Codimension-two Bifurcation Research Articles

Strong Resonance Bifurcations in a Discrete-Time In-Host Model With a Saturating Infection Rate

Abstract Viral blips are a recurrent pattern observed in many persistent infections such as the human immunodeficiency virus (HIV). The main goal of this research is to present a comprehensive analytical study of a two-dimensional discrete-time in-host infection model, that exhibits viral blips, with a saturating infection rate. We examine the interactions between the population densities of infected and uninfected CD4+ T cells by discussing the model's dynamics in the long run. The local asymptotic stability of fixed points of the model is investigated. The model undergoes both flip and Neimark–Sacker bifurcations. Moreover, codimension-two bifurcations of the endemic fixed point are discussed using bifurcation theory and normal forms. The model exhibits 1:2, 1:3, and 1:4 strong resonances. Numerical simulations are performed to verify our analysis. In addition, bifurcations of higher iterations are extracted from the numerical continuation.

Discontinuous codimension-two bifurcation in a Vlasov equation.

In a Vlasov equation, the destabilization of a homogeneous stationary state is typically described by a continuous bifurcation characterized by strong resonances between the unstable mode and the continuous spectrum. However, when the reference stationary state has a flat top, it is known that resonances drastically weaken and the bifurcation becomes discontinuous. In this article we analyze one-dimensional spatially periodic Vlasov systems, using a combination of analytical tools and precise numerical simulations to demonstrate that this behavior is related to a codimension-two bifurcation, which we study in detail.

Bifurcation and Chaos of a Nonlinear Discrete-Time Predator-Prey Model Involving the Nonlinear Allee Effect

The complex dynamics of a nonlinear discretized predator-prey model with the nonlinear Allee effect in prey and both populations are investigated. First, the rigorous results are derived from the existence and stability of the fixed points of the model. Second, we establish a model with the Allee effect in prey undergoing codimension-one bifurcations (flip bifurcation and Neimark–Sacker bifurcation) and codimension-two bifurcation associated with 1 : 2 strong resonance by using center manifold theorem and bifurcation theory, and the direction of bifurcations is also evaluated. In particular, chaos in the sense of Marotto is proved at some certain conditions. Third, numerical simulations are performed to illustrate the effectiveness of the theoretical results and other complex dynamical behaviors, such as the period-3, 4, 6, 8, 9, 30, and 43 orbits, attracting invariant cycles, coexisting chaotic sets, and so forth. Of most interest is the finding of coexisting attractors and multistability. Moreover, a moderate Allee effect in predators can stabilize the dynamical behavior. Finally, the hybrid feedback control strategy is implemented to stabilize chaotic orbits existing in the model.

Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor.

In this paper, we propose a spatiotemporal prey-predator model with fear and Allee effects. We first establish the global existence of solution in time and provide some sufficient conditions for the existence of non-negative spatially homogeneous equilibria. Then, we study the stability and bifurcation for the non-negative equilibria and explore the bifurcation diagram, which revealed that the Allee effect and fear factor can induce complex bifurcation scenario. We discuss that large Allee effect-driven Turing instability and pattern transition for the considered system with the Holling-Ⅰ type functional response, and how small Allee effect stabilizes the system in nature. Finally, numerical simulations illustrate the effectiveness of theoretical results. The main contribution of this work is to discover that the Allee effect can induce both codimension-one bifurcations (transcritical, saddle-node, Hopf, Turing) and codimension-two bifurcations (cusp, Bogdanov-Takens and Turing-Hopf) in a spatiotemporal predator-prey model with a fear factor. In addition, we observe that the circular rings pattern loses its stability, and transitions to the coldspot and stripe pattern in Hopf region or the Turing-Hopf region for a special choice of initial condition.

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

<p style='text-indent:20px;'>In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of the center manifold reduction technique and normal form theory, we present a new algorithm for computing normal forms associated with the codimension-two double Hopf bifurcation. The theoretical results are applied to a predator-prey model, and complex dynamic behaviors such as spatially nonhomogeneous periodic oscillations and spatially nonhomogeneous quasi-periodic oscillations could occur.</p>

Complex dynamics of a discrete-time SIR model with nonlinear incidence and recovery rates

In this paper, a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method. The existence and stability of fixed points in this model are well studied. The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters. We discuss in detail some codimension-one bifurcations such as transcritical, period-doubling and Neimark–Sacker bifurcations, and a codimension-two bifurcation with 1:2 resonance. In addition, the phase portraits, bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis. It is found that the numerical results are consistent with the theoretical analysis. More interestingly, we also found other bifurcations in the model during the numerical simulation, such as codimension-two bifurcations with 1:1 resonance, 1:3 resonance and 1:4 resonance, generalized period-doubling and fold-flip bifurcations. The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model. Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases, but also make a great contribution to the prevention and control of infectious diseases.

