Multiple Periodic Orbits Research Articles

BIFURCATION ANALYSIS OF AN SIR MODEL WITH SATURATED INCIDENCE RATE AND STRONG ALLEE EFFECT

In this paper, rich dynamics and complex bifurcations of an SIR epidemic model with saturated incidence rate and strong Allee effect are investigated. First, the existence of disease-free and endemic equilibria is explored, and we prove that the system has at most three positive equilibria, which exhibit different types such as hyperbolic saddle and node, degenerate unstable saddle (node) of codimension 2, degenerate saddle-node of codimension 3 at disease-free equilibria, and cusp, focus, and elliptic types Bogdanov–Takens singularities of codimension 3 at endemic equilibria. Second, bifurcation analysis at these equilibria are investigated, and it is found that the system undergoes a series of bifurcations, including transcritical, saddle node, Hopf, degenerate Hopf, homoclinic, cusp type Bogdanov–Takens of codimensional 2, and focus and elliptic type Bogdanov–Takens bifurcation of codimension 3 which are composed of some bifurcations with lower codimension. The system shows very rich dynamics such as the coexistence of multiple periodic orbits and homoclinic loops. Finally, numerical simulations are conducted on the theoretical results.

Theoretical analysis of co-existing periodic orbits in sparse binary neural networks

A sparse binary neural network is a discrete-time recurrent neural network characterized by local binary connections and signum-type neuron models. The dynamics is described by an autonomous difference equation of binary state variables. Depending on the parameters, the network generates various periodic orbits of binary vectors. Multiple periodic orbits can co-exist and the network exhibits one of them depending on initial condition. Real/potential applications of the periodic orbits include control signals of switching circuits and basic signals for time-series approximation. The network is well suited for theoretical analysis and simple hardware implementation. This paper considers shift-type periodic orbits in the networks. As the main theoretical results, we clarify the number, period, and stability of the periodic orbits for key parameters. As a first step to the applications in the future, we present an FPGA-based hardware prototype and confirm typical periodic orbits experimentally.

Relative prevalence-based dispersal in an epidemic patch model.

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov-Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.

Bifurcation analysis of an SIR model considering hospital resources and vaccination

In this paper, an SIR epidemic model considering hospital resources and vaccination is established and the rich dynamics and complex bifurcations are investigated. Firstly, the existence of disease-free equilibrium and endemic equilibria is explored. It is founded that when the vaccination rate is not high, the number of endemic equilibrium points changed easily with the number of hospital resources and vaccination, resulting in transcritical bifurcation and saddle–node bifurcations. Secondly, different types singularities such as degenerate saddle–node of codimension 1 at the disease-free equilibrium, and cusp or focus type Bogdanov–Takens singularities of codimension 3 at endemic equilibria are presented. Thirdly, bifurcation analysis at these equilibria is investigated, and it is found that the system undergoes a sequence of bifurcations, including Hopf, degenerate Hopf bifurcation, homoclinic bifurcation, the cusp type Bogdanov–Takens bifurcation of codimension 2, and the focus type Bogdanov–Takens bifurcation of codimension 3 which are the organizing centers for a series of bifurcations with lower codimension. And the system shows very rich dynamics such as the existence of multiple coexistent periodic orbits, homoclinic loops. Finally, numerical simulations are presented to verify the theoretical results.

Bifurcations in Holling-Tanner model with generalist predator and prey refuge

Refuge provides an important mechanism for preserving many ecosystems. Prey refuges directly benefit prey but also indirectly benefit predators in the long term. In this paper, we consider the complex dynamics and bifurcations in Holling-Tanner model with generalist predator and prey refuge. It is shown that the model admits a nilpotent cusp or focus of codimension 3, a nilpotent elliptic singularity of codimension at least 4, and a weak focus with order at least 3 for different parameter values. As the parameters vary, the model can undergo three types degenerate Bogdanov-Takens bifurcations of codimension 3 (cusp, focus and elliptic cases), and degenerate Hopf bifurcation of codimension 3. The system can exhibit complex dynamics, such as multiple coexistent periodic orbits and homoclinic loops. Moreover, our results indicate that the constant prey refuge prevents prey extinction and causes global coexistence. A preeminent finding is that refuge can induce a stable, large-amplitude limit cycle enclosing one or three positive steady states. Numerical simulations are provided to illustrate and complement our theoretical results.

An SIRS model with nonmonotone incidence and saturated treatment in a changing environment.

