Search results for: hopf bifurcation

Authors: Ankit Kumar, Balram Dubey

Abstract:

This study proposes a three dimensional prey-predator model with additional food, provided to predator individuals, including gestation delay in predators and delay in supplying the additional food to predators. It is assumed that the interaction between prey and predator is followed by Holling type-II functional response. We discussed the steady states and their local and global asymptotic behavior for the non-delayed system. Hopf-bifurcation phenomenon with respect to different parameters has also been studied. We obtained a range of predator’s tendency factor on provided additional food, in which the periodic solutions occur in the system. We have shown that oscillations can be controlled from the system by increasing the tendency factor. Moreover, the existence of periodic solutions via Hopf-bifurcation is shown with respect to both the delays. Our analysis shows that both delays play an important role in governing the dynamics of the system. It changes the stability behavior into instability behavior. The direction and stability of Hopf-bifurcation are also investigated through the normal form theory and the center manifold theorem. Lastly, some numerical simulations and graphical illustrations have been carried out to validate our analytical findings.

Keywords: additional food, gestation delay, Hopf-bifurcation, prey-predator

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Authors: Matthaios Katsanikas

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We study the structure of the invariant manifolds of complex unstable periodic orbits of a family of periodic orbits, in a 3D autonomous Hamiltonian system of galactic type, after a transition of this family from stability to complex instability (Hamiltonian Hopf bifurcation). We consider the case of a supercritical Hamiltonian Hopf bifurcation. The invariant manifolds of complex unstable periodic orbits have two kinds of structures. The first kind is represented by a disk confined structure on the 4D space of section. The second kind is represented by a complicated central tube structure that is associated with an extended network of tube structures, strips and flat structures of sheet type on the 4D space of section.

Keywords: dynamical systems, galactic dynamics, chaos, phase space

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Authors: Anuraj Singh

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The complex dynamics is explored in a prey predator system with multiple delays. The predator dynamics is governed by Leslie-Gower scheme. The existence of periodic solutions via Hopf bifurcation with respect to delay parameters is established. To substantiate analytical findings, numerical simulations are performed. The system shows rich dynamic behavior including chaos and limit cycles.

Keywords: chaos, Hopf bifurcation, stability, time delay

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Authors: Burak Yildirim, Muhsin Tunay Gencoglu

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This publication shows the effects of Microgrid (MG) integration on the power systems oscillating stability. Generated MG model power systems were applied to the IEEE 14 bus test system which is widely used in stability studies. Stability studies were carried out with the help of eigenvalue analysis over linearized system models. In addition, Hopf bifurcation point detection was performed to show the effect of MGs on the system loadability margin. In the study results, it is seen that MGs affect system stability positively by increasing system loadability margin and has a damper effect on the critical modes of the system and the electromechanical local modes, but they make the damping amount of the electromechanical interarea modes reduce.

Keywords: Eigenvalue analysis, microgrid, Hopf bifurcation, oscillatory stability

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Authors: Amadou Banda Ndione, Charles Awono Onana

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In this paper the compartmental approach is applied to build a macroeconomic model characterized by countries. We consider a total of N countries that are subdivided into three compartments according to their economic status: D(t) denotes the compartment of developing countries at time t, E(t) stands for the compartment of emerging countries at time t while A(t) represents advanced countries at time t. The model describes the process of economic development and includes the notion of openness through collaborations between countries. Two delays appear in this model to describe the average time necessary for collaborations between countries to become efficient for their development process. Our model represents the different stages of development. It further gives the conditions under which a country can change its economic status and demonstrates the short-term positive effect of openness on economic growth. In addition, we investigate bifurcation by considering the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations from a positive equilibrium. Numerical simulations are provided in order to illustrate the theoretical part and to support discussion.

Keywords: compartmental systems, delayed dynamical system, economic development, fiscal policy, hopf bifurcation

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Authors: Qamar J. A. Khan, Fatma Ahmed Al Kharousi

Abstract:

A prey-predator eco-epidemiological model is studied where transmission of the disease between infected and uninfected prey is nonlinear. The interaction of the predator with infected and uninfected prey species depend on their numerical superiority. Harvesting of both uninfected and infected prey is considered. Stability analysis is carried out for equilibrium values. Using the parameter µ, the death rate of infected prey as a bifurcation parameter it is shown that Hopf bifurcation could occur. The theoretical results are compared with numerical results for different set of parameters.

