Normal Form Theory Research Articles

A diffusive plant-sulphide model: spatio-temporal dynamics contrast between discrete and distributed delay

Abstract This paper studies the spatio-temporal dynamics of a diffusive plant-sulphide model with toxicity delay. More specifically, the effects of discrete delay and distributed delay on the dynamics are explored, respectively. The deep analysis of eigenvalues indicates that both diffusion and delay can induce Hopf bifurcations. The normal form theory is used to set up an exact formula that determines the properties of Hopf bifurcation in a diffusive plant-sulphide model. A sufficiently small discrete delay does not affect the stability and a sufficiently large discrete delay destabilizes the system. Nonetheless, a sufficiently small or large distributed delay does not affect the stability. Both delays cause instability by inducing Hopf bifurcation rather than Turing bifurcation.

On the bifurcations in a quadrotor unmanned aerial vehicle dynamical system using normal form theory

On the bifurcations in a quadrotor unmanned aerial vehicle dynamical system using normal form theory

Complex dynamics of an SIHR epidemic model with variable hospitalization rate depending on unoccupied hospital beds

In this paper, we propose an susceptible-infectious-hospitalized-recovered (SIHR) epidemic model with a nonlinear hospitalization rate depending on the number of unoccupied hospital beds. Note that the number of all hospital beds is used as a measure of all available medical resources. The basic reproduction number is calculated using the next-generation matrix method. We analyze the existence of endemic equilibria and discuss the global stability of the disease-free equilibrium. Existence and stability of endemic equilibria indicate possible occurrences of bifurcations. We confirm the appearance of backward bifurcation, saddle–node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation using normal form theory and central manifold theory. Numerical simulations show that the dynamic behavior of the model undergoes a transition from forward bifurcation to backward bifurcation and saddle–node bifurcation when the number of total hospital beds is reduced. Our findings suggest that when the number of total hospital beds falls below a threshold, backward bifurcation will occur, meaning that the disease cannot be eliminated even if the basic reproduction number is below unity. Therefore, the number of hospital beds should be increased beyond the bed threshold during an outbreak of a disease, which has important implications for disease control.

Learning the latent dynamics of fluid flows from high-fidelity numerical simulations using parsimonious diffusion maps

We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of high-fidelity Navier–Stokes simulations with a focus on the two-dimensional (2D) fluidic pinball problem. By varying the Reynolds number Re, different flow regimes emerge, ranging from steady symmetric flows to quasi-periodic asymmetric and chaos. The proposed non-linear manifold learning scheme identifies in a crisp manner the expected intrinsic dimension of the underlying emerging dynamics over the parameter space. In particular, PDMs estimate that the emergent dynamics in the oscillatory regime can be captured by just two variables, while in the chaotic regime, the dominant modes are three as anticipated by the normal form theory. On the other hand, proper orthogonal decomposition/principal component analysis (POD/PCA), most commonly used for dimensionality reduction in fluid mechanics, does not provide such a crisp separation between the dominant modes. To validate the performance of PDMs, we also compute the reconstruction error, by constructing a decoder using geometric harmonics (GHs). We show that the proposed scheme outperforms the POD/PCA over the whole Re number range. Thus, we believe that the proposed scheme will allow for the development of more accurate reduced order models for high-fidelity fluid dynamics simulators, relaxing the curse of dimensionality in numerical analysis tasks such as bifurcation analysis, optimization, and control.

Stability and Hopf Bifurcation Analysis of a Predator–Prey Model with Weak Allee Effect Delay and Competition Delay

The purpose of this paper is to study a predator–prey model with Allee effect and double time delays. This research examines the dynamics of the model, with a focus on positivity, existence, stability and Hopf bifurcations. The stability of the periodic solution and the direction of the Hopf bifurcation are elucidated by applying the normal form theory and the center manifold theorem. To validate the correctness of the theoretical analysis, numerical simulations were conducted. The results suggest that a weak Allee effect delay can promote stability within the model, transitioning it from instability to stability. Nevertheless, the competition delay induces periodic oscillations and chaotic dynamics, ultimately resulting in the population’s collapse.

