Normal Form Analysis Research Articles

Pattern dynamics analysis of a time-space discrete FitzHugh-Nagumo (FHN) model based on coupled map lattices

This paper investigates the dynamics of a discrete FitzHugh-Nagumo (FHN) model with self-diffusion on two-dimensional coupled map lattices. The primary objective is to analyze the complex dynamics of neuronal systems in a discrete setting. Through the application of central manifold and normal form analysis, it has been demonstrated that the system is capable of undergoing Neimark-Sacker and flip bifurcations even in the absence of diffusion. Additionally, when influenced by diffusion, the system can manifest pure Turing instability, Neimark-Sacker-Turing instability, and Flip-Turing instability. In the numerical simulation section, the path from bifurcation to chaos is explored by calculating the maximum Lyapunov exponent and drawing the bifurcation diagram. The interconversion between various Turing instabilities is simulated by varying the values of the time step and the self-diffusion coefficient. This study contributes to a deeper understanding of the complexity of neural network systems.

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.

Normal form analysis in the presence of SVC and STATCOM and the effect of their siting on damping of rotor swings

Normal form analysis is used to assess the nonlinear behavior; it can also be used for siting controllers. Normal form analysis is straight-forward for systems governed by explicit ordinary differential equations. However, the analysis becomes complicated if the governing equations are differential–algebraic. The presence of SVC/STATCOM results in the governing equations for which the normal form analysis is not as straight-forward as that for explicit differential equations. In this paper, it is shown how semi-explicit differential–algebraic equations of index 1 are amenable to normal form analysis. In particular, methods are proposed for normal form analysis in the presence of SVC and STATCOM. Normal form analysis can be used to find the location of SVC and STATCOM that results in the best damping of rotor swings; the results are analyzed by case studies and also compared with the performance obtained by locating SVC and STATCOM using linearization methods.

Saturation correction for a piezoelectric shunt absorber based on 2:1 internal resonance using a cubic nonlinearity

In this study, we present a theoretical and experimental analysis of an antiresonance detuning correction for a nonlinear piezoelectric shunt absorber based on a two-to-one internal resonance. Thanks to this purely nonlinear feature, the oscillations of the primary system become independent of the forcing at a particular antiresonance frequency, thus creating an efficient reduction of the vibration. Past works of the literature present the design of the piezoelectric shunt and show that it is subjected to a softening behavior that detunes the antiresonance frequency as a function of the amplitude and thus degrades the performance. It is also shown that this softening behavior is caused by some non-resonant terms present in the equations, linked to the piezoelectric coupling. To counteract this undesired effect, we propose in this work to add a cubic nonlinearity in the shunt circuit, in addition to the quadratic one already present. Its tuning is based on a normal form analysis already published, which shows how cubic nonlinearities can cancel the effect of quadratic non-resonant terms. The present article describes the main features of the theory and focuses on the experimental proof of concept of this antiresonance detuning correction as well as the analysis of its range of validity. It is applied to the damping of the first bending mode of a hydrodynamic foil structure.

Approximate localised dihedral patterns near a turing instability

Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular patterns from a quiescent state. A key issue is that standard techniques for one-dimensional patterns have proven insufficient for understanding localisation in higher dimensions. In this work, we present a comprehensive approach to this problem by using techniques developed in the study of radially-symmetric patterns. Our analysis covers localised planar patterns equipped with a wide range of dihedral symmetries, thereby avoiding a restriction to solutions on a predetermined lattice. The context in this paper is a theory for the emergence of such patterns near a Turing instability for a general class of planar reaction-diffusion equations. Posing the reaction-diffusion system in polar coordinates, we carry out a finite-mode Fourier decomposition in the angular variable to yield a large system of coupled radial ordinary differential equations. We then utilise various radial spatial dynamics methods, such as invariant manifolds, rescaling charts, and normal form analysis, leading to an algebraic matching condition for localised patterns to exist in the finite-mode reduction. This algebraic matching condition is nontrivial, which we solve via a combination of by-hand calculations and Gröbner bases from polynomial algebra to reveal the existence of a plethora of localised dihedral patterns. These results capture the essence of the emergent localised hexagonal patterns witnessed in experiments. Moreover, we combine computer-assisted analysis and a Newton–Kantorovich procedure to prove the existence of localised patches with 6 m-fold symmetry for arbitrarily large Fourier decompositions. This includes the localised hexagon patches that have been elusive to analytical treatment.

