We propose a new hybrid modelling approach that combines a mechanistic model with a machine-learnt model to predict the limit cycle oscillations of physical systems with a Hopf bifurcation. The mechanistic model is an ordinary differential equation normal-form model capturing the bifurcation structure of the system. A data-driven mapping from this model to the experimental observations is then identified based on experimental data using machine learning techniques. The proposed method is first demonstrated numerically on a Van der Pol oscillator and a three-degree-of-freedom aeroelastic model. It is then applied to model the behaviour of a physical aeroelastic structure exhibiting limit cycle oscillations during wind tunnel tests. The method is shown to be general, data-efficient and to offer good accuracy without any prior knowledge about the system other than its bifurcation structure.
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…
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…
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.
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 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 a previous paper, the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like $λ^{1/2}$, $λ^{1/6}$, $λ^{1/18}$, etc. Such amplified Hopf branches were previously found by others in a subclass of feed-forward networks with three cells, first under a normal form assumption and later by ex…
We present an unfolding of the codimension-two scenario of the simultaneous occurrence of a discontinuous bifurcation and an Andronov-Hopf bifurcation in a piecewise-smooth, continuous system of autonomous ordinary differential equations in the plane. We find the Hopf cycle undergoes a grazing bifurcation that may be very shortly followed by a saddle-node bifurcation of the orbit. We derive scaling laws for the bifurcation curves that emanate from the codimension-two bifurcation.
A diffusive ratio-dependent Holling-Tanner system subject to Neumann boundary conditions is considered. The existence of multiple bifurcations, including Turing-Hopf bifurcation, Turing-Truing bifurcation, Hopf-double-Turing bifurcation and triple-Turing bifurcation, are given. Among them, the Turing-Hopf bifurcation are carried out in details by the normal form method. We theoretically prove that the system exists various spatio-temporal patterns, such as, non-constant steady state, the spatially inhomogeneous periodic or quasi-periodic solution, etc. Numerical simulations are presented to il…
Our concern is the study of degenerate Hopf bifurcation of smooth planar dynamical systems near isolated singular points. To do so, we propose to split up the definition of degeneracy into two types. Degeneracy of first kind shall means that no limit cycle surrounding the steady state can emerge after or before the critical point, with the possible emergence of limit cycles surrounding the point at infinity. Degeneracy of second kind shall means that either several limit cycles or semistable cycles as a limiting case, emerge surrounding the steady state super or subcritically. In degenerate bi…
We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.
In this paper are studied the codimensions one, two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients. Algebraic expressions obtained with computer assisted calculations are displayed.
We present a computer-assisted approach to prove the existence of Hopf bubbles and degenerate Hopf bifurcations in ordinary and delay differential equations. We apply the method to rigorously investigate these nonlocal bifurcation structures in the FitzHugh- Nagumo equation, the extended Lorenz-84 model and a time-delay SI model.
In this work we study the periodic orbits which bifurcate from a zero-Hopf bifurcations that a Lorenz-Haken system in R 4 can exhibit. The main tool used is the averaging theory.
In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulation…
Nonresonant Hopf-Hopf singularity in neutral functional differential equation (NFDE) is considered. An algorithm for calculating the third-order normal form is established by using the formal adjoint theory, center manifold theorem and the traditional normal form method for RFDE. Van der Pol's equation with extended delay feedback is studied as an example. The unfoldings near the Hopf-Hopf bifurcation point is given by applying this algorithm. Periodic solutions, quasi-periodic solutions are found via theoretical bifurcation diagram and numerical illustrations. The Hopf-Hopf bifurcation diagra…
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.
This paper presents an analysis on nonstandard generalized Hopf bifurcation in a class of switched systems where the lost of stability of linearized systems is not due to the crossing of their complex conjugate eigenvalues but relevant to the switching laws between the subslystems. Thus is remarkably different from the mechanism of the Hopf bifurcation and the generalized Hopf bifurcation studied in the literature.