Bifurcations and Homoclinic Orbits of a Model Consisting of Vegetation-Prey-Predator Populations.

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.

Dynamics and control of two-dimensional discrete-time biological model incorporating weak Allee's effect.

Incorporating a weak Allee effect in a two-dimensional biological model in ℜ2, the study delves into the application of bifurcation theory, including center manifold and Ljapunov-Schmidt reduction, normal form theory, and universal unfolding, to analyze nonlinear stability issues across various engineering domains. The focus lies on the qualitative dynamics of a discrete-time system describing the interaction between prey and predator. Unlike its continuous counterpart, the discrete-time model exhibits heightened chaotic behavior. By exploring a biological Mmdel with linear functional prey response, the research elucidates the local asymptotic properties of equilibria. Additionally, employing bifurcation theory and the center manifold theorem, the analysis reveals that, for all α1 (i.e., intrinsic growth rate of prey), ð1˙ (i.e., parameter that scales the terms yn), and m (i.e., Allee effect constant), the model exhibits boundary fixed points A1 and A2, along with the unique positive fixed point A∗, given that the all parameters are positive. Additionally, stability theory is employed to explore the local dynamic characteristics, along with topological classifications, for the fixed points A1, A2, and A∗, considering the impact of the weak Allee effect on prey dynamics. A flip bifurcation is identified for the boundary fixed point A2, and a Neimark-Sacker bifurcation is observed in a small parameter neighborhood around the unique positive fixed point A∗=(mð1˙-1,α1-1-α1mð1˙-1). Furthermore, it implements two chaos control strategies, namely, state feedback and a hybrid approach. The effectiveness of these methods is demonstrated through numerical simulations, providing concrete illustrations of the theoretical findings. The model incorporates essential elements of population dynamics, considering interactions such as predation, competition, and environmental factors, along with a weak Allee effect influencing the prey population.

Stability, bifurcation, and chaos in a class of scalar quartic polynomial delay systems.

In this paper, a class of scalar quartic polynomial delay systems is investigated. We found rich dynamics in this system through numerical simulation, including chaotic attractors, chaotic saddles, and intermittent chaos. Moreover, this chaotic quartic system may serve as an approximation, through Taylor expansion, for a wide class of scalar delay differential equations. Thus, these nonlinear systems may exhibit chaotic behaviors, and the studies in our paper may provide an insight into the emergence of chaos in other time-delay nonlinear systems. We also conduct a detailed theoretical analysis of the system, including the stability of equilibria and Hopf bifurcation analysis based on the theory of normal form and center manifold. Additionally, a numerical analysis is provided to give numerical evidence for the existence of chaos.

Controlling chaos and codimension-two bifurcation in a discrete fractional-order Brusselator model

This paper explores the qualitative behavior of a discrete fractional–order Brusselator model. We analyze the local dynamics of the model around its fixed point and determine its topological classification. We perform the bifurcation analysis for both codimension-one and codimension-two cases to examine the system behavior near critical parameter values. Using normal form theory and center manifold theorem (CMT), we prove that the model exhibits period-doubling bifurcation around its interior fixed point. We also study the existence and direction of Neimark–Sacker bifurcation using normal form theory. For codimension-two bifurcation, we show that the model undergoes 1:2, 1:3, and 1:4 resonances by applying normal form theory and suitable affine transformations. The system displays a rich variety of bifurcations, including quasi–periodicity, periodic orbits, chaotic behavior, and resonance bifurcation. Furthermore, the existence of chaos is discussed in the sense of Marotto, and a novel chaos control method is proposed for discrete Brusselator model using an extended pole–placement approach. This modified approach is more suitable for codimension-two bifurcation situations. Numerical simulations are used to illustrate the theoretical discussion.

