Time Delay in the Swing Equation: A Variety of Bifurcations
Tessina H. Scholl, Lutz Gröll, Veit Hagenmeyer
备注：
12 pages, 6 figures, "The following article has been accepted by 'Chaos: An Interdisciplinary Journal of Nonlinear Science'. After it is published, it will be found at https://aip.scitation.org/journal/cha."
The present paper addresses the swing equation with additional delayed damping as an example for pendulum-like systems. In this context, it is proved that recurring sub- and supercritical Hopf bifurcations occur if time delay is increased. To this end, a general formula for the first Lyapunov coefficient in second order systems with additional delayed damping and delay-free nonlinearity is given. In so far the paper extends results about stability switching of equilibria in linear time delay systems from Cooke and Grossman. In addition to the analytical results, periodic solutions are numerically dealt with. The numerical results demonstrate how a variety of qualitative behaviors is generated in the simple swing equation by only introducing time delay in a damping term.
In this paper, the dynamics of a modified Leslie-Gower predator-prey system with two delays and diffusion is considered. By calculating stability switching curves, the stability of positive equilibrium and the existence of Hopf bifurcation and double Hopf bifurcation are investigated on the parametric plane of two delays. Taking two time delays as bifurcation parameters, the normal form on the center manifold near the double Hopf bifurcation point is derived, and the unfoldings near the critical points are given. Finally, we obtain the complex dynamics near the double Hopf bifurcation point, i…
For systems of delay differential equations the Hopf bifurcation was investigated by several authors. The problem we consider here is that of the possibility of emergence of a codimension two bifurcation, namely the Bautin bifurcation, for some of such systems.
Let us give a two dimensional family of real vector fields. We suppose that there exists a stationary point where the linearized vector field has successively a stable focus, an unstable focus and an unstable node. When the parameter moves slowly, a bifurcation delay appears due to the Hopf bifurcation. The studied question in this article is the continuation of the delay after the focus-node bifurcation.
We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the $S^1$-equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation of the periodic oscillations.
Recently, the influence of leakage delay on the dynamics of integer-order neural networks has been investigated extensively. It has been confirmed that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we study the existence of Hopf bifurcation in a six-neuron fractional bidirectional associative memory (BAM) neural network with leakage delay. By selecting two appropriate bifurcation parameters and analyzing corresponding characteristic equations, it is verified that the delayed fractional neural network generates a Hopf bifu…
In this paper, we consider a continuous-time model with discrete and dis-tributed delays to describe how two pieces of information interact in online social networks. Sufficient conditions are carried out to illustrate the stability of each e-quilibrium. Taking time delay as a bifurcation parameter, the system undergoes a sequence of Hopf bifurcation when this parameter passes through a critical value. By methods of multiple scales, we prove that the direction of Hopf bifurcation is depending on the condition which is related to delay.
Some errors contained in the author's previous article "An example of Bautin-type bifurcation in a delay differential equation", JMAA, 329(2007), 777-789, are listed and corrected. Original abstract: In a previous paper we gave sufficient conditions for a system of delay differential equations to present Bautin-type bifurcation. In the present work we present an example of delay equation that satisfies these conditions.
In this paper, we investigate a delayed reaction-diffusion-advection equation, which models the population dynamics in the advective heterogeneous environment. The existence of the nonconstant positive steady state and associated Hopf bifurcation are obtained. A weighted inner product associated with the advection rate is introduced to compute the normal forms, which is the main difference between Hopf bifurcation for delayed reaction-diffusion-advection model and that for delayed reaction-diffusion model. Moreover, we find that the spatial scale and advection can affect Hopf bifurcation in th…
In this paper we investigate the impact of delayed tax revenues on the fiscal policy out-comes. Choosing the delay as a bifurcation parameter we study the direction and the stability of the bifurcating periodic solutions. With respect to the delay we show when the system is stable. Some numerical examples are finally given for justifying the theoretical results.
In this paper, we show the existence of Hopf bifurcation of a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the "local" Hopf bifurcation values over all patches. On the other hand, when the dispersal rate tends to infinity, the Hopf bifurcation value tends to that of the "average" model.
In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretica…
In this paper, we investigate a reaction-diffusion-advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our result implies that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction-diffusion-advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation.
We study the two state model which describes the balance equation for carbon dioxide and oxygen. These are nonlinear parameter dependent and because of the transport delay in the respiratory control system, they are modeled with delay differential equation. So, the dynamics of a two state one delay model are investigated. By choosing the delay as a parameter, the stability and Hopf bifurcation conditions are obtained. We notice that as the delay passes through its critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. The stable region of the system with dela…
In the context of a spatially extended model for the electrical activity in a pituitary lactotroph cell line, we establish that two delayed bifurcation phenomena from ODEs ---folded node canards and slow passage through Hopf bifurcations--- persist in the presence of diffusion. For canards, the single cell (ODE) model exhibits canard-induced bursting. Numerical simulations of the PDE reveal rich spatio-temporal canard dynamics, and the transitions between different bursts are mediated by spatio-temporal maximal canards. The ODE model also exhibits delayed loss of stability due to slow passage …
In this paper, we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical unde…
It is known that Lotka - Volterra type differential equations with delays or distributed delays have an important role in modeling ecological systems. In this paper we study the effects of distributed delay on the dynamics of the harvested one predator - two prey model. Using the expectation of the distribution of the delay as a bifurcation parameter, we show that the equilibrium that was asymptotic stable becomes unstable and Hopf bifurcation can occur as the expectation crosses some critical values.
