Normal Form Method Research Articles

Asymptotic behavior of solutions of nonlinear discrete systems in critical cases

AbstractThe paper is dedicated to the analysis of the asymptotic behavior of essentially nonlinear discrete systems whose linearization possesses eigenvalues on the unit circle. For these systems, the paper establishes sufficient conditions for the asymptotic stability regardless of terms higher than the third order. Moreover, it derives an asymptotic stability criterium for a reduced critical subsystem. The proof incorporates a constructive approach for deriving Lyapunov functions using the center manifold reduction and normal form method. The key result of the paper asserts that, given the obtained stability conditions, the solutions of the system converge to the origin with a polynomial decay rate. The paper also introduces a method for evaluating the coefficients of the decay rate estimate. To illustrate these findings, we apply the obtained result to analyzing nonlinear business cycle and predator-pray models.

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

Hopf-Hopf bifurcation, period [formula omitted] solutions, slow-fast phenomena, and chimera of an optoelectronic reservoir computing system with single delayed feedback loop

In this paper, we investigate the co-dimension two bifurcations and complicated dynamics of an optoelectronic reservoir computing (RC) system with single delayed feedback loop. We focuses primarily on its underlying system’s Hopf-Hopf bifurcation. Firstly, we apply DDE-BIFTOOL built in Matlab to sketch the bifurcation diagrams with respect to two bifurcation parameters, namely feedback strength β and time delay τ, and find the existence of the Hopf-Hopf bifurcation points. Then, using the multiple scales method, we obtain their normal forms, and using the normal form method, we unfold and classify their local dynamics. Then numerical simulations are conducted to verify these results. We discover rich dynamical behaviors of the system in specific regions. Besides, other complicated dynamics, such as fast-slow phenomena, Period n solutions, and chimera, are found in the system. All these rich dynamical phenomena can provide excellent performance potentially for this optoelectronic reservoir computing system with single delayed feedback loop.

Bifurcation Analysis of a Class of Two-Delay Lotka–Volterra Predation Models with Coefficient-Dependent Delay

In this paper, a class of two-delay differential equations with coefficient-dependent delay is studied. The distribution of the roots of the eigenequation is discussed, and conditions for the stability of the internal equilibrium and the existence of Hopf bifurcation are obtained. Additionally, using the normal form method and the central manifold theory, the bifurcation direction and the stability for the periodic solution of Hopf bifurcation are calculated. Finally, the correctness of the theory is verified by numerical simulation.

Dynamics of a discrete one‐predator two‐prey system with Michaelis–Menten‐type prey harvesting and prey refuge

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system.

Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle

AbstractWe consider a class of Hamiltonian Klein–Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small initial data of size $$\varepsilon $$ ε showing that the corresponding solutions remain close to oscillatory motions over a time scale $$\varepsilon ^{{-\frac{9}{4}+\delta }}$$ ε - 9 4 + δ for any $$\delta >0$$ δ > 0 . The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach.

Integration of a degenerate system of ODEs

The integrability of a two-dimensional autonomous polynomial system of ordinary differential equations (ODEs) with a degenerate singular point at the origin that depends on six parameters is investigated. The integrability condition for the first quasihomogeneous approximation allows one of these parameters to be fixed on a countable set of values. The further analysis is carried out for this value and five free parameters. Using the power geometry method, the system is reduced to a non-degenerate form through the blowup process. Then, the necessary conditions for its local integrability are calculated using the method of normal forms. In other words, the conditions for the parameters under which the original system is locally integrable near the degenerate stationary point are found. By resolving these conditions, we find seven twoparameter families in the five-dimensional parametric space. For parameter values from these families, the first integrals of the system are found. The cumbersome calculations that occur in the problem under consideration are carried out using computer algebra.

Poincaré’s method of normal forms in the study of factors of Production

As the main factors of production, the paper considers physical capital and labor resources. The study of the behavior of the main factors of production is based on the analysis of the volume of investment in them, the depreciation of physical capital and changes in the volume of labor resources. Most of the studies on this topic do not address issues related to distortions in financing and their possible consequences for the main factors of production. To assess these consequences, formulas were derived that include partial derivatives of investment volumes for each of the main factors. The use of these formulas helps to develop an investment algorithm that contributes to the sustainable growth of the main factors of production. The method of Poincaré normal forms was chosen as the main research method, which allows us to simplify the initial problem and reduce it to an analysis in a linear form. A classification of possible variants of eigenvalues, the resulting linear form was carried out. The behavior of the main factors of production depending on the found eigenvalues is determined. This method allows you to get more understandable results and draw more accurate conclusions. The results obtained were tested in a number of federal districts of Russia and a number of EU countries. Testing showed a qualitative match with the real state of the main factors of production. This suggests that the developed investment algorithm can be applied in practical conditions and contribute to the achievement of stable growth in various regions of Russia. This study is important for understanding and optimizing production processes and economic development. It provides a basis for developing effective investment and decision-making strategies that promote both a balanced and directed development of the main factors of production.

Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known “Benjamin-Feir resonances”. We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.

Bifurcation analysis on a river population model with varying boundary conditions

A delayed reaction-diffusion-advection equation subject to constant-flux and free-flow boundary conditions is considered, which models single population dynamics in a river. At first, we show the existence of a nonconstant steady state induced by the change of constant flux value. Then by analyzing the distribution of eigenvalues, the stability of the constant and nonconstant steady states and the existence of Hopf bifurcations are obtained. And an algorithm for determining the direction and stability of Hopf bifurcations is derived by applying the center manifold theory and normal form method for PFDEs. Finally, the effects of advection and downstream boundary condition on periodic oscillations are discussed theoretically and numerically.

Stability and Hopf bifurcation of an intraguild prey-predator fishery model with two delays and Michaelis-Menten type predator harvest.

In this paper, we have proposed and investigated an intraguild predator-prey system incorporating two delays and a harvesting mechanism based on the Michaelis-Menten principle, and it was assumed that the two species compete for a shared resource. Firstly, we examined the properties of the relevant characteristic equations to derive sufficient conditions for the asymptotical stability of equilibria in the delayed model and the existence of Hopf bifurcation. Using the normal form method and the central manifold theorem, we analyzed the stability and direction of periodic solutions arising from Hopf bifurcations. Our theoretical findings were subsequently validated through numerical simulations. Furthermore, we explored the impact of harvesting on the quantity of biological resources and examined the critical values associated with the two delays.

The dynamics of a delayed predator-prey model with square root functional response and stage structure

<abstract> <p>In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results.</p> </abstract>

INFLUENCE OF EXTERNAL CUBIC VISCOSITY ON THE DYNAMICS OF A CANTILEVER ROD LOADED WITH A TRACKING FORCE

The analysis of the influence of external nonlinear cubic viscosity on the behavior of a cantilever rod loaded with a compressive tracking force is performed. The Bernoulli – Euler beam model is used for the describing of the rod bending deformations. After the transition to dimensionless variables, we obtained the form of notation in which the coefficients of the equation do not depend on the coefficient of cubic viscous friction. This indicates the independence of the critical load and system frequency characteristics from the level of nonlinear attenuation. Using the standard procedure of the Bubnov – Galerkin method, taking into account the first two forms of eigen oscillations, the problem is reduced to the analysis of a fourth order nonlinear system of ordinary differential equations with respect to generalized coordinates and generalized velocities. For the linear version of the equations with low attenuation, the values of the critical load are obtained in dependence on the ratio of the damping coefficients of the eigenforms. The behavior of the system is analyzed taking into account the considered cubic nonlinearity using the method of nonlinear normal forms. The problem is reduced to the analysis of a nonlinear dynamical system with one degree of freedom. There are constructive analytical methods of qualitative and approximate quantitative research for this class of problems. The critical value of the compressive force was determined by the analysis of the dependence of the first Lyapunov magnitude on the load parameter. It turned out to be less than the value corresponding to the system without damping (the destabilizing effect of external nonlinear friction). A bifurcation occurs in the system at the critical load value. Therefore, the zero equilibrium becomes an unstable complex focus of the first order and a limit cycle is born. The function of the self-oscillations amplitude of the load magnitude is constructed using the harmonic linearization method.

Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System

The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.

Dynamics of a delayed reaction–diffusion predator–prey model with nonlocal competition and double Allee effect in prey

In this paper, we study the effects of the nonlocal competition and double Allee effect in prey on a diffusive predator–prey model. We investigate the local stability of coexistence equilibrium in the predator–prey model by analyzing the eigenvalue spectrum. We study the consequence of double Allee effect on the prey population. Also, we discuss the existence of Hopf bifurcation under different parameters by using the gestation time delay of predators as a bifurcation parameter and analyzing the distribution of eigenvalues. By utilizing the normal form method and center manifold theorem, we have given some conditions that could determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Through our research, we obtain that the double Allee effect can affect the coexistence of prey and predator and induce periodic oscillations of the densities of prey and predator. In addition, the nonlocal competition can also affect the stability and homogeneity of the solutions, but may receive the consequences of the Allee effect as well as time delay. In the numerical simulation part, we further demonstrate the correctness of this conclusion by comparison.

Properties of Hopf bifurcation to a reaction-diffusion population model with nonlocal delayed effect

This paper is concerned with a reaction-diffusion population model with nonlocal delayed effect and zero-Dirichlet boundary condition. Under the condition when the delayed feedback control is dominant, the normal form for spatially nonhomogeneous Hopf bifurcation from the sufficiently small positive equilibrium is computed by means of the normal form method and the center manifold theorem for partial functional differential equations. It is revealed that Hopf bifurcations appearing at the small positive equilibrium are forward and all bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. To verify the validity of the obtained theoretical results, numerical simulations are also provided.