Global dynamics and bifurcation for a discontinuous oscillator with irrational nonlinearity

The objective is to carry out thorough analysis of global dynamics and bifurcation for a discontinuous oscillator, showing the transition of dynamics between the piecewise smooth system (with irrational nonlinearity) and the piecewise linear system. The existence of one or three equilibrium sets is proven by the approach of Filippov. Based on non-smooth Lyapunov functions and their set-valued derivatives, asymptotical stability especially finite-time asymptotical stability is shown for the equilibrium sets. In particular, some conditions are established guaranteeing the uniqueness of an equilibrium set and its global finite-time asymptotical stability. Making use of the parametric representation, codimension-one bifurcation curves and a codimension-two bifurcation point are obtained. We give numerical simulations and bifurcation diagram to illustrate the transition of dynamics and verify the results. We further clarify some previous misconceptions in Li et al. (2016).

Nonintegrability of Dynamical Systems Near Degenerate Equilibria

We prove that general three- or four-dimensional systems %of differential equations are real-analytically nonintegrable near degenerate equilibria in the Bogoyavlenskij sense under additional weak conditions when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of the corresponding Poincare-Dulac normal forms and further to that of simple planar systems, and use a novel approach for proving the analytic nonintegrability of planar systems. Our result also implies that general three- and four-dimensional systems exhibiting fold-Hopf and double-Hopf codimension-two bifurcations, respectively, are real-analytically nonintegrable under the weak conditions. To demonstrate these results, we give two examples for the Rossler system and coupled van der Pol oscillators.

Soret effect on the onset of viscous dissipation thermal instability for Poiseuille flows in binary mixtures

We investigate numerically the Soret effect on the linear instability properties in convection due to viscous dissipation in a horizontal channel filled with a binary fluid mixture. Two sets of boundary conditions of experimental interest are considered. Both have no-slip boundaries for the velocity and no mass flux through them. The lower boundary is considered adiabatic, while the upper boundary is isothermal for case A and inversely for case B. As no external temperature or concentration difference is imposed on the layer, the cause of thermal instability is the flow rate through the volumetric heating induced by the viscous dissipation and the Soret effect inherent to binary mixtures. It is found that longitudinal rolls (LR) represent the preferred mode for the onset of convection. For case A, both oscillatory and steady-state LR may develop depending on the value of the separation ratio ψ, which represents the ratio between the mass contribution and the temperature contribution to buoyancy forces. The dependence of the instability thresholds on the separation ratio is discussed near and far from the codimension-two bifurcation point. For case B, the basic state remains stable for positive separation ratios, while it loses its stability via a stationary bifurcation with zero wave number for negative values of the separation ratio. The relevance of the theoretical results for the observability of such instability in real systems is discussed. Finally, we suggest a protocol to determine Soret coefficients by using the stability diagrams obtained in the current paper.

When Hopf meets saddle: bifurcations in the diffusive Selkov model for glycolysis

We study the linear instabilities and bifurcations in the Selkov model for glycolysis with diffusion. We show that this model has a zero wave-vector, finite frequency Hopf bifurcation, which is a forward or supercritical bifurcation, to a growing oscillatory but spatially homogeneous state and a saddle-node bifurcation, which is a backward or subcritical bifurcation, to a growing inhomogeneous state with a steady pattern characterised by a finite wavevector. We further demonstrate that by tuning the relative diffusivity of the two concentrations, it is possible to make both the instabilities to occur at the same point in the parameter space, leading to an unusual type of codimension-two bifurcation. We then show that in the vicinity of this codimension-two bifurcation, the initial conditions decide whether a spatially uniform oscillatory or a spatially periodic steady pattern emerges in the long time limit. It is also possible to form a co-existing patterned and time-periodic state by fine-tuning the diffusivity ratio for moderate values, in qualitative agreement with recent experimental studies.

Codimension-one and codimension-two bifurcations in a new discrete chaotic map based on gene regulatory network model

Codimension-one and codimension-two bifurcations in a new discrete chaotic map based on gene regulatory network model

Nonlinear Analysis of an Injection-Locked Oscillator Coupled to an External Resonator

This work investigates an injection-locked power oscillator inductively coupled to an external resonator for wireless power transfer. The system allows a high transfer efficiency, while ensuring a constant oscillation frequency versus the coupling factor, unlike free-running implementations. An analytical formulation provides insight into the impact of the coupling factor on the locked-operation ranges. Two types of qualitative behavior, delimited by a codimension-two bifurcation, are distinguished. The investigation is extended to a Class-E oscillator at 13.56 MHz, analyzed with a new harmonic balance (HB) method that provides the family of locked-solution curves in a single simulation. Very good agreement is obtained with the measurement results.

Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed.