In this work we study local oscillations in delay differential equations with a frequency domain methodology. The main result is a bifurcation equation from which the existence and expressions of local periodic solutions can be determined. We present an iterative method to obtain the bifurcation equation up to a fixed arbitrary order. It is shown how this method can be implemented in symbolic math programs.
We show that a simple piecewise-linear system with time delay and periodic forcing gives rise to a rich bifurcation structure of torus bifurcations and Arnold tongues, as well as multistability across a significant portion of the parameter space. The simplicity of our model enables us to study the dynamical features analytically. Specifically, these features are explained in terms of border-collision bifurcations of an associated Poincaré map. Given that time delay and periodic forcing are common ingredients in mathematical models, this analysis provides widely applicable insight.
In this paper, we consider the dynamics of a delayed reaction-diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the directi…
We investigate a diffusive predator-prey model by incorporating the fear effect into prey population, since the fear of predators could visibly reduce the reproduction of prey. By introducing the mature delay as bifurcation parameter, we find this makes the predator-prey system more complicated and usually induces Hopf and Hopf-Hopf bifurcations. The formulas determining the properties of Hopf and Hopf-Hopf bifurcations by computing the normal form on the center manifold are given. Near the Hopf-Hopf bifurcation point we give the detailed bifurcation set by investigating the universal unfoldin…
This article provides an example of fast-slow system such that most orbits remain as close as possible to the unstable manifold of the fast dynamics for an arbitrarily long time.
We present the simplest criterion that determines the direction of the Hopf bifurcations of the delay differential equation $x'(t)=-μf(x(t-1))$, as the parameter $μ$ passes through the critical values $μ_k$. We give a complete classification of the possible bifurcation sequences. Using this information and the Cooke-transformation, we obtain local estimates and monotonicity properties of the periods of the bifurcating limit cycles along the Hopf-branches. Further, we show how our results relate to the often required property that the nonlinearity has negative Schwarzian derivative.
Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the…
In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov-Takens bifurcation in classical delay differential equations (DDEs). Using a generalization of the Lindstedt-Poincaré method to approximate the homoclinic solution allows us to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm al…
This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. H…
The aim of this work is to investigate the qualitative behaviour of a financial dynamical system which contains a time delay. We investigate the dynamic response of this system of which variables are interest rate, investment demand, price index and average profit margin. We perform a stability analysis at the fixed points and show that the system undergoes a Hopf bifurcation. The bifurcation analyses are supported by numerical simulations.
DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with discrete constant and state-dependent delays. The package supports continuation and stability analysis of steady state solutions and periodic solutions. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. To analyse the stability of steady state solutions, approximations are computed to the r…
The spatiotemporal patterns of a reaction diffusion mussel-algae system with a delay subject to Neumann boundary conditions is considered. The paper is a continuation of our previous studies on delay-diffusion mussel-algae model. The global existence and positivity of solutions are obtained. The stability of the positive constant steady state and existence of Hopf bifurcation and Turing bifurcation are discussed by analyzing the distribution of eigenvalues. Furthermore, the dynamic classifications near the Turing-Hopf bifurcation point are obtained in the dimensionless parameter space by calcu…
This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of delay differential equations (DDEs) arising in applications, including those that feature state-dependent delays. We discuss the present capabilities of the most recent release of DDE-BIFTOOL. They include the numerical continuation of steady states, periodic orbits and their bifurcations of codimension one, as well as the detection of certain bifurcations of…
Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction-diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. We find that the dynamical behavior near bifurcation points are proved to be governed by twelve disti…
The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete de…
Pseudospectral approximation reduces DDE (delay differential equations) to ODE (ordinary differential equations). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyse certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connectio…
Omnivory is defined as feeding on more than one trophic level. An example of this is the so-called intraguild predation (IG) which includes a predator and its prey that share a common resource. IG predation models are known to exhibit interesting dynamics including chaos. This work considers a three-species food web model with omnivory, where the interactions between the basal resource, the IG prey, and the IG predator are of Lotka-Volterra type. In the absence of predation, the basal resource follows a delayed logistic equation or popularly known as Hutchinson's equation. Conditions for the e…
The Turing-Hopf type spatiotemporal patterns in a diffusive Holling-Tanner model with discrete time delay is considered. A global Turing bifurcation theorem for $τ=0$ and a local Turing bifurcation theorem for $τ>0$ are given by the method of eigenvalue analysis and prior estimation. Further considering the degenerated situation, the existence of Bogdanov-Takens bifurcation and Turing-Hopf bifurcation are obtained. The normal form method is used to study the explicit dynamics near the Turing-Hopf singularity, and we establish the existence of various self-organized spatiotemporal patterns, suc…
We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spa…
Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions…
In a previous work we investigated the existence of Hopf degenerate bifurcation points for a differential delay equation modeling leukemia and we actually found Hopf points of codimension two for the considered problem. If around the parameters corresponding to such a point we vary two parameters (the considered problem has five parameters), then a Bautin bifurcation should occur. In this work we chose a Hopf point of codimension two for the considered problem and perform numerical integration for parameters chosen in a neighborhood of the bifurcation point parameters. The results show that, i…
A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Sinc…
In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit…
For delayed reaction-diffusion Schnakenberg systems with Neumann boundary conditions, critical conditions for Turing instability are derived, which are necessary and sufficient. And existence conditions for Turing, Hopf and Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at Turing-Hopf singularity of codimension 2, are derived. By investigating Turing-Hopf bifurcation, the parameter regions for the stability of a periodic solution, a pair of spatially inhomogeneous steady states and a pair of spatially inhomogeneous periodic solutions, are derived in $(τ,\varepsilon…
We consider the scalar delay differential equation $$ \dot{x}(t)=-x(t)+f_{K}(x(t-1)) $$ with a nondecreasing feedback function $f_{K}$ depending on a parameter $K$, and we verify that a saddle-node bifurcation of periodic orbits takes place as $K$ varies. The nonlinearity $f_{K}$ is chosen so that it has two unstable fixed points (hence the dynamical system has two unstable equilibria), and these fixed points remain bounded away from each other as $K$ changes. The generated periodic orbits are of large amplitude in the sense that they oscillate about both unstable fixed points of $f_{K}$.