Transient Stability Analysis for a Multi-VSC Parallel System Based on the CFND Method

Because of the control complexity of voltage source converters (VSCs), transient stability analysis of multi-VSC parallel systems is challenging, and there are still no effective methods to solve this problem. Inspired by the decoupling principle and combined with the normal form method, an innovative coupling-factor-based nonlinear decoupling (CFND) method is proposed. According to the coupling factors that can be used to evaluate the nonlinear coupling degree among different state variables, the CFND method approximately transforms a high-order nonlinear multi-VSC parallel system into multiple decoupled low-order modes. Thus, the transient stability of the original high-order multi-VSC system can be reflected indirectly by the mature inversing trajectory method and the phase plane method. The CFND method has universality, flexibility, and insensitivity to system order, and no need to construct corresponding Lyapunov functions for different nonlinear systems, breaking through the inherent limitations of traditional analysis methods. Furthermore, this paper derives a reduced-order large-signal model and the corresponding truncated model for a single VSC grid-connected system. The effectiveness of the reduced-order model is verified through simulation waveforms and ROAs partitioning. Subsequently, a generalized model of the multi-VSC grid-connected system is developed. Finally, taking the grid-connected system with three VSCs as an example, the proposed CFND method is used to analyze typical operation cases, and the conclusions of transient stability analysis are verified using the hardware-in-loop (HIL) experiments.

Dynamics of Chains of Many Oscillators with Unidirectional and Bidirectional Delay Coupling

Chains of Van der Pol equations with a large delay in coupling are considered. It is assumed that the number of chain elements is also sufficiently large. In a natural manner, a chain is replaced by a Van der Pol equation with an integral term in the space variable and with periodic boundary conditions. Primary attention is given to the local dynamics of chains with unidirectional and bidirectional coupling. For sufficiently large values of the delay parameter, parameters are explicitly determined for which critical cases occur in the stability problem for the zero equilibrium state. It is shown that the problems under consideration have an infinite-dimensional critical case. The well-known methods of invariant integral manifolds and the methods of normal forms are inapplicable in these problems. Proposed by this paper’s author, the method of infinite normalization—the method of quasi-normal forms—is used to show that the leading terms of the asymptotics of the original system are determined by solutions of (nonlocal) quasi-normal forms, i.e., special nonlinear boundary value problems of the parabolic type. As the main results, corresponding quasi-normal forms are constructed for the considered chains.

Integral equations of the second kind in dynamic analysis of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material dependencies

Introduction. Development of computational methods for nonlinear systems possesses a significant potential, considering that linear theory sometimes fails to accurately describe the properties of dynamic systems, and the linear approximation gives only a very rough idea of real processes for a number of cases.Aim. Calculating linear systems and deriving the resulting equations for nonlinear systems involves impulse response and transfer functions of linear “generating” systems of differential equations. Such an approach in comparison with the traditional method of the so-called normal forms enables the calculation algorithm to be simplified considerably, avoiding several stages and presenting the solution by means of the normal mode method for linear systems directly with respect to the generalized coordinates.Materials and methods. The paper presents a method and an algorithm developed for the calculation of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material nonlinearity. Systems of nonlinear differential equations were reduced to nonlinear integral equations of the second kind, considered as resulting equations. The solution was developed in time steps, the value of which, among other things, determines the accuracy of the solution and the nature of the computational algorithm.Results. The paper presents main computational dependencies in a generalized form, convenient for numerical simulation. The author provides solutions for a nonlinear system with one degree of freedom and a cubic reactiondisplacement relation, as well as for a system with one and two degrees of freedom with a viscous damper. In both cases, the developed solution contains all properties of nonlinear systems, including the jump (transition) from the upper ascending branch to the lower, stable one, and the associated excitation of free oscillations.Conclusions. According to the calculations, the occurrence of nonlinear effects in oscillating systems makes positive impact on their behavior, in resonant modes in particular.

Codimension-One and Codimension-Two Bifurcations of a Fractional-Order Cubic Autocatalator Chemical Reaction System

This article delves into an investigation of the dynamic behavior exhibited by a fractional order cubic autocatalator chemical reaction model. Specifically, our focus lies on exploring codimension-one bifurcations associated with period-doubling bifurcation and Neimark-Sacker bifurcation. Additionally, we undertake an analysis of codimension-two bifurcations linked to resonances of the types 1:2, 1:3, and 1:4. To achieve these outcomes, we employ the normal form method and bifurcation theory. The results are presented through comprehensive numerical simulations, encompassing visual representations such as phase portraits, two-parameter bifurcation diagrams, and maximum Lyapunov exponents diagrams. These simulations aptly examine the behavior of a system governed by two distinct parameters that vary within a three-dimensional space. Furthermore, the simulations effectively illustrate the theoretical findings while providing valuable insights into the underlying dynamics.