Nonmonotone incidence and saturated treatment are incorporated into an SIRS model under constant and changing environments. The nonmonotone incidence rate describes the psychological or inhibitory effect: when the number of the infected individuals exceeds a certain level, the infection function decreases. The saturated treatment function describes the effect of infected individuals being delayed for treatment due to the limitation of medical resources. In a constant environment, the model undergoes a sequence of bifurcations including backward bifurcation, degenerate Bogdanov-Takens bifurcation of codimension 3, degenerate Hopf bifurcation as the parameters vary, and the model exhibits rich dynamics such as bistability, tristability, multiple periodic orbits, and homoclinic orbits. Moreover, we provide some sufficient conditions to guarantee the global asymptotical stability of the disease-free equilibrium or the unique positive equilibrium. Our results indicate that there exist three critical values r_1, r_2 and r_3 for the treatment rate r: (i) when rge max {r_1, r_2}, the disease will disappear; (ii) when r<min {r_1, r_3}, the disease will persist. In a changing environment, the infective population starts along the stable disease-free state (or an endemic state) and surprisingly continues tracking the unstable disease-free state (or a limit cycle) when the system crosses a bifurcation point, and eventually tends to the stable endemic state (or the stable disease-free state). This transient tracking of the unstable disease-free state when {mathscr {R}}_0>1 predicts regime shifts that cause the delayed disease outbreak in a changing environment. Furthermore, the disease can disappear in advance (or belatedly) if the rate of environmental change is negative and large (or small). The transient dynamics of an infectious disease heavily depend on the initial infection number and rate or the speed of environmental change. Supplementary InformationThe online version contains supplementary material available at 10.1007/s00285-022-01787-3.

Power expectation as a unified metric for the evaluation of vibration energy harvesters

Vibration energy harvesters (VEHs) allow the extraction of ambient mechanical energy. The proposed analysis focuses on bistable harvesters with Duffing nonlinearity. Bistable harvesters exhibit larger frequency bandwidth than linear monostable harvesters, but they exhibit complex behaviors (i.e., multiple periodic orbits and chaos), making their evaluation challenging. To assess the quality of a VEH, it is necessary to evaluate its energy performance generically. In this paper, we offer a new quantitative metric that can be used to evaluate bistable harvesters while taking into account the richness and diversity of their dynamics. This metric – called power expectation – is based on the average harvested power of each existing orbit weighted by its occurrence probability. The value of this power expectation depends on the harvester characteristics and potential orbit jump strategy implemented. Finally, we define a new figure of merit (FoM) based on the integral of the power expectation over all vibration frequencies. This FoM can be used for a generic evaluation of nonlinear vibration energy harvesters (NVEHs), by taking into account the various orbits and their respective probability of occurrence depending on the orbit jump strategy used. The proposed FoM also makes it possible to ensure a fair comparison with linear VEHs and quantitatively assess the effectiveness of orbit jump strategies for a given VEH.

Finite Element Analysis of the Dynamics of Power-Law Fluid around an Obstacle in a Channel

Control of fluid forces is an emerging area of research with numerous engineering applications. The uneven wake behind an obstacle causes undesirable structural oscillations, which can lead to fatigue or structural failure. Controlling the wake phenomena could directly benefit a wide range of engineering applications, including skyscrapers, naval risers, bridges, columns, and a few sections of airplanes. This study is concerned with the time dependent simulations in a channel in presence of an obstacle aiming to compute fluid forces. The underlying mathematical model is based on nonstationary Navier–Stokes equations coupled with the constitutive relations of power law fluids. Because the representative equations are complex, an effective computing strategy based on the finite element approach is used. To achieve higher accuracy, a hybrid computational grid at a very fine level is used. Theℙ2−ℙ1elements based on the shape functions of the second and first-order polynomials were used to approximate the solution. The discrete nonlinear system arising from this discretization is linearized by Newton’s method and then solved through a direct linear solver PARADISO. The code validation study is also performed for Newtonian fluids as a special case, and then the study is extended to compute drag and lift forces for other cases of viscosity as described by the power law index. When looking at the phase plot, it can be seen that for the Newtonian case n = 1, there is only one closed orbit after the steady state is reached, whereas forn=0.5, there are multiple periodic orbits. Moreover, the effects of shear rate on the drag-lift phase plot are also discussed.

Adding species to chemical reaction networks: Preserving rank preserves nondegenerate behaviours

We show that adding new chemical species into the reactions of a chemical reaction network (CRN) in such a way that the rank of the network remains unchanged preserves its capacity for multiple nondegenerate equilibria and/or periodic orbits. One consequence is that any bounded nondegenerate behaviours which can occur in a CRN can occur in a CRN with bounded stoichiometric classes. The main result adds to a family of theorems which tell us which enlargements of a CRN preserve its capacity for nontrivial dynamical behaviours. It generalises some earlier claims, and complements similar claims involving the addition of reactions into CRNs. The result gives us information on how ignoring some chemical species, as is common in biochemical modelling, might affect the allowed dynamics in differential equation models of CRNs. We demonstrate the scope and limitations of the main theorem via several examples. These illustrate how we can use the main theorem to predict multistationarity and oscillation in CRNs enlarged with additional species; but also how the enlargements can introduce new behaviours such as additional periodic orbits and new bifurcations.

