Normal Form Coefficients Research Articles

Periodic Solutions and Bifurcations in a Long-Timescale Oceanic Carbon Cycle Model

We study the dynamics of a recent model of Rothman for long timescale carbon cycle. We reproduce and extend variousresults of the Rothman model. We present numerical results showing that the model exhibits both stable and unstable limitcycles via Hopf bifurcations as the parameters are varied. We numerically find normal forms of Bautin bifurcations to confirmtheir criticality. We also extend the analysis of the normal form coefficients to identify where the fold limit cycle bifurcationoccurs.

Computation of Normal Form and Unfolding of Codimension-3 Zero-Hopf–Hopf Bifurcation

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics.

Wake dynamics in buoyancy-driven flows: Steady-state-Hopf-mode interaction with O(2) symmetry revisited.

We present a detailed mathematical study of a truncated normal form relevant to the bifurcations observed in wake flow past axisymmetric bodies, with and without thermal stratification. We employ abstract normal form analysis to identify possible bifurcations and the corresponding bifurcation diagrams in parameter space. The bifurcations and the bifurcation diagrams are interpreted in terms of symmetry considerations. Particular emphasis is placed on the presence of attracting robust heteroclinic cycles in certain parameter regimes. The normal form coefficients are computed for several examples of wake flows behind buoyant disks and spheres, and the resulting predictions compared with the results of direct numerical flow simulations. In general, satisfactory agreement is obtained.

Turing-Hopf bifurcation and bi-stable spatiotemporal periodic orbits in a delayed predator-prey model with predator-taxis

When Turing-Hopf bifurcation occurs, it is naturally worth discussing the types of spatiotemporal patterns and their stabilities. In this paper, for partial functional differential equations with nonlinear diffusion, we establish the explicit formulae for the coefficients of normal form associated with Turing-Hopf bifurcation. This provides an explicit connection between the original system and the third-order ordinary differential equations (i.e. normal form) restricted on center manifold. Subsequently, with the aid of extended formulae, we discuss a delayed predator-prey model with predator-taxis and find that Turing-Hopf bifurcation leads to the emergence of a pair of stable single-peak spatially inhomogeneous steady states, a pair of stable multi-peak spatially inhomogeneous periodic orbits and so on. Also, the parameter regions in which these phenomena arise are given quantitatively. It is further shown that the pair of bistable spatially inhomogeneous periodic orbits arise due to the interaction of predator-taxis and time delay, which won't emerge in the same system without predator-taxis.

Bifurcation analysis of Duffing oscillator with a fifth-order nonlinear factor

ABSTRACT This paper aims to investigate and analyze the dynamic behavior of a Duffing equation with a non-linear factor of the fifth-order, which is similar to the one developed by Zhao and Zhang but with a more general bifurcation analysis. While Zhao and Zhang use numerical methods to explore the behavior of the model, this paper employs both analytical and numerical approaches to analyze the bifurcation behavior. The paper determines normal form coefficients of codim-1 bifurcation points, including fold, pitchfork, and Hopf, as well as codim-2 bifurcation points such as Bogdanov-Takens and cusp. The sub-criticality or super-criticality of these bifurcation points is also calculated analytically. The analytical results are then confirmed through numerical simulation and continuations, which reveal more dynamic behaviors of the system. Furthermore, the paper investigates the external forcing effect on the model and extracts its dynamic behaviors.

Degenerate codimension-2 cusp of limit cycles in a Holling–Tanner model with harvesting and anti-predator behavior

In this paper, a Holling–Tanner predator–prey model with constant-yield prey harvesting and anti-predator behavior is investigated. It is shown that the codimension of a nilpotent cusp is 3 and a degenerate Bogdanov–Takens bifurcation of codimension 3 acts as an organizing center for rich dynamical behaviors including saddle–node bifurcation, Hopf bifurcation, saddle–node bifurcation of limit cycles, and homoclinic bifurcation. Moreover, a Hopf bifurcation of codimension 2 is found, a codimension-2 cusp of limit cycles is firstly observed, and the corresponding normal form coefficients are also obtained, which indicate that there exists an acute angle region of three coexistent limit cycles. In particular, a reversed S-shaped saddle–node bifurcation curve of limit cycles is found. In addition, we observe that the anti-predator behavior may cause nearly extinction of the predator population while the prey population reaches its maximum which is less than the carrying capacity due to the effect of harvesting; however, the constant-yield prey harvesting may drive both species to extinction suddenly. Finally, numerical simulations including bifurcation diagrams and phase portraits are given to illustrate the theoretical results.