Stability and Hopf Bifurcation in a Modified Sprott C System

Abstract In this article, a Modified Sprott C system is considered. The stability of equilibrium points and the occurrence of Hopf bifurcation in the system are investigated. It has been proved that the system displays a Hopf bifurcation at α = 0. Additionally, by applying normal form theory, the stability, direction and increase (or decrease) of the period of bifurcating periodic solutions of the system are illustrated. It has been shown that the solutions of bifurcating periodic solutions at the bifurcation value α = 0 are unstable. The type of Hopf bifurcation is subcritical and the periods of the bifurcating periodic solutions increase.

Prey group defense and hunting cooperation among generalist-predators induce complex dynamics: a mathematical study.

Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed 'bubble' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of "saltatory equilibria" in ecological systems and capture the dynamics observed for lion-herbivore interactions.

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Emergence of pathological beta oscillation and its uncertainty quantification in a time-delayed feedback Parkinsonian model

It has been experimentally found that transmission delays play an important role in the emergence of excessively synchronized beta oscillations (13–30 Hz) related to the onset of Parkinson’s symptoms. In order to clarify the dynamical mechanism underlying beta oscillations, we generalize a conventional resonance model based on the subthalamic nucleus (STN)-external segment of globus pallidus (GPe)-cortex circuit to a feedback Parkinsonian model by incorporating the transmission delay within the basal ganglia (STN-GPe circuit) and the transmission delay from the STN to the cortex. By combining the theory of center manifolds and normal forms with numerical simulations, it is revealed how the pathological beta oscillations occur as the transmission delays increase beyond critical values. Furthermore, by regarding the transmission delays and the connection weights as random variables, variance-based sensitivity analysis is performed based on polynomial chaos expansion. It is identified that the transmission delays and connection strength in the long STN-cortex circuit are more significant than those in the STN-GPe circuit for beta oscillations. Our results also suggest that the severity of the Parkinsonian symptoms could be alleviated if a clinical therapy, such as deep brain stimulation, can enlarge the transmission delays in the STN-GPe and STN-cortex circuits simultaneously.

Hopf Bifurcation and Turing Instability of a Delayed Diffusive Zooplankton–Phytoplankton Model with Hunting Cooperation

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results.

On solitary-wave solutions of Rosenau-type equations

The present paper is concerned with the existence of solitary wave solutions of Rosenau-type equations. By using two standard theories, Normal Form Theory and Concentration-Compactness Theory, some results of existence of solitary waves of three different forms are derived. The results depend on some conditions on the speed of the waves with respect to the parameters of the equations. They are discussed for several families of Rosenau equations present in the literature. The analysis is illustrated with a numerical study of generation of approximate solitary-wave profiles from a numerical procedure based on the Petviashvili iteration.

Bogdanov–Takens Bifurcation of Kermack–McKendrick Model with Nonlinear Contact Rates Caused by Multiple Exposures

In this paper, we consider the influence of a nonlinear contact rate caused by multiple contacts in classical SIR model. In this paper, we unversal unfolding a nilpotent cusp singularity in such systems through normal form theory, we reveal that the system undergoes a Bogdanov-Takens bifurcation with codimension 2. During the bifurcation process, numerous lower codimension bifurcations may emerge simultaneously, such as saddle-node and Hopf bifurcations with codimension 1. Finally, employing the Matcont and Phase Plane software, we construct bifurcation diagrams and topological phase portraits. Additionally, we emphasize the role of symmetry in our analysis. By considering the inherent symmetries in the system, we provide a more comprehensive understanding of the dynamical behavior. Our findings suggest that if this occurrence rate is applied to the SIR model, it would yield different dynamical phenomena compared to those obtained by reducing a 3-dimensional dynamical model to a planar system by neglecting the disease mortality rate, which results in a stable nilpotent cusp singularity with codimension 2. We found that in SIR models with the same occurrence rate, both stable and unstable Bogdanov-Takens bifurcations occur, meaning both stable and unstable limit cycles appear in this system.

