$, which is highest known to date. Perturbation method has been properly established in making the maximal number of limit cycles for each class. To check the implementation of the newly developed method, some examples are also presented. By considering all these facts it can be concluded that presented methods are new, authentic and novel.

Complex dynamics in a delay differential equation with two delays in tick growth with diapause

We consider a delay differential equation for tick population with diapause, derived from an age-structured population model, with two time lags due to normal and diapause mediated development. We derive threshold conditions for the global asymptotic stability of biologically important equilibria, and give a general geometric criterion for the appearance of Hopf bifurcations in the delay differential system with delay-dependent parameters. By choosing the normal development time delay as a bifurcation parameter, we analyze the stability switches of the positive equilibrium, and examine the onset and termination of Hopf bifurcations of periodic solutions from the positive equilibrium. Under some technical conditions, we show that global Hopf branches are bounded and connected by a pair of Hopf bifurcation values. This allows us to show that diapause can lead to the occurrence of multiple stability switches, coexistence of two stable limit cycles, among other rich dynamical behaviours.

A nonlinear optimization shooting method for bifurcation tracking of nonlinear systems

A continuation method for bifurcation tracking is presented based on the proposed optimization problem formulation which is designed to locate the bifurcation periodic solution. The bifurcation detection problem is formulated as a constrained optimization problem. The nonlinear constraints of the optimization problem are imposed on the shooting function and bifurcation conditions derived from the Floquet theory whereas the objective function associated with the pseudo-arclength correlation equation is devised to solution continuation. The proposed optimization formulation is integrated with the prediction–correction strategy to achieve bifurcation tracking. Two numerical examples about the Jeffcott rotor and the nonlinear tuned vibration absorber are illustrated to validate the effectiveness of the proposed methodology. Numerical results have demonstrated that the proposed method offers a convenient scheme to follow bifurcation periodic solution.

Bifurcation of Periodic Solutions and Its Maximum Number in a Circular Mesh Antenna System with 1:2 Internal Resonance

In this paper, we study the bifurcation of periodic solutions for a four-dimensional deployable circular mesh antenna system. The tools for proving these results are the averaging theory and Brouwer degree theory. Based on constructing displacement maps, we study the bifurcation of the periodic solutions of linear center, and to discuss the maximum number of periodic solutions in certain parameter control conditions. The results in this paper are helpful to the study of nonlinear dynamic characteristics and vibration control of deployable circular mesh antenna model.

Bifurcations in periodic integrodifference equations in C(\Omega) I: Analytical results and applications

<p style='text-indent:20px;'>We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional discrete dynamical systems arise in theoretical ecology as models to describe the spatial dispersal of species having nonoverlapping generations. Our explicit criteria allow us to identify branchings of fold- and crossing curve-type, which include the classical transcritical-, pitchfork- and flip-scenario as special cases. Indeed, not only tools to detect qualitative changes in models from e.g. spatial ecology and related simulations are provided, but these critical transitions are also classified. In addition, the bifurcation behavior of various time-periodic integrodifference equations is investigated and illustrated. This requires a combination of analytical methods and numerical tools based on Nyström discretization of the integral operators involved.

Periodic forcing on degenerate Hopf bifurcation

<p style='text-indent:20px;'>This paper is devoted to the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by Poincaré map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods.

Hopf bifurcation control for the main drive delay system of rolling mill

In this work, the Hopf bifurcation of the main drive delay system of rolling mill is controlled and analyzed by designing a nonlinear controller. The time-delay is selected as a bifurcation parameter, and the following conclusions are obtained through analysis: (1) in the absence of state feedback control, the system will generate the Hopf bifurcation at the expense of its stability when the bifurcation parameter exceeds the threshold value; (2) in the state under feedback control, the occurrence of Hopf bifurcation is effectively delayed and the stable region of the system is also well extended. More importantly, we can change the properties of bifurcation periodic solutions by selecting the appropriate gain parameters. Some numerical simulations reveal that under the nonlinear feedback control, the vibration amplitude of the system can be effectively reduced.

Investigation of quasi-periodic response of a buckled beam under harmonic base excitation with an “unexplained” sideband structure

The aim of this paper is to investigate quasi-periodic response of a fixed–fixed buckled beam experimentally studied in detail by Kreider and Nayfeh (Nonlinear Dyn 15(2):155–177, 1998. https://doi.org/10.1023/A:1008231012968) with an “unexplained” sideband structure. After a description of the equation of motion of the buckled beam with quadratic and cubic nonlinearities, it is shown that the proposed approach, the incremental harmonic balance (IHB) method with two time scales, can be used to find the quasi-periodic response of the buckled beam with the “unexplained” sideband structure and its understanding is revealed. The traditional IHB method with a single time scale is first used to automatically track periodic and period-doubling solutions of nonlinear responses of the buckled beam, and stability and bifurcations of periodic solutions are investigated using the Floquet theory. In the case of 1:1 internal resonance between the first two modes of the buckled beam, it is found that anti-symmetrical modes cannot be excited due to small excitation amplitudes. However, the anti-symmetrical modes are excited via period-doubling bifurcation with an increased excitation amplitude. By continuously increasing the excitation amplitude, Hopf bifurcation occurs, which leads to quasi-periodic response whose spectrum contains uniformly spaced sidebands around integer multiples of a half of the excitation frequency, where the uniformly spaced sidebands were called an “unexplained” sideband structure in Kreider and Nayfeh (1998). The results obtained from the IHB method with two time scales are shown to well correlate with the experimental results in Kreider and Nayfeh (1998). Moreover, results obtained from the IHB method with two time scales are in excellent agreement with those from numerical integration using the fourth-order Runge–Kutta method.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Grobner bases and for the work with polynomial ideals in the computer algebra system Maple.