Dynamical analysis of a discrete-time SIR epidemic model

In this paper, a discrete-time seasonally forced SIR epidemic model is investigated for different types of bifurcations. Although, many researchers already suggested numerically that this model can exhibit chaotic dynamics but not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, the one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Secondly, the flip, Neimark–Sacker, and strong resonances bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. Some graphical representations of the model are presented to verify the obtained results.

Period-Doubling Bifurcation of Cycles in Retarded Functional Differential Equations

AbstractA rigorous description of a period-doubling bifurcation of limit cycles in retarded functional differential equations based on tools of functional analysis and singularity theory is presented. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. We also prove the exchange of stability in the case of a non-degenerate period-doubling bifurcation. The approach concerns Fredholm operators, Lyapunov–Schmidt reduction and recognition problem for pitchfork bifurcation.

Analytical bifurcation and strong resonances of a discrete Bazykin–Berezovskaya predator–prey model with Allee effect

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and [Formula: see text], [Formula: see text], [Formula: see text] strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM.

Complex dynamics of a discrete‐time seasonally forced SIR epidemic model

In this paper, a discrete‐time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark–Sacker, and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results.

Dynamics and bifurcations of a discrete‐time Lotka–Volterra model using nonstandard finite difference discretization method

A newly disclosed nonstandard finite difference method has been used to discretize a Lotka–Volterra model to investigate the critical normal form coefficients of bifurcations for both one‐parameter and two‐parameter bifurcations. The discrete‐time prey–predator model exhibits a variety of local bifurcations such as period‐doubling, Neimark–Sacker, and strong resonances. Critical normal form coefficients are determined to reveal dynamical scenarios corresponding to each bifurcation point. We also investigate the complex dynamics of the model numerically by Matlab package using MatcotM based on numerical continuation technique. The numerical continuation validates the theoretical analysis, which is discussed from an ecological perspective.

Dynamical Behaviors of an SIR Epidemic Model with Discrete Time

Analytically and numerically, the study examines the stability and local bifurcations of a discrete-time SIR epidemic model. For this model, a number of bifurcations are studied, including the transcritical, flip bifurcations, Neimark–Sacker bifurcations, and strong resonances. These bifurcations are checked, and their non-degeneracy conditions are determined by using the normal form technique (computing of critical normal form coefficients). We use the MATLAB toolbox MatcontM, which is based on the numerical continuation method, to confirm the obtained analytical results and specify more complex behaviors of the model. Numerical simulation is employed to present a closed invariant curve emerging from a Neimark–Sacker point and its breaking down to several closed invariant curves and eventually giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

Computational Methods for Nonlinear Analysis of Hopf Bifurcations in Power System Models

Hopf bifurcations can occur in power systems when a system mode experiences low damping because of changes in system operating conditions and they can lead to the emergence of limit cycles and oscillations. There are two types of Hopf bifurcations, namely, supercritical and subcritical, and they are determined by the sign of a cubic normal form coefficient. This paper discusses the two types of Hopf phenomena in test power system models where both types could be seen under changes in system and control parameters. The paper proposes an efficient computational method for carrying out the higher-order center manifold and normal form calculations for a general power system model and discusses the implications of the normal form coefficients for power system dynamics. Distinguishing between subcritical versus supercritical is important since they lead to very different types of oscillatory phenomena, divergence versus sustained oscillations, respectively.

Codimension-2 bifurcation in a discrete predator–prey system with constant yield predator harvesting

This work investigates the bifurcation analysis in a discrete-time Leslie–Gower predator–prey model with constant yield predator harvesting. The stability analysis for the fixed points of the discretized model is shown briefly. In this study, the model undergoes codimension-1 bifurcation such as fold bifurcation (limit point), flip bifurcation (period-doubling) and Neimark–Sacker bifurcation at a positive fixed point. Further, the model exhibits codimension-2 bifurcations, including Bogdanov–Takens bifurcation and generalized flip bifurcation at the fixed point. For each bifurcation, by using the critical normal form coefficient method, various critical states are calculated. To validate our analytical findings, the bifurcation curves of fixed points are drawn by using MATCONTM. The system exhibits interesting rich dynamics including limit cycles and chaos. Moreover, it has been shown that the predator harvesting may control the chaos in the system.