Keywords: bifurcation, optimal harvesting, predator, prey, stability

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Authors: J. C. Moctezuma, V. Breña-Medina, Jose Luis Nunez-Yanez, Joseph P. McGeehan

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In this work we make a bifurcation analysis for a single compartment representation of Traub model, one of the most important conductance-based models. The analysis focus in two principal parameters: current and leakage conductance. Study of stable and unstable solutions are explored; also Hop-bifurcation and frequency interpretation when current varies is examined. This study allows having control of neuron dynamics and neuron response when these parameters change. Analysis like this is particularly important for several applications such as: tuning parameters in learning process, neuron excitability tests, measure bursting properties of the neuron, etc. Finally, a hardware implementation results were developed to corroborate these results.

Keywords: Traub model, Pinsky-Rinzel model, Hopf bifurcation, single-compartment models, bifurcation analysis, neuron modeling

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Authors: Paolo Russu

Abstract:

This paper investigates the economic and ecological dynamics that emerge in Protected Areas (PAs) due to interactions between visitors and the animals that live there. The PAs contain two species whose interactions are determined by the Lotka-Volterra equations system. Visitors' decisions to visit PAs are influenced by the entrance cost required to enter the park and the chance of witnessing the species living there. Visitors have contradictory effects on the species and thus on the sustainability of the protected areas: on the one hand, an increase in the number of tourists damages the natural habitat of the regions and thus the species living there; on the other hand, it increases the total amount of entrance fees that the managing body of the PAs can use to perform defensive expenditures that protect the species from extinction. For a given set of parameter values, saddle-node bifurcation, Hopf bifurcation, homoclinic orbits, and a Bogdanov–Takens bifurcation of codimension two has been investigated. The system displays periodic doubling and chaotic solutions, as numerical examples demonstrate. Pontryagin's Maximum Principle was utilised to develop an optimal admission charge policy that maximised social gain and ecosystem conservation.

Keywords: chaos, bifurcation points, dynamical model, optimal control

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Authors: Muhammad Fiaz

Abstract:

The article in hand is the study of complex features like Zero Hopf Bifurcation, Chaos and Synchronization of integer and fractional order version of a new 3D finance system. Trusted tools of averaging theory and active control method are utilized for investigation of Zero Hopf bifurcation and synchronization for both versions respectively. Inventiveness of the paper is to find the answer of a question that is it possible to find a chaotic system which can be synchronized with any other of the same dimension? Based on different examples we categorically develop a theory that if a couple of master and slave chaotic dynamical system is synchronized by selecting a suitable gain matrix with special conditions then the master system is synchronized with any chaotic dynamical system of the same dimension. With the help of this study we developed generalized theorems for synchronization of n-dimension dynamical systems for integer as well as fractional versions. it proposed that this investigation will contribute a lot to control dynamical systems and only a suitable gain matrix with special conditions is enough to synchronize the system under consideration with any other chaotic system of the same dimension. Chaotic properties of fractional version of the new finance system are also analyzed at fractional order q=0.87. Simulations results, where required, also provided for authenticity of analytical study.

Keywords: complex analysis, chaos, generalized synchronization, control dynamics, fractional order analysis

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Authors: Shao-Yuan Huang

Abstract:

We study the shape of the bifurcation curve of positive solutions for the semipositone problem with the Minkowski-curvature operator. The Minkowski-curvature problem plays an important role in certain fundamental issues in differential geometry and in the special theory of relativity. In addition, it is well known that studying the multiplicity of positive solutions is equivalent to studying the shape of the bifurcation curve. By the shape of the bifurcation curve, we can understand the change in the multiplicity of positive solutions with varying parameters. In this paper, our main technique is a time-map method used in Corsato's PhD Thesis. By this method, studying the shape of the bifurcation curve is equivalent to studying the shape of a certain function T with improper integral. Generally speaking, it is difficult to study the shape of T. So, in this paper, we consider two cases that the nonlinearity is convex or concave. Thus we obtain the following results: (i) If f''(u) < 0 for u > 0, then the bifurcation curve is C-shaped. (ii) If f''(u) > 0 for u > 0, then there exists η>β such that the bifurcation curve does not exist for 0 η. Furthermore, we prove that the bifurcation is C-shaped for L > η under a certain condition.