Nonlinear Normal Modes of Vibrating Mechanical Systems: 10 Years of Progress

Abstract This paper contains a review of the theory and applications of nonlinear normal modes, which have been developed during last decade. This review has more than 200 references. It is a continuation of two previous review papers by the same authors. The following theoretical issues of nonlinear normal modes are treated: basic concepts and definitions; application of the normal form theory for nonlinear modes construction; nonlinear modes in finite degrees-of-freedom systems; resonances and bifurcations; reduced-order modeling; nonlinear modes in stochastic dynamical systems; numerical methods; identification of mechanical systems using nonlinear modes. The following applied issues of this theory are treated in this review: experimental measurement of nonlinear modes; nonlinear modes in continuous systems; engineering applications (aerospace engineering, power engineering, piecewise-linear systems and structures with dry friction); nonlinear modes in nanostructures and physical systems; targeted energy transfer and absorption problem.

Data-driven model reduction for pipes conveying fluid via spectral submanifolds

Pipes conveying fluid have non-linearizable dynamics when the flow velocity is large enough. Developing analytical models that accurately capture these highly nonlinear and complex dynamics is challenging. Efficient analysis and predictions of these nonlinear dynamics are essential in the design and control of pipe systems in engineering applications. To address these challenges, here we construct low-dimensional reduced-order models in a data-driven setting. In particular, we perform model reduction on spectral submanifolds (SSMs) via SSMLearn. This open-source package automates the identification of low-dimensional SSMs and their associated reduced-order models (ROMs). Our reduction via SSMs treats pipes with various boundary conditions and flow velocities in a unified way. It achieves a significant reduction because the ROMs are two-dimensional, which enables efficient and even analytical predictions on the nonlinear vibration of the pipes, paving the way for real-time prediction of the non-linearizable dynamics. In addition, our SSM-based reduction shows remarkable extrapolation capability because it is built on invariant manifolds and normal form theory. We demonstrate that the ROMs trained on unforced responses can be directly extrapolated to make reliable predictions on forced vibrations under various excitation frequencies and amplitudes, including periodic and quasi-periodic orbits and their bifurcations.

Crucial impact of component Allee effect in predator-prey system

Allee effect in models in interacting species of predator prey system has great significance in ecological context. Allee effect plays crucial role in population dynamics in ecology, where it is the challenging fact that per capita population growth rate is positively dependent on the population density of a species. In this paper, we inspect the famous Hastings and Powell (HP) (Hastings and Powell 1991 Ecology 72 896–903) model incorporating component Allee effect on top predator’s reproduction. We analyse the updated model with the help of both analytical and numerical phenomena. Considering cost of Allee effect, half-saturation constant of prey as the key parameters, the Hopf bifurcations are also analysed. The directions of Hopf bifurcations at the critical values of Allee parameter and half-saturation constant of prey are studied theoretically by using normal form theory introduced by Hassard et al (1981 Theory and Applications of Hopf Bifurcation vol 41 (CUP Archive)). The formulated model indicates that the system demonstrates chaotic, periodic and stable dynamics in the variation of key parameters. The chaos can be controlled for proper application of the large values of parameter used as the cost of Allee effect and also for small values of the parameter used as the half saturation constant of prey population. The results of this study are applicable in the field of marine and wild ecosystem dynamics.

A comprehensive study of spatial spread and multiple time delay in an eco-epidemiological model with infected prey

This paper studies the dynamics of interacting Tilapia fish and Pelican bird population in the Salton Sea. We assume that the diseases spread in Tilapia fish follows the Holling type II response function, and the interaction between Tilapia and Pelican follows the Beddington–DeAngelis response function. The dynamics of diffusive and delayed system are discussed separately. Analytically, all the feasible equilibria and their stability are discussed. The criterion for Turing instability is derived. Based on the normal form theory and center manifold arguments, the existence of stability criterion and the direction of Hopf bifurcation are obtained. Numerical simulation shows the occurrence Hopf bifurcation, double Hopf bifurcation and transcritical bifurcation scenarios. The snap shot shows the spot, spot-strip mix patterns in the whole domain. Further, the stability switching phenomena is observed in the delayed system. Our comprehensive study highlights the effect of different parameters, multiple time delay and extinction in Pelican populations.