Well-Posedness for the Dispersive Hunter–Saxton Equation

Abstract This article represents a 1st step toward understanding the well-posedness of the dispersive Hunter–Saxton equation, which arises in the study of nematic liquid crystals. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character. Further, the lack of spatial decay obstructs access to dispersive tools, including local smoothing estimates. Here, we give the 1st proof of local and global well-posedness for the Cauchy problem. Secondly, we improve our well-posedness results with respect to the low regularity of the initial data. The key techniques we use include constructing modified energies to realize a normal form analysis in our quasilinear setting, and frequency envelopes to prove continuous dependence with respect to the initial data.

Hydroelastic lumps in shallow water

Hydroelastic solitary waves propagating on the surface of a three-dimensional ideal fluid through the deformation of an elastic sheet are studied. The problem is investigated based on a Benney–Luke-type equation derived via an explicit non-local formulation of the classic water wave problem. The normal form analysis is carried out for the newly developed equation, which results in the Benney–Roskes–Davey–Stewartson (BRDS) system governing the coupled evolution of the envelope of a carrier wave and the wave-induced mean flow. Numerical results show three types of free solitary waves in the Benney–Luke-type equation all of which are predicted by the BRDS system: plane solitary wave, lump (i.e., fully localized traveling waves in three dimensions), and transversally periodic solitary wave. They are linked together by a dimension-breaking bifurcation where plane solitary waves and lumps can be viewed as two limiting cases, and transversally periodic solitary waves serve as intermediate states. The stability and interaction of solitary waves are investigated via a numerical time integration of the Benney–Luke-type equation. For a localized load moving on the elastic sheet with a constant speed, it is found that there exists a transcritical regime of forcing speed for which there are no steady solutions. Instead, periodic shedding of lumps can be observed if the forcing moves at speed in this range.

Dynamics in a Predator–Prey Model with Cooperative Hunting and Allee Effect

This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p=p#, by investigating stable and unstable manifolds of saddles. When p>p#, both species go extinct, and when p<p#, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays.

Accuracy of Normal Form Approximation for Power System Nonlinear Analysis

This paper presents some results on the accuracy of normal form analysis for large deviations from the stable equilibrium point (SEP) in a power system. The results of studies on the region of convergence of Neumann series, which is required for normal form analysis, are presented. A polynomial function approximation is proposed to model the generator exciter limiter. Studies on the accuracy of higher order normal form analysis for large deviations from SEP are presented. Use of Prony analysis in addition to simulation is proposed for the verification of accuracy.

Direct normal form analysis of oscillators with different combinations of geometric nonlinear stiffness terms

Nonlinear oscillators with geometric stiffness terms can be used to model a range of structural elements such as cables, beams and plates. In particular, single-degree-of-freedom (SDOF) systems are commonly studied in the literature by means of different approximate analytical methods. In this work, an analytical study of nonlinear oscillators with different combinations of geometric polynomial stiffness nonlinearities is presented. To do this, the method of direct normal forms (DNF) is applied symbolically using Maple software. Closed form (approximate) expressions of the corresponding frequency-amplitude relationships (or backbone curves) are obtained for both e and e2 expansions, and a general pattern for e truncation is presented in the case of odd nonlinear terms. This is extended to a system of two degrees-of-freedom, where linear and nonlinear cubic and quintic coupling terms exist. Considering the non-resonant case, an example is shown to demonstrate how the single mode backbone curves of the two degree-of-freedom system can be computed in an analogous manner to the approach used for the SDOF analysis. Numerical verifications are also presented using COCO numerical continuation toolbox in Matlab for the SDOF examples.

Numerical analysis of a degenerate generalized Hopf bifurcation

In this paper, we present a numeric bifurcation analysis of the normal form of degenerate Hopf bifurcation truncated up to seventh order with an equilibrium point located at the origin. By applying the genericity nondegenerate conditions and normal form theory, we study the bifurcation analysis of the codimension-3 Takens–Hopf bifurcation for the difficult case, where a rich bifurcation scenario is displayed. The third Lyapunov coefficient is used to distinguish the different cases of a codimension-3 Takens–Hopf bifurcation point, which can be efficiently computed with the aid of a software program based on the symbolic package Maple, presented in Appendix A. The normal form analysis results can be used to depict the complete bifurcation diagrams and phase portraits. In order to investigate the mechanism of the transitions between equilibrium and limit cycles, the methods of two scales in frequency domain are employed to study the evolutions.