Keywords: bifurcation curve, Minkowski-curvature problem, positive solution, time-map method

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Authors: Sanjeev Kumar Sharma, Arnab Mondal, Ranjit Kumar Upadhyay

Abstract:

Excitable cells often produce different oscillatory activities that help us to understand the transmitting and processing of signals in the neural system. We consider a FitzHugh-Rinzel (FH-R) model and studied the different dynamics of the model by considering the parameter c as the predominant parameter. The model exhibits different types of neuronal responses such as regular spiking, mixed-mode bursting oscillations (MMBOs), elliptic bursting, etc. Based on the bifurcation diagram, we consider the three regimes (MMBOs, elliptic bursting, and quiescent state). An analytical treatment for the occurrence of the supercritical Hopf bifurcation is studied. Further, we extend our study to a network of a hundred neurons by considering the bi-directional synaptic coupling between them. In this article, we investigate the alternation of spiking propagation and bursting phenomena of an uncoupled and coupled FH-R neurons. We explore that the complete graph of heterogenous desynchronized neurons can exhibit different types of bursting oscillations for certain coupling strength. For higher coupling strength, all the neurons in the network show complete synchronization.

Keywords: excitable neuron model, spiking-bursting, stability and bifurcation, synchronization networks

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Authors: Paolo Russu

Abstract:

This paper investigates the economic and ecological dynamics that emerge in Protected Areas (PAs) as a result of interactions between visitors to the area and the animals that live there. We suppose that the PAs contain two species whose interactions are determined by the Lotka-Volterra equations system. Visitors' decisions to visit PAs are influenced by the entrance cost required to enter the park as well as the chance of witnessing the species that live there. Visitors have contradictory effects on the species and thus on the sustainability of the protected areas: on the one hand, an increase in the number of tourists damages the natural habitat of the areas and thus the species living there; on the other hand, it increases the total amount of entrance fees that the managing body of the PAs can use to perform defensive expenditures that protect the species from extinction. For a given set of parameter values, the existence of saddle-node bifurcation, Hopf bifurcation, homoclinic orbits, and a Bogdanov–Takens bifurcation of codimension two has been investigated. The system displays periodic doubling and chaotic solutions, as demonstrated by numerical examples. Pontryagin's Maximum Principle was utilized to develop an optimal admission charge policy that maximized both social gain and ecosystem conservation.

Keywords: environmental preferences, singularities point, dynamical system, chaos

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Authors: Amin Behradfar

Abstract:

‎Tourism has a direct impact on the national revenue for all touristic countries. It creates work opportunities‎, ‎industries‎, ‎and several investments to serve and raise nations performance and cultures. ‎This paper is devoted to analyze dynamical behaviour of a four-dimensional non-linear tourism-based social-ecological system by using the codimension two bifurcation theory‎. ‎In fact we investigate the cusp bifurcation of that‎. ‎Implications of our mathematical results to the tourism‎ ‎industry are discussed‎. Moreover, profitability‎, ‎compatibility and sustainability of the tourism system are shown by the aid of cusp bifurcation and numerical techniques‎.

Keywords: tourism-based social-ecological dynamical systems, cusp bifurcation, center manifold theory, profitability, ‎compatibility, sustainability

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Authors: Sarita Kumari, Ranjit Kumar Upadhyay

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This paper deals with plankton-fish dynamics where the fish population is growing logistically and nonlinearly harvested. The interaction between phytoplankton and zooplankton population is considered to be Crowley-Martin type functional response. It has been assumed that phytoplankton grows logistically and is affected by a space-dependent growth rate. Conditions for the existence of a positive equilibrium point and their stability analysis (both local and global) have been discussed for the non-spatial system. We have discussed maximum sustainable yields as well as optimal harvesting policy for maximizing the economic gain. The stability and existence of Hopf –bifurcation analysis have been discussed for the spatial system. Different conditions for turning pattern formation have been established through diffusion-driven instability analysis. Numerical simulations have been carried out for both non-spatial and spatial models. Phase plane analysis, the largest Lyapunov exponent, and bifurcation theory are used to numerically analyzed the non-spatial system. Our study shows that spatial heterogeneity, the mortality rate of phytoplankton, and constant harvesting of the fish population each play an important role in the dynamical behavior of the marine system.