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…
Singular Hopf bifurcation occurs in generic families of vector-fields with two slow variables and one fast variable. Normal forms for this bifurcation depend upon several parameters, and the dynamics displayed by the normal forms is intricate. This paper analyzes a normal form for this bifurcation. It presents extensive diagrams of bifurcations of equilibrium points and periodic orbits that are close to singular Hopf bifurcation. In addition, parameters are determined where there is a tangency between invariant manifolds that are important in the appearance of mixed-mode oscillations in system…
A Hopf bifurcation criterion of fractional-order systems with order 1 < α < 2 is established in this paper, in which all conditions are explicitly expressed by parameters without solving the roots of the relevant characteristic polynomial of Hopf bifurcation conditions. It avoids the problem that existing methods may fail due to the computational complexity in the multi-parameter situation. The bifurcation hyper-surface of multi-parameter can be obtained directly.
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…
This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber $α$, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking $α\approx1$, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, previous calculations seem to indicate that the bifurcating quasi-periodic flows are stable and go backwards with respect to the Reynolds number, $Re$. By improving the precision of previous works we find that t…
In this paper we study the Lyapunov stability and Hopf bifurcation in a system coupling a Watt-centrifugal-governor with a steam-engine. Sufficient conditions for the stability of the equilibrium state in terms of the physical parameters and of the bifurcating periodic orbit at most critical parameters on the bifurcation surface are given.
In this paper we study the Lyapunov stability and Hopf bifurcation in a biological system which models the biological control of parasites of orange plantations.
A Hopf bifurcation in the Kuramoto-Daido model is investigated based on the generalized spectral theory and the center manifold reduction for a certain class of frequency distributions. The dynamical system of the order parameter on a four-dimensional center manifold is derived. It is shown that the dynamical system undergoes a Hopf bifurcation as the coupling strength increases, which proves the existence of a periodic two-cluster state of oscillators.
This paper is devoted to the study of a predator-prey model with predator-age structure that involves Michaelis-Menten type ratio-dependent functional response. We study some dynamical properties of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain. The existence of Hopf bifurcation is established by regarding the biological maturation period $τ$ as the bifurcation parameter. The computer simulations and sensitivity analysis on parameters are also performed to illustrate the conclusions.
It had been shown that the transition from a rigidly rotating spiral wave to a meandering spiral wave is via a Hopf bifurcation. Many studies have shown that these bifurcations are supercritical, but we present numerical studies which show that subcritical bifurcations are also present within FitzHugh-Nagumo. Furthermore, we present a evidence that this bifurcation is highly sensitive to initial conditions, and it is possible to convert one solution in the hysteresis loop to the other.
This Letter outlines 20 geometric mechanisms by which limit cycles are created locally in two-dimensional piecewise-smooth systems of ODEs. These include boundary equilibrium bifurcations of hybrid systems, Filippov systems, and continuous systems, and limit cycles created from folds and by the addition of hysteresis or time-delay. Scaling laws for the amplitude and period of the limit cycles are compared to (classical) Hopf bifurcations.
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.
This paper pursues the study carried out by the authors in "Stability and Hopf bifurcation in a hexagonal governor system", focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor differential system. Here are studied the codimension two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, ilustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found an open region in the parameter space where two attracting periodic orbits coexist with an…
We investigate the stability loss of invariant n-dimensional quasi-periodic tori during a double Hopf bifurcation, where at bifurcation the two normal frequencies are in normal-normal resonance. Invariants are used to analyse the normal form approximations in a unified manner. The corresponding dynamics form a skeleton for the dynamics of the original system. Here both normal hyperbolicity and KAM theory are being used.
In this paper we analyze a simple mathematical model which describes the interaction between the proteins p53 and Mdm2. For the stationary state we discuss the local stability and the existence of the Hopf bifurcation. Choosing the delay as a bifurcation parameter we study the direction and stability of the bifurcating periodic solutions. Some numerical examples and the conclusions are finally made.