Codimension-one and -two bifurcation analysis of a two-dimensional coupled logistic map

This paper devotes to a detailed bifurcation analysis of a two-dimensional non-invertible map, obtained using a symmetric coupling between one-dimensional logistic maps. The critical normal form coefficients method is employed to detect bifurcations and to explore further critical conditions without explicit reduction to the center manifold. The results show that the two-dimensional map undergoes codimension-one (codim-1) bifurcations such as transcritical, pitchfork, period-doubling, Neimark–Sacker, and codim-2 bifurcations including transcritical-flip, pitchfork-flip, strong resonances 1:2, 1:3, 1:4. For each bifurcation, the critical normal form coefficients are calculated to check the non-degeneracy conditions and predict the bifurcation scenarios around the bifurcation points. To validate the theoretical results, all bifurcation curves of fixed points are plotted with the aid of the numerical continuation method. Weak resonances are also specified by the isoclines on the bi-parameter plane. The results will help in understanding the occurrence and the structure of bifurcation cascades observed in many coupled discrete systems.

Subharmonic parametric instability in nearly brimful circular cylinders: a weakly nonlinear analysis

In labscale Faraday experiments, meniscus waves respond harmonically to small-amplitude forcing without threshold, hence potentially cloaking the instability onset of parametric waves. Their suppression can be achieved by imposing a contact line pinned at the container brim with static contact angle $\theta _s=90^{\circ }$ (brimful condition). However, tunable meniscus waves are desired in some applications as those of liquid-based biosensors, where they can be controlled adjusting the shape of the static meniscus by slightly underfilling/overfilling the vessel ( $\theta _s\ne 90^{\circ }$ ) while keeping the contact line fixed at the brim. Here, we refer to this wetting condition as nearly brimful. Although classic inviscid theories based on Floquet analysis have been reformulated for the case of a pinned contact line (Kidambi, J. Fluid Mech., vol. 724, 2013, pp. 671–694), accounting for (i) viscous dissipation and (ii) static contact angle effects, including meniscus waves, makes such analyses practically intractable and a comprehensive theoretical framework is still lacking. Aiming at filling this gap, in this work we formalize a weakly nonlinear analysis via multiple time scale method capable of predicting the impact of (i) and (ii) on the instability onset of viscous subharmonic standing waves in both brimful and nearly brimful circular cylinders. Notwithstanding that the form of the resulting amplitude equation is in fact analogous to that obtained by symmetry arguments (Douady, J. Fluid Mech., vol. 221, 1990, pp. 383–409), the normal form coefficients are here computed numerically from first principles, thus allowing us to rationalize and systematically quantify the modifications on the Faraday tongues and on the associated bifurcation diagrams induced by the interaction of meniscus and subharmonic parametric waves.

Birkhoff normal form and twist coefficients of periodic orbits of billiards* *X J is supported in part by the NSF Grant DMS-1854232.

In this paper we study the Birkhoff normal form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist coefficients in terms of the geometric parameters of the billiard table. As an application, we obtain characterizations of the nonlinear stability and local analytic integrability of the billiards around the elliptic periodic points.

Complex dynamics of a discrete-time Bazykin–Berezovskaya prey-predator model with a strong Allee effect

The present paper investigates the critical normal form coefficients for the one-parameter and two-parameter bifurcations of a discrete-time Bazykin–Berezovskaya prey-predator model. Based on the critical coefficients, it can be determined which scenario corresponds to each bifurcation. Further, for a better representation of the study, the complex dynamics of the model are investigated theoretically and numerically using MatcotM, which is a Matlab package. Some graphical representations of the model are presented to verify the obtained results. The outcome of the study reveals that the model undergoes multiple bifurcations including period-doubling, Neimark–Sacker, and strong resonance bifurcations.

Multiple Bifurcations of a Discrete-Time Prey–Predator Model with Mixed Functional Response

In this paper, the one- and two-parameter bifurcations of a discrete-time prey–predator model with a mixed functional response are investigated by computing their critical normal form coefficients. The flip, Neimark–Sacker and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complex dynamical behavior of the model up to the 16th iteration is investigated.

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.