Variational existence proof for multiple periodic orbits in the planar circular restricted three-body problem

The restricted three-body problem (R3BP) is an important research area that deals with significant issues in celestial mechanics, such as analyzing asteroid movement behavior and orbit design for space probes. We aim to show the existence of periodic orbits in the planar circular R3BP. To find these orbits, we adapt a variational approach and symmetry.

Projective Multi-Synchronization of Fractional-order Complex-valued Coupled Multi-stable Neural Networks with Impulsive Control

In this paper, we study a projective multi-synchronization problem for fractional-order complex-valued coupled multi-stable neural networks (FCVCMNNs) with time-delays. Using a complex decomposition approach, FCVCMNNs are divided into their real and imaginary components. Our method uses certain conditions for each subnetwork to achieve the multiple local equilibrium points or stable periodic orbits that are exponentially stable, which, when combined with the Lyapunov functions method, result in FCVCMNNs that are projectively multi-synchronized. The FCVCMNNs, on the other hand, are examined directly through the use of the Lyapunov functional method and linear matrix inequality (LMI). Various new sufficient conditions in the form of complex-valued LMIs are presented for the projective multi-synchronization of the considered FCVCMNNs. As a final step, we provide two numerical simulations to verify the effectiveness of the main results derived in this paper.

Dynamic binary neural networks with time-variant parameters and switching of desired periodic orbits

This paper studies 3-layer dynamic binary neural networks characterized by binary connection parameters and the signum activation function. The connection parameters and the number of hidden neurons are time-variant. The dynamics is described by a non-autonomous difference equation of binary state variables and the network can generate various periodic orbits of binary vectors. First, we present a simple synthesis method that guarantees storage of desired multiple periodic orbits and switching of the periodic orbits through desired entrances. Second, as a basic step to engineering applications, we present an FPGA based hardware prototype. The hardware can realize switching of various periodic orbits experimentally and typical examples are demonstrated. The switching of periodic orbits is applicable to switching of walking patterns in robotics and variable output of switching power converters.

Global analysis in Bazykin's model with Holling II functional response and predator competition

In this paper, we study the well-known Bazykin's model with Holling II functional response and predator competition. A detailed bifurcation analysis, depending on all four parameters, reveals a rich bifurcation structure, including supercritical and subcritical Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension at most 2, and a focus type degenerate Bogdanov-Takens bifurcation of codimension 3, originating from a nilpotent focus of codimension 3 which acts as the organizing center for the bifurcation set. Moreover, some sufficient conditions to guarantee the global asymptotical stability of the semi-trivial equilibrium or the unique positive equilibrium are also given. Our analysis indicates that we can classify the long-time dynamics of the model with a threshold value c0 for the natural mortality rate c of predators, in detail, the following are true. (i) When c≥c0, the prey will persist and predators will eventually go extinct for all positive initial populations. (ii) When c<c0, the prey and predators will coexist, for all positive initial populations, in the form of multiple positive equilibria or multiple periodic orbits. Our results can be seen as a complement to the work by Bazykin et al. [2–5], Hainzl [22,23], Kuznetsov [30].

Bifurcation Analysis of a Dynamical Model for the Innate Immune Response to Initial Pulmonary Infections

It has been reported that COVID-19 patients had an increased neutrophil count and a decreased lymphocyte count in the severe phase and neutrophils may contribute to organ damage and mortality. In this paper, we present the bifurcation analysis of a dynamical model for the initial innate system response to pulmonary infection. The model describes the interaction between a pathogen and neutrophilis (also known as polymorphonuclear leukocytes). It is shown that the system undergoes a sequence of bifurcations including subcritical and supercritical Bogdanov–Takens bifurcations, Hopf bifurcation, and degenerate Hopf bifurcation as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent periodic oscillations, homoclinic orbits, bistability and tristability, etc. Numerical simulations are presented to explain the theoretical results.