Codimension-Two Bifurcations and Bifurcation Controls in a Discrete Biological System with Weak Allee Effect

In this paper, the codimension-two bifurcations and bifurcation controls of a discrete predatory system with weak Allee effect on prey are investigated. After analyzing the qualitative properties of all the fixed points, we explore the existences of 1:2 and 1:3 strong resonances at the unique interior fixed point. Combining the adjoint operator method with an approximate identity transformation, we obtain the normal forms of these strong resonances. Meanwhile, we calculate the approximate expressions of bifurcation curves around the resonance point, which include Neimark–Sacker and period doubling bifurcation curves. And we give the conditions under which these bifurcations are supercritical or subcritical. To study global bifurcations near the resonance point, we use an ordinary differential equation to approximate the original discrete system by the time-one approximation flow theory. In biology, bifurcations and unstable fluctuations may be harmful for the breeding of biological population. Then we extend the hybrid control strategy to control 1:2 and 1:3 strong resonances. Numerical simulations are in good agreement with our theoretical results. By using the MatContM package, we perform numerical bifurcation analyses to further support our theoretical results, in which the bifurcation points and the normal form coefficients are completely consistent with our theories. According to the results, we finally analyze the impact of weak Allee effect on the system. And a reasonable biological explanation is given for these conclusions.

Bifurcation analysis of mixed-mode oscillations and Farey trees in an extended Bonhoeffer–van der Pol oscillator

The two-variable Bonhoeffer–van der Pol oscillator, which is equivalent to the FitzHugh–Nagumo model, can be represented by a natural circuit, i.e., a circuit consisting only of simple two-terminal elements. An extended Bonhoeffer–van der Pol (BVP) oscillator is a circuit extended to a three-variable system from a two-variable BVP oscillator; this is obtained by adding an inductor–resistor branch to the original BVP oscillator. The extended BVP oscillator is known to generate mixed-mode oscillations. In this study, we classify the bifurcations of the simple sequences and their daughters, which constitute asymmetric Farey trees. Because the extended BVP oscillator is an extremely simple circuit, the parents–daughter processes can be quite precisely explained. We confirm that simple sequences 1s (s≥0) are born via saddle–node bifurcations and can be basic parents, which correspond to each stable fixed point of a Poincaré return map, where 1s represents one large excursion followed by a number s of small peaks. The two basic parents 1s and 1s+1 generate daughters [1s+1,1s×n] sequentially for the successive values of n via mixed-mode oscillation-incrementing bifurcations. The terms [1s+1,1s×n] indicate that 1s+1 is followed by 1sn-times. The daughters are born in a similar fashion to period-adding bifurcations generated by the circle map, and the parents–daughter processes satisfy Farey arithmetic and are terminated by a saddle–node bifurcation through which 1s appears. We create multiple two-parameter bifurcation diagrams and confirm that these phenomena are stably observed in these diagrams over broad ranges, i.e., complex bifurcations, such as codimension-two bifurcations or cusps, do not appear in the diagrams. Furthermore, the theoretical results are confirmed using laboratory measurements and experiments.

Firing patterns and bifurcation analysis of Purkinje cell dendrite model

This paper mainly studied firing patterns and related bifurcations in the Purkinje cell dendrite model. Based on the methods of equivalent potentials and time scale analysis, the initial six-dimensional (6D) dendrite model is reduced to a 3D form to facilitate the calculation. We numerically show that the dendrite model could exhibit period-adding bifurcation and four bursting patterns for several vital parameters. Then the bifurcation mechanisms and transition of these four bursting patterns are discussed by phase plane analysis, and two-parameter bifurcation analysis of the fast subsystem, respectively. Moreover, we computed the first Lyapunov coefficient to determine the stability of Hopf bifurcation. Ultimately, we analyzed the codimension-two bifurcation of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation.

The existence of codimension-two bifurcations in a discrete-time SIR epidemic model

<abstract><p>In this paper, we consider a discrete-time SIR epidemic model. Codimension-two bifurcations associated with 1:2, 1:3 and 1:4 strong resonances are analyzed by using a series of affine transformations and bifurcation theory. Numerical simulations are carried out to verify and illustrate these theoretical results. More precisely, two kinds of high-resolution stability phase diagrams are exhibited to describe how the system's complexity unfolds with control parameters varying.</p></abstract>

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

Bistability, bifurcations and chaos in the Mackey-Glass equation

<p style='text-indent:20px;'>Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equation for moderate values of the delay close to the onset of chaos. In particular a cusp bifurcation of periodic orbits and resulting branches of folds of periodic orbits effectively partition the parameter space into regions where different behaviours are seen. The cusp bifurcation leads directly to bistability between periodic orbits, and subsequently to bistability between a periodic orbit and a chaotic attractor. This leads to two different mechanisms by which the chaotic attractor is destroyed in a global bifurcation with a periodic orbit in either an interior crisis or a boundary crisis. In another part of parameter space a sequence of subcritical period-doublings is found to give rise to bistability between a periodic orbit and a chaotic attractor. Torus bifurcations, and a codimension-two fold-flip bifurcation are also identified, and Lyapunov exponent computations are used to determine chaotic regions and attractor dimension.</p>

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

<p style='text-indent:20px;'>The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> and the detuning <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the <inline-formula><tex-math id="M4">\begin{document}$ (f, \delta) $\end{document}</tex-math></inline-formula>-plane is organized by two points of codimension-two bifurcations——a <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}_2 $\end{document}</tex-math></inline-formula>-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the <inline-formula><tex-math id="M6">\begin{document}$ (f, \delta) $\end{document}</tex-math></inline-formula>-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.</p>