We investigate a diffusive, stage-structured epidemic model with the maturation delay and freely-moving delay. Choosing delays and diffusive rates as bifurcation parameters, the only possible way to destabilize the endemic equilibrium is through Hopf bifurcation. The normal forms of Hopf bifurcations on the center manifold are calculated, and explicit formulae determining the criticality of bifurcations are derived. There are two different kinds of stable oscillations near the first bifurcation: on one hand, we theoretically prove that when the diffusion rate of infected immature individuals i…
The aim of this paper is to study the steady states of the mathematical models with delay kernels which describe pathogen-immune dynamics of many kinds of infectious diseases. In the study of mathematical models of infectious diseases it is important to predict whether the infection disappears or the pathogens persist. The delay kernel is described by the memory function that reflects the influence of the past density of pathogen in the blood. By using the coefficient of kernel k, as a bifurcation parameter, the models are found to undergo a sequence of Hopf bifurcation. The direction and the …
We discuss a bifurcation scenario which creates periodic pulsating solutions in slow-fast delayed systems through a cascade of almost simultaneous Hopf bifurcations. This scenario has been previously associated with formation of pulses in a delayed model of mode-locked semiconductor lasers. In this work, through a case study of several examples, we establish that a cascade of Hopf bifurcations can produce periodic pulses, with a period close to the delay time, in population dynamics models and explore the conditions that ensure the realization of this scenario near a transcritical bifurcation …
Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent delay differential equation is studied in detail and periodic singular solutions are constructed with one and two local max…
In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and a discrete intracellular delay: the delay corresponds to the time between infection of a infected target hepatocytes and production of new HCV particles. We establish the global stability of the infection-free equilibrium and existence, uniqueness, local and global stabilities of the infected equilibrium, also we establish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. The unique infected equilibrium is…
The memory-based diffusion systems have wide applications in practice. Hopf bifurcations are observed from such systems. To meet the demand for computing the normal forms of the Hopf bifurcations of such systems, we develop an effective new algorithm where the memory delay is treated as the perturbation parameter. To illustrate the effectiveness of the algorithm, we consider a diffusive predator-prey system with memory-based diffusion and Holling type-II functional response. By employing this newly developed procedure, we investigate the direction and stability of the delay-induced mode-1 and …
This paper generalizes the existing minimal model of the hypothalamic-pituitary-adrenal (HPA) axis in a realistic way, by including memory terms: distributed time delays, on one hand and fractional-order derivatives, on the other hand. The existence of a unique equilibrium point of the mathematical models is proved and a local stability analysis is undertaken for the system with general distributed delays. A thorough bifurcation analysis for the distributed delay model with several types of delay kernels is provided. Numerical simulations are carried out for the distributed delays models and f…
Time-delay chaotic systems refer to the hyperchaotic systems with multiple positive Lyapunov exponents. It is characterized by more complex dynamics and a wider range of applications as compared to those non-time-delay chaotic systems. In a three-dimensional general Lorenz chaotic system, time delays can be applied at different positions to build multiple heterogeneous Lorenz systems with a single time delay. Despite the same equilibrium point for multiple heterogeneous Lorenz systems with single time delay, their stability and Hopf bifurcation conditions are different due to the difference in…
Dynamics in delayed differential equations (DDEs) is a well studied problem mainly because DDEs arise in models in many areas of science including biology, physiology, population dynamics and engineering. The change of the nature in the solutions in the parameter space for a network of Phase-Locked Loop oscillators was studied in Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators. Communications in Nonlinear Science and Numerical Simulation, Elsevier BV, 2014, (on-line version), where the existence of Hopf bifurcations for both cases, sym…