Multiple periodic orbits from Hopf bifurcation in a hierarchical neural network with [formula omitted]-symmetry and delays

This paper concerns a model of hierarchical neural network with Dn×Dn-symmetry and time delays. The linear stability of the steady state is obtained via an artful factorization technique of the characteristic equation corresponding to the linearized system. And we also perform a delay-induced Hopf bifurcation analysis involving spatio-temporal patterns of bifurcated periodic solutions relying on the equivariant Hopf bifurcation theory for retarded functional differential equations and the representation method of Lie groups. Some numerical simulations for a simple example are carried out to demonstrate the application of our theoretical conclusion that multiple periodic orbits could occur simultaneously from Hopf bifurcation in the model.

Bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method

We investigate a class of n-dimensional periodic differential equations with multiple small parameters by averaging method, and obtain sufficient conditions for the existence of multiple periodic orbits. As applications of the main results, a cubic system is studied.

Phononic canonical quasicrystalline waveguides

The dynamic behavior of the class of periodic waveguides whose unit cells are generated through a quasicrystalline sequence can be interpreted geometrically in terms of a trace map that embodies the recursive rule obeyed by traces of the transmission matrices. We introduce the concept of canonical quasicrystalline waveguides, for which the orbits predicted by the trace map at specific frequencies, called canonical frequencies, are periodic. In particular, there exist three families of canonical waveguides. The theory reveals that for those (i) the frequency spectra are periodic and the periodicity depends on the canonical frequencies, (ii) a set of multiple periodic orbits exists at frequencies that differ from the canonical ones, and (iii) perturbation of the periodic orbit and linearization of the trace map define a scaling parameter, linked to the golden ratio, which governs the self-similar structure of the spectra. The periodicity of the waveguide responses is experimentally verified on finite specimens composed of selected canonical unit cells.

Dynamic stability and bifurcation phenomena of an axially loaded flexible shaft-disk system supported by flexible bearing

The present research studies the vibration and bifurcation analysis of a spinning rotor-disk-bearing system to carefully scrutinize the dynamic stability under extrinsic mass unbalance and pulsating axial load. The shaft is flexible and taken into account the geometrical nonlinearities due to large elastic deformation in bending. The rotor is supported by flexible bearings, which are modeled as an equivalent spring-damper system having linear and nonlinear stiffness elements. Equation of motion of the rotating system, which includes flexible shaft, rigid disk, and flexible bearing, is derived using extended Hamilton principle with the assumption of the Euler's beam theory. We studied initially the modal analysis to determine the modal parameters, i.e. natural frequency and mode shapes prior to the investigation of the dynamics of the system. Further, we developed the bifurcation diagram for steady-state solutions and to study the subsequent dynamic stability and verify with the findings solved numerically. The interactive behavior among the nonlinear shaft-bearing, axial load and an unbalance has been analyzed. Numerical simulation tools, i.e. frequency–response characteristics, time history, phase trajectories, and Poincaré map have been used to highlight the presence of nonlinear phenomena and its important role towards evaluating the system dynamics and subsequent stability. This current research showed that flexible bearings stabilize the system as a result of increasing the restoring force. Analyzing the bifurcation diagrams of pulsating axial load, we found that the system exhibits complex phenomena as multiple but stable periodic orbits leading to period doubling. The present system is highly vulnerable to catastrophic failure due to the S–N and Pitchfork bifurcation. The present research enables the notability of axial load and mechanical unbalance on the overall system behavior and stability in real working conditions.

Phase-amplitude reduction far beyond the weakly perturbed paradigm.

While phase reduction is a well-established technique for the analysis of perturbed limit cycle oscillators, practical application requires perturbations to be sufficiently weak thereby limiting its utility in many situations. Here, a general strategy is developed for constructing a set of phase-amplitude reduced equations that is valid to arbitrary orders of accuracy in the amplitude coordinates. This reduction framework can be used to investigate the behavior of oscillatory dynamical systems far beyond the weakly perturbed paradigm. Additionally, a patchwork phase-amplitude reduction method is suggested that is useful when exceedingly large magnitude perturbations are considered. This patchwork method incorporates the high-accuracy phase-amplitude reductions of multiple nearby periodic orbits that result from modifications to nominal parameters. The proposed method of high-accuracy phase-amplitude reduction can be readily implemented numerically and examples are provided where reductions are computed up to fourteenth order accuracy.

Multiple periodic orbits of high-dimensional differential delay systems

In this paper, we consider differential delay systems of the formx′(t)=−∑s=12k−1(−1)s+1∇F(x(t−s)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x'(t)=-\\sum_{s=1}^{2k-1}(-1)^{s+1} \\nabla F \\bigl(x(t-s) \\bigr), $$\\end{document} in which the coefficients of the nonlinear terms with different hysteresis have different signs. Such systems have not been studied before. The multiplicity of the periodic orbits is related to the eigenvalues of the limit matrix. The results provide a theoretical basis for the study of differential delay systems.