Keywords: optimal harvesting, pattern formation, spatial heterogeneity, Crowley-Martin functional response

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Authors: Amit K. Singh, Subhankar Sen

Abstract:

The bifurcation curve of separation in steady two-dimensional viscous flow past an elliptic cylinder is studied by varying the angle of incidence (α) with different aspect ratio (ratio of minor to major axis). The solutions are based on numerical investigation, using finite element analysis, of the Navier-Stokes equations for incompressible flow. Results are presented for Reynolds number up to 50 and angle of incidence varies from 0° to 90°. Range of aspect ratio (Ar) is from 0.1 to 1 (in steps of 0.1) and flow is considered as unbounded flow. Bifurcation curve represents the locus of Reynolds numbers (Res) at which flow detaches or separates from the surface of the body at a given α and Ar. In earlier studies, effect of Ar on laminar separation curve or bifurcation curve is limited for Ar = 0.1, 0.2, 0.5 and 0.8. Some results are also available at α = 90° and 45°. The present study attempts to provide a systematic data and clear understanding on the effect of Ar at bifurcation curve and its point of maxima. In addition, issues regarding location of separation angle and maximum ratio of coefficient of lift to drag are studied. We found that nature of curve, separation angle and maximum ratio of lift to drag changes considerably with respect to change in Ar.

Keywords: aspect ratio, bifurcation curve, elliptic cylinder, GMRES, stabilized finite-element

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Authors: Leonard Mike McNelly Longwa, Divine Kusosa Musiku, D. Nahum Kabeya

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In this paper the bifurcation analysis of oilfield in Democratic Republic of Congo is presented in order to enhance petroleum production in an intense tectonic evolution characterized by distinct compressive and extensive phases and the digenetic transformation in the reservoirs during burial geological configuration. The use of porous media in Makelekese MS-25 field has been established to simulate the boundaries within 3 sedimentary basins open to exploration including the coastal basin with an area of 5992 km2, a central basin with an area of 800,000 km2, the western branch of the East African Rift in which there are 50,000 km2. The fractal characterization of complex hydro-dynamic fractures in oilfield is developed to facilitate oil production process based on reservoirs bifurcation model.

Keywords: reservoir bifurcation, fractal characterisation, permeability, conductivity, skin effect

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Authors: Leonard Mike McNelly Longwa, Divine Kusosa Musiku, Dieudonne Nahum Kabeya

Abstract:

In this paper, the bifurcation analysis of oilfields in the Democratic Republic of Congo is presented in order to enhance petroleum production in an intense tectonic evolution characterized by distinct compressive and extensive phases and the digenetic transformation in the reservoirs during burial geological configuration. The use of porous media in the Makelekese MS-25 field has been established to simulate the boundaries within 3 sedimentary basins open to exploration including the coastal basin with an area of 5992 km², a central basin with an area of 800,000 km², the western branch of the East African Rift in which there are 50,000 km². The fractal characterization of complex hydro-dynamic fractures in oilfields is developed to facilitate the oil production process based on the reservoirs bifurcation model.

Keywords: reservoir bifurcation, fractal characterization, permeability, conductivity, skin effect

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Authors: Lamia Bourdache, Amar Djema

Abstract:

The effect of non-Newtonian property (power law index n) on traveling waves of thin layer of power law fluid flowing over an inclined plane is investigated. For this, a simplified second-order two-equation model (SM) is used. The complete model is second-order four-equation (CM). It is derived by combining the weighted residual integral method and the lubrication theory. This is due to the fact that at the beginning of the instability waves, a very small number of waves is observed. Using a suitable set of test functions, second order terms are eliminated from the calculus so that the model is still accurate to the second order approximation. Linear, spatial, and temporal stabilities are studied. For travelling waves, a particular type of wave form that is steady in a moving frame, i.e., that travels at a constant celerity without changing its shape is studied. This type of solutions which are characterized by their celerity exists under suitable conditions, when the widening due to dispersion is balanced exactly by the narrowing effect due to the nonlinearity. Changing the parameter of celerity in some range allows exploring the entire spectrum of asymptotic behavior of these traveling waves. The (SM) model is converted into a three dimensional dynamical system. The result is that the model exhibits bifurcation scenarios such as heteroclinic, homoclinic, Hopf, and period-doubling bifurcations for different values of the power law index n. The influence of the non-Newtonian parameter on the nonlinear development of these travelling waves is discussed. It is found at the end that the qualitative characters of bifurcation scenarios are insensitive to the variation of the power law index.