Bifurcation Analysis and Periodic Solutions of the HD 191408 System with Triaxial and Radiative Perturbations

The nonlinear orbital dynamics of a class of the perturbed restricted three-body problem is studied. The two primaries considered here refer to the binary system HD 191408. The third particle moves under the gravity of the binary system, whose triaxial rate and radiation factor are also considered. Based on the dynamic governing equation of the third particle in the binary HD 191408 system, the motion state manifold is given. By plotting bifurcation diagrams of the system, the effects of various perturbation factors on the dynamic behavior of the third particle are discussed in detail. In addition, the relationship between the geometric configuration and the Jacobian constant is discussed by analyzing the zero-velocity surface and zero-velocity curve of the system. Then, using the Poincaré–Lindsted method and numerical simulation, the second- and third-order periodic orbits of the third particle around the collinear libration point in two- and three-dimensional spaces are analytically and numerically presented. This paper complements the results by Singh et al. [Singh et al., AMC, 2018]. It contains not only higher-order analytical periodic solutions in the vicinity of the collinear equilibrium points but also conducts extensive numerical research on the bifurcation of the binary system.

Analysis of Special Cases in the Study of Bifurcations of Periodic Solutions of the Ikeda Equation

Analysis of Special Cases in the Study of Bifurcations of Periodic Solutions of the Ikeda Equation

Global dynamics of an immunosuppressive infection model with stage structure.

In this paper, we propose an immunosuppressive infection model incorporating natural mortality of immune cells during the time lag needed for the expansion of immune cells. Starting from a stage structure model for the immune cells with various ages, we use the method of characteristic lines to derive a delay differential equation for the population of mature immune cells. Then, we use Lyapunov functional techniques to obtain the global dynamics of the model system. Specifically, we show that the virus dominant equilibrium is globally asymptotically stable when the delay is large. Next, we conduct local and global Hopf bifurcation analysis for the proposed model via Hopf bifurcation theory of delay differential equations. We choose the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from the immune control equilibrium. We also prove that the model has only a finite number of Hopf bifurcation values, and the periodic solutions with specific oscillation frequencies occur only in bounded delay intervals. Under some technical conditions, we show that two global Hopf branches bifurcated from different Hopf bifurcation values may connect to each other and thus be bounded. However, unlike the global Hopf bifurcation results in the existing literature, the Hopf branches for our model system are not necessarily bounded, though the delay components are always bounded. Numerical simulation suggests that bounded and unbounded Hopf branches may co-exist in the bifurcation diagram. Moreover, we observe a new interesting phenomenon that a global Hopf branch may have uniformly bounded periodic solutions, bounded delays, but unbounded periods.

Numerical inclusion of exact periodic solutions for time delay Duffing equation

Numerical inclusion results for exact periodic solutions are presented for the time delay autonomous Duffing equations. Constructive implicit function theorem is used for including one dimensional solution manifolds consisting of exact periodic solutions. A conjecture of a lower bound for a number of periodic solutions is given as a function of the time delay. If the delay time is less than 30, we have proved this conjecture using verified numerical computations.Theory for proving the existence of periodic solutions of the forced delay Duffing equation is proposed based on the verified numerical computations. The forced term is sinusoidal waves. Stress is on a study of the bifurcation of periodic solutions synchronizing to the external forces. A rich bifurcation phenomena of periodic solutions are reported taking the delay time as parameters. Especially, a kind of fractal structure is observed concerning resonance peaks.

Bifurcation Features of Periodic Solutions of the Mackey–Glass Equation

Bifurcations of periodic solutions of the well-known Mackey–Glass equation from its unique equilibrium state under varying equation parameters are considered. The equation is used as a mathematical model of variations in the density of white blood cells (neutrophils). Written in dimensionless variables, the equation contains a small parameter multiplying the derivative, which makes this equation singular. It is shown that the behavior of solutions to the equation with initial data from a fixed neighborhood of the equilibrium state in the equation phase space is described by a countable system of nonlinear ordinary differential equations. This system has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. One fast variable and a countable number of slow variables can be extracted from this system of equations. As a result, the averaging method can be applied to the obtained system. It is shown that the equilibrium states of the averaged system of equations in slow variables are associated with periodic solutions of the same stability type in the original equation. The possibility of simultaneous bifurcation of a large number of periodic solutions (multistability bifurcation) is shown. It is also shown that, with a further increase in the bifurcation parameter, each of the periodic solutions exhibits the transition to a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Mackey–Glass equation is characterized by chaotic multistability.