For the reaction networks with zero-one stoichiometric coefficients (or simply zero-one networks), we prove that if a network admits a Hopf bifurcation, then the rank of the stoichiometric matrix is at least four. As a corollary, we show that if a zero-one network admits a Hopf bifurcation, then it contains at least four species and five reactions. As applications, we show that there exist rank-four subnetworks, which have the capacity for Hopf bifurcations/oscillations, in two biologically significant networks: the MAPK cascades and the ERK network. We provide a computational tool for computi…
Nonsmooth formulations of physical models are common, particularly in climate modeling. However, in many of these models, there is little justification for this modeling choice, and no mathematical indication that the resulting behavior in the nonsmooth system is consistent with behavior that would arise in a similar smooth system. I explore this question by analyzing the relationship between a smooth and nonsmooth Hopf bifurcation in one particular climate example. I show that the nonsmooth bifurcation perturbs to the smooth bifurcation under small changes in a limiting parameter.
In this paper, we study degenerate Hopf bifurcations in a class of parametrized retarded functional differential equations. Specifically, we are interested in the case where the eigenvalue crossing condition of the classical Hopf bifurcation theorem is violated. Our approach is based on center manifold reduction and Poincare-Birkhoff normal forms, and a singularity theoretical classification of this degenerate Hopf bifurcation. Our results are then applied to a recently developed SIS model incorporating a delayed behavioral response. We show that the phenomenon of endemic bubbles, which is cha…
For finite-dimensional bifurcation problems, it is well-known that it is possible to compute normal forms which possess nice symmetry properties. Oftentimes, these symmetries may allow for a partial decoupling of the normal form into a so-called ``radial'' part and an ``angular'' part. Analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its unfoldings. In this paper, we are interested in the case where such bifurcations occur in retarded functional differential equations, and we revisit the realizability and restrictions problem for th…
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…
The global behavior of solutions is described for the smallest chemical reaction system that exhibits a Hopf bifurcation, discovered in \cite{WH1}. This three-dimensional system is a competitive system and a monotone cyclic feedback system. The Poincaré-Bendixson theory extends to such systems \cite{MS,H0,HS,S} and a Bendixson criterion exists to rule out periodic orbits \cite{LM}.
In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicab…
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…
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.
In this paper we study the dynamics of the monoscale Lorenz-96 model using both analytical and numerical means. The bifurcations for positive forcing parameter $F$ are investigated. The main analytical result is the existence of Hopf or Hopf-Hopf bifurcations in any dimension $n\geq4$. Exploiting the circulant structure of the Jacobian matrix enables us to reduce the first Lyapunov coefficient to an explicit formula from which it can be determined when the Hopf bifurcation is sub- or supercritical. The first Hopf bifurcation for $F>0$ is always supercritical and the periodic orbit born at this…
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…
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…
We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto-Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto-Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton-Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution b…
We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part. The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li-Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability.
The equivariant Hopf bifurcation dynamics of a class of system of partial differential equations is carefully studied. The connections between the current dynamics and fundamental concepts in hyperbolic conservation laws are explained. The unique approximation property of center manifold reduction function is used in the current work to determine certain parameter in the normal form. The current work generalizes the study of the second author ([J. Yao, $O(2)$-Hopf bifurcation for a model of cellular shock instability, Physica D, 269 (2014), 63-75.]) and supplies a class of examples of $O(2)$ H…
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…
In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.
In this paper, we consider the diffusive Nicholson's blowflies model in spatially heterogeneous environment when the diffusion rate is large. We show that the ratio of the average of the maximum per capita egg production rate to that of the death rate affects the dynamics of the model. The unique positive steady state is locally asymptotically stable if the ratio is less than a critical value. However, when the ratio is greater than the critical value, large time delay can make the unique positive steady state unstable through Hopf bifurcation. Especially, the first Hopf bifurcation value tend…
We focus on the qualitative analysis of a reaction-diffusion with spatial heterogeneity. The system is a generalization of the well known FitzHugh-Nagumo system in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of an Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon.
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…
A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are $\pm ωi\neq 0$ and $0$. In general for a such equilibrium there is no theory for knowing when from it bifurcates some small-amplitude limit cycle moving the parameters of the system. Here we study the zero-Hopf bifurcation using the averaging theory. We apply this theory to a Chua system depending on $6$ parameters, but the way followed for studying the zero-Hopf bifurcation can be applied to any other differential system in dimension $3$ or higher. In this paper first we show that there are three $4$-parameter fami…