Domain Walls for the Bénard–Rayleigh Convection Problem with “Rigid–Free” Boundary Conditions

We prove the existence of domain walls for the Bénard–Rayleigh convection in the case of “rigid–free” boundary conditions. In the recent work (Haragus and Iooss in Arch Ration Mech Anal 239:733–781, 2021), we studied this bifurcation problem in the cases of “rigid–rigid” and “free–free” boundary conditions. In the three cases, for the existence proof we use a spatial dynamics approach in which the governing equations are written as an infinite-dimensional dynamical system. A center manifold theorem shows that bifurcating domain walls lie on a 12-dimensional center manifold, and can be constructed as heteroclinic solutions connecting periodic solutions of the restriction of the dynamical system to this center manifold. The existence proof for these heteroclinic connections then relies upon a normal form analysis, the construction of a leading order heteroclinic connection, and the implicit function theorem. The main difference between the case of “rigid–free” boundary conditions considered here and the two cases in Haragus and Iooss (2021), is the loss of a vertical reflection symmetry of the governing equations. This symmetry was exploited in Haragus and Iooss (2021) to show that bifurcating domain walls lie on an 8-dimensional invariant submanifold of the center manifold. Consequently, the heteroclinic connections were found as solutions of an 8-dimensional, instead of a 12-dimensional, dynamical system.

Emergence of Beta Oscillations of a Resonance Model for Parkinson's Disease.

In Parkinson's disease, the excess of beta oscillations in cortical-basal ganglia (BG) circuits has been correlated with normal movement suppression. In this paper, a physiologically based resonance model, generalizing an earlier model of the STN-GPe circuit, is employed to analyze critical dynamics of the occurrence of beta oscillations, which correspond to Hopf bifurcation. With the experimentally measured parameters, conditions for the occurrence of Hopf bifurcation with time delay are deduced by means of linear stability analysis, center manifold theorem, and normal form analysis. It is found that beta oscillations can be induced by increasing synaptic transmission delay. Furthermore, it is revealed that the oscillations originate from interaction among different synaptic connections. Our analytical results are consistent with the previous experimental and simulating findings, thus may provide a more systematic insight into the mechanisms underlying the transient beta bursts.

Temporal Forcing Induced Pattern Transitions Near the Turing–Hopf Bifurcation in a Plankton System

We explore the impact of time-periodic forcing on pattern transitions in a plankton system of reaction–diffusion type. Here, we mainly focus on the forced states near the Turing–Hopf bifurcation. A normal form analysis leads to the finding that weak forcing exhibits a destabilizing effect on the dynamics by exciting the transitions from a spatially homogeneous stationary state to a periodic oscillation in time. The results are obtained by studying the amplitude equations derived using weakly nonlinear analysis in the presence of forcing, which enables us to calculate the changes of states. Examples are given to confirm the theoretical results.

Betatron frequency and the Poincaré rotation number

Symplectic maps are routinely used to describe single-particle dynamics in circular accelerators. In the case of a linear accelerator map, the rotation number (the betatron frequency) can be easily calculated from the map itself. In the case of a nonlinear map, the rotation number is normally obtained numerically, by iterating the map for given initial conditions, or through a normal form analysis, a type of a perturbation theory for maps. Integrable maps, a subclass of symplectic maps, allow for an analytic evaluation of their rotation numbers. In this paper we propose an analytic expression to determine the rotation number for integrable symplectic maps of the plane and present several examples, relevant to accelerators. These new results can be used to analyze the topology of the accelerator Hamiltonians as well as to serve as the starting point for a perturbation theory for maps.

Normal-form analysis of the cusp-transcritical interaction: applications in population dynamics

Bistability, the presence of alternative stable states, is an important feature of population models as it indicates that long-term predictions are dependent on the current population density. Two distinct kinds of bistability re-occur in population modelling studies, Allee Bistability and Positive Bistability. In this article, we show that a novel codimension-3 bifurcation, the cusp-transcritical interaction, can act as an organising centre for ordinary differential equations that exhibit both Allee Bistability and Positive Bistability. We first show how a normal form for cusp-transcritical interactions emerges from the unfolding of a particular one-dimensional degeneracy. We then illustrate the ecological relevance of the cusp-transcritical interaction. Finally, we provide a comprehensive example of normal-form analysis of an existing population model that demonstrates the occurrence of the codimension-3 bifurcation. We note that Allee Bistability and Positive Bistability may manifest unexpectedly in complex, ecological models, and therefore, this bifurcation-focused approach can provide valuable insight into the behaviour of newly developed ecosystem models.

Bifurcations of a Bouncing Ball Dynamical System

A deceptively simple difference system of a bouncing ball is investigated. Applying the center manifold theorem and the normal form analysis, we first give the parameter conditions for the flip bifurcation. Then we discuss the 1:2 resonance. Because the nondegeneracy conditions are not satisfied, we use the approximation of mappings by a flow and change this difference system into an ordinary differential system with dimension two. With the aid of the Melnikov method, the homoclinic bifurcation and the Poincaré bifurcation are analyzed, which imply the existence of the invariant circles for the difference system. Furthermore, we compute the normal forms to provide the parameter conditions for the Chenciner bifurcation and also obtain the stability of the fixed point. Finally, some numerical simulations are presented to verify the obtained results.