Keywords: inclined plane, nonlinear stability, non-Newtonian, thin film

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Authors: Midiya Khademi, Ali Nikoo, Shabnam Rahimnezhad Baghche Jooghi

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Cardiovascular diseases (CVDs) are the main cause of death globally. Most CVDs can be prevented by avoiding habitual risk factors. Separate from the habitual risk factors, there are some inherent factors in each individual that can increase the risk potential of CVDs. Vessel shapes and geometry are influential factors, having great impact on the blood flow and the hemodynamic behavior of the vessels. In the present study, the influence of bifurcation angle on blood flow characteristics is studied. In order to approach this topic, by simplifying the details of the bifurcation, three models with angles 30°, 45°, and 60° were created, then by using CFD analysis, the response of these models for stable flow and pulsatile flow was studied. In the conducted simulation in order to eliminate the influence of other geometrical factors, only the angle of the bifurcation was changed and other parameters remained constant during the research. Simulations are conducted under dynamic and stable condition. In the stable flow simulation, a steady velocity of 0.17 m/s at the inlet plug was maintained and in dynamic simulations, a typical LAD flow waveform is implemented. The results show that the bifurcation angle has an influence on the maximum speed of the flow. In the stable flow condition, increasing the angle lead to decrease the maximum flow velocity. In the dynamic flow simulations, increasing the bifurcation angle lead to an increase in the maximum velocity. Since blood flow has pulsatile characteristics, using a uniform velocity during the simulations can lead to a discrepancy between the actual results and the calculated results.

Keywords: coronary artery, cardiovascular disease, bifurcation, atherosclerosis, CFD, artery wall shear stress

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Authors: Nabila Jaman, K. E. Hoque, S. Sawall, M. Ferdows

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Coronary artery disease (CAD) is one of the most lethal diseases of the cardiovascular diseases. Coronary arteries stenosis and bifurcation angles closely interact for myocardial infarction. We want to use computer-aided design model coupled with computational hemodynamics (CHD) simulation for detecting several types of coronary artery stenosis with different locations in an idealized model for identifying virtual fractional flow reserve (vFFR). The vFFR provides us the information about the severity of stenosis in the computational models. Another goal is that we want to imitate patient-specific computed tomography coronary artery angiography model for constructing our idealized models with different left anterior descending (LAD) and left circumflex (LCx) bifurcation angles. Further, we want to analyze whether the bifurcation angles has an impact on the creation of narrowness in coronary arteries or not. The numerical simulation provides the CHD parameters such as wall shear stress (WSS), velocity magnitude and pressure gradient (PGD) that allow us the information of stenosis condition in the computational domain.

Keywords: CAD, CHD, vFFR, bifurcation angles, coronary stenosis

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Authors: Paolo Russu

Abstract:

The purpose of the present paper is to highlight some features of an economic growth model with environmental negative externalities, giving rise to a three-dimensional dynamic system. In particular, we show that the economy, which is based on a Perfect-Substitution Technologies function of production, has no neither indeterminacy nor poverty trap. This implies that equilibrium select by economy depends on the history (initial values of state variable) of the economy rather than on expectations of economies agents. Moreover, by contrast, we prove that the basin of attraction of locally equilibrium points may be very large, as they can extend up to the boundary of the system phase space. The infinite-horizon optimal control problem has the purpose of maximizing the representative agent’s instantaneous utility function depending on leisure and consumption.

Keywords: Hopf bifurcation, open-access natural resources, optimal control, perfect-substitution technologies, Poincarè compactification

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Authors: S. M. Abdul Khader, Anurag Ayachit, Raghuvir Pai, K. A. Ahmed, V. R. K Rao, S. Ganesh Kamath

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The numerical simulation has made tremendous advances in investigating the blood flow phenomenon through elastic arteries. Such study can be useful in demonstrating the disease progression and haemodynamics of cardiovascular diseases such as atherosclerosis. In the present study, patient specific case diagnosed with partially stenosed complete right ICA and normal left carotid bifurcation without any atherosclerotic plaque formation is considered. 3D patient specific carotid bifurcation model is generated based on CT scan data using MIMICS-4.0 and numerical analysis is performed using FSI solver in ANSYS-14.5. The blood flow is assumed to be incompressible, homogenous and Newtonian, while the artery wall is assumed to be linearly elastic. The two-way sequentially-coupled transient FSI analysis is performed using FSI solver for three pulse cycles. The haemodynamic parameters such as flow pattern, Wall Shear Stress, pressure contours and arterial wall deformation are studied at the bifurcation and critical zones such as stenosis. The variation in flow behavior is studied throughout the pulse cycle. Also, the simulation results reveals that there is a considerable increase in the flow behavior in stenosed carotid in contrast to the normal carotid bifurcation system. The investigation also demonstrates the disturbed flow pattern especially at the bifurcation and stenosed zone elevating the haemodynamics, particularly during peak systole and later part of the pulse cycle. The results obtained agree well with the clinical observation and demonstrates the potential of patient specific numerical studies in prognosis of disease progression and plaque rupture.

Keywords: fluid-structure interaction, arterial stenosis, wall shear stress, carotid artery bifurcation

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Authors: Chethan Purushothama

Abstract:

Coronary artery disease is one of the major causes of death in developing countries. The coronary arteries show wide range of variations and these variations have not been dealt with different population groups. The present study aims to focus on the pattern and variations of coronary artery in south Indian population. The study was performed to analyze the coronary artery dominance and divisions of main trunk of left coronary artery in 81 isolated adult human cadaveric hearts of South Indian population. The specimens were fixed in 10% formalin and were dissected manually. In our specimens, 74.1% of the hearts were right dominant, 11.1% were left dominant, and 14.8% were co-dominant. Bifurcation, trifurcation, and quadrifurcation of main trunk of left coronary artery were seen in 49.4%, 48.1%, and 2.5% cases respectively. The right dominant hearts had bifurcation, trifurcation and quadrifurcation of main trunk of left coronary artery in 46.7%, 50% and 3.3% hearts respectively. The left dominant hearts had bifurcation and trifurcation of main trunk of left coronary artery in 55.6% and 44.4% cases respectively. The co-dominant hearts had bifurcation and trifurcation of main trunk of left coronary artery in 58.3% and 41.7% respectively. Quadrifurcation of main trunk of left coronary artery were seen only in right dominant hearts. We believe that the data obtained from the present study are important to the interventional cardiologists and radiologists. The details obtained will also be helpful for the clinical anatomists.

Keywords: bifurcation, coronary artery, trifurcation, quadrifurcation

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Authors: C. E. Madubueze, S. C. Madubueze, S. Ajama

Abstract:

Cholera is a disease that is predominately common in developing countries due to poor sanitation and overcrowding population. In this paper, a deterministic model for the dynamics of cholera is developed and control measures such as health educational message, therapeutic treatment, and vaccination are incorporated in the model. The effective reproduction number is computed in terms of the model parameters. The existence and stability of the equilibrium states, disease free and endemic equilibrium states are established and showed to be locally and globally asymptotically stable when R0 < 1 and R0 > 1 respectively. The existence of backward bifurcation of the model is investigated. Furthermore, numerical simulation of the model developed is carried out to show the impact of the control measures and the result indicates that combined control measures will help to reduce the spread of cholera in the population

Keywords: backward bifurcation, cholera, equilibrium, dynamics, stability

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Authors: Sanjeet Patra, Soham Roychowdhury

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In recent times, functionally graded dielectric elastomer (FGDE) has gained significant attention within the realm of soft actuation due to its dual capacity to exert highly localized stresses while maintaining its compliant characteristics on application of electro-mechanical loading. Nevertheless, the full potential of dielectric elastomer (DE) has not been fully explored due to their susceptibility to instabilities when subjected to electro-mechanical loads. As a result, study and analysis of such instabilities becomes crucial for the design and realization of dielectric actuators. Prismatic bifurcation is a type of instability that has been recognized in a DE tube. Though several studies have reported on the analysis for prismatic bifurcation in an isotropic DE tube, there is an insufficiency in studies related to prismatic bifurcation of FGDE tubes. Therefore, this paper aims to determine the onset of prismatic bifurcations on an incompressible FGDE tube when subjected to electrical loading across the thickness of the tube and internal pressurization. The analysis has been conducted by imposing two axial boundary conditions on the tube, specifically axially free ends and axially clamped ends. Additionally, the rigidity modulus of the tube has been linearly graded in the direction of thickness where the inner surface of the tube has a lower stiffness than the outer surface. The static equilibrium equations for deformation of the axisymmetric tube are derived and solved using numerical technique. The condition for prismatic bifurcation of the axisymmetric static equilibrium solutions has been obtained by using the linearized incremental constitutive equations. Two modes of bifurcations, corresponding to two different non-circular cross-sectional geometries, have been explored in this study. The outcomes reveal that the FGDE tubes experiences prismatic bifurcation before the Hessian criterion of failure is satisfied. It is observed that the lower mode of bifurcation can be triggered at a lower critical voltage as compared to the higher mode of bifurcation. Furthermore, the tubes with larger stiffness gradient require higher critical voltages for triggering the bifurcation. Moreover, with the increase in stiffness gradient, a linear variation of the critical voltage is observed with the thickness of the tube. It has been found that on applying internal pressure to a tube with low thickness, the tube becomes less susceptible to bifurcations. A thicker tube with axially free end is found to be more stable than the axially clamped end tube at higher mode of bifurcation.