Rich dynamics in a delayed HTLV-I infection model: Stability switch, multiple stable cycles, and torus

In this paper, we investigate the impact of time delay in CTL immune response on a HTLV-I infection model. By defining basic reproduction number for viral infection R0 and basic reproduction number for CTL response RCTL, we characterize the model dynamics according to whether these two threshold values are greater than one. Especially, we obtain the global dynamics if R0≤1 or RCTL≤1<R0, as well as infection persistent result when R0>1. However, the model dynamics become much richer when RCTL<1. In this case, we use the time delay as a bifurcation parameter to obtain stability switch result on the positive equilibrium and global bifurcation diagrams for the model system. We also conduct higher-order normal form analysis and apply center manifold theory to classify the rich model dynamics near the double Hopf bifurcation points. Our analysis indicates that time delay in CTL immune response can induce not only Hopf bifurcation and double Hopf bifurcation, but also quasi-periodic orbits (torus) and coexistence of multiple stable periodic solutions.

Gröbner bases and multi-dimensional persistent bifurcation diagram classifications

The work presented here is motivated by our ongoing project on symbolic bifurcation analysis of multidimensional germs of singularities, say [EQUATION] where z and λ denote the state variables and a distinguished parameter, respectively. A local solution Z (λ) for f ( Z (λ), λ) = 0 is called a bifurcation diagram. Any two bifurcation diagrams Z 1 (λ) and Z 2 (λ) of f ( z , λ) are called qualitatively equivalent when there exists a diffeomorphism (Φ( Z 1 , λ 1 ), Λ(λ 1 )) transforming the bifurcation diagram Z 1 (λ 1 ) into Z 2 (λ 2 ) = Φ( Z 1 (Λ -1 (lD 2 )), Λ -1 (λ 2 )). Our proposed bifurcation analysis often corresponds to steady state bifurcations of PDEs, static and dynamical systems and occur in many real life and engineering control problems [5]. This will be integrated into the Singularity library [3, 4] as MultiDimSingularities module. Our library provides systematic tools for symbolic treatment of bifurcation analysis of such systems. The classical approach uses root finding and parametric continuation methods through numerical normal form analysis for differential systems with low codimension (degeneracy). The latter fails on comprehensive analysis for a parametric system and for systems with moderate degeneracies. The goal here is to classify the qualitatively different bifurcation diagrams of an unfolding germ [EQUATION], for f ( z , λ), where α = (α 1 ,...,α p ) are unfolding parameters and F ( z , λ, 0) = f ( z , λ). This classification is feasible via the concept of persistent bifurcation diagrams. A diagram is called persistent when it is qualitatively self-equivalent under arbitrarily small perturbations. The idea is to find non-persistent sources of bifurcations. Non-persistent sources are divided into three categories: bifurcation B, hysteresis H and double limit point D ; see [6, Page 410]. The set [EQUATION] is called a transition local variety , that is a codimension-1 hyperplane in the parameter space. A local variety refers to a neighborhood subset of zeros of a polynomial system. The complement space of this hyperplane consists of a finite number of connected components, say C 1 , ..., C n . For any i and α, β ∈ C i , the bifurcation diagrams Z (λ, α) and Z (λ, β) are qualitatively equivalent. Hence, a choice of bifurcation diagram from each connected component gives rise to a complete list of persistent bifurcation diagram classifications for F ( z , λ, α). The RegularChains library in Maple provides a tool through the command CylindricalAlgebraicDecompose enabling us to generate this list; see [1]. However, there usually exist multiple bifurcation diagrams from each connected component C i in the generating list. In this abstract paper, we merely focus on the use of Gröbner basis for multi-dimensional persistent bifurcation diagram classifications.

Normal form analysis of bouncing cycles in isotropic rotor stator contact problems

This work considers analysis of sustained bouncing responses of rotating shafts with nonlinear lateral vibrations due to rotor stator contact. The insight that this is an internal resonance phenomena makes this an ideal system to be studied with the method of normal forms, which assumes that a system may be modelled primarily in terms of just its resonant response components. However, the presence of large non smooth nonlinearities due to impact and rub mean that the method must be extended. This is achieved here by incorporating an alternating frequency/time (AFT) step to capture nonlinear forces. Furthermore, the presence of gyroscopic terms leads to the need to handle complex modal variables, and a rotating coordinate frame must be used to obtain periodic responses. The process results in an elegant formulation that can provide reduced order models of a wide variety of rotor systems, with potentially many nonlinear degrees of freedom. The method is demonstrated by comparing against time simulation of two example rotors, demonstrating high precision on a simple model and approximate precision on a larger model.