Keywords: critical voltage, functionally graded dielectric elastomer, linearized incremental approach, modulus of rigidity, prismatic bifurcation

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Authors: Lana Horvat Dmitrović

Abstract:

The main purpose of this paper is to study the box dimension as fractal property of bifurcations of discrete dynamical systems in higher dimensions. The paper contains the fractal analysis of the orbits near the hyperbolic and non-hyperbolic fixed points in discrete dynamical systems. It is already known that in one-dimensional case the orbit near the hyperbolic fixed point has the box dimension equal to zero. On the other hand, the orbit near the non-hyperbolic fixed point has strictly positive box dimension which is connected to the non-degeneracy condition of certain bifurcation. One of the main results in this paper is the generalisation of results about box dimension near the hyperbolic and non-hyperbolic fixed points to higher dimensions. In the process of determining box dimension, the restriction of systems to stable, unstable and center manifolds, Lipschitz property of box dimension and the notion of projective box dimension are used. The analysis of the bifurcations in higher dimensions with one multiplier on the unit circle is done by using the normal forms on one-dimensional center manifolds. This specific change in box dimension of an orbit at the moment of bifurcation has already been explored for some bifurcations in one and two dimensions. It was shown that specific values of box dimension are connected to appropriate bifurcations such as fold, flip, cusp or Neimark-Sacker bifurcation. This paper further explores this connection of box dimension as fractal property to some specific bifurcations in higher dimensions, such as fold-flip and flip-Neimark-Sacker. Furthermore, the application of the results to the unit time map of continuous dynamical system near hyperbolic and non-hyperbolic singularities is presented. In that way, box dimensions which are specific for certain bifurcations of continuous systems can be obtained. The approach to bifurcation analysis by using the box dimension as specific fractal property of orbits can lead to better understanding of bifurcation phenomenon. It could also be useful in detecting the existence or nonexistence of bifurcations of discrete and continuous dynamical systems.

Keywords: bifurcation, box dimension, invariant manifold, orbit near fixed point

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Authors: Piotr Muzyk, Beata Morawiec, Mariusz Opara, Andrzej Tomasik, Brygida Przywara-Chowaniec, Wojciech Jachec, Ewa Nowalany-Kozielska, Damian Kawecki

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Background: Coronary bifurcation is one of the most complex lesion in patients with coronary ar-tery disease. Provisional T-stenting is currently one of the recommended techniques. The aim was to assess optimal methods of treatment in the era of drug-eluting stents (DES). Methods: The regis-try consisted of data from 1916 patients treated with coronary percutaneous interventions (PCI) using either first- or second-generation DES. Patients with bifurcation lesion entered the analysis. Major adverse cardiac and cardiovascular events (MACCE) were assessed at one year of follow-up and comprised of death, acute myocardial infarction (AMI), repeated PCI (re-PCI) of target ves-sel and stroke. Results: Of 1916 registry patients, 204 patients (11%) were diagnosed with bifurcation lesion >50% and entered the analysis. The most commonly used technique was provi-sional T-stenting (141 patients, 69%). Optimization with kissing-balloons technique was performed in 45 patients (22%). In 59 patients (29%) second-generation DES was implanted, while in 112 pa-tients (55%), first-generation DES was used. In 33 patients (16%) both types of DES were used. The procedure success rate (TIMI 3 flow) was achieved in 98% of patients. In one-year follow-up, there were 39 MACCE (19%) (9 deaths, 17 AMI, 16 re-PCI and 5 strokes). Provisional T-stenting resulted in similar rate of MACCE to other techniques (16% vs. 5%, p=0.27) and similar occurrence of re-PCI (6% vs. 2%, p=0.78). The results of post-PCI kissing-balloon technique gave equal out-comes with 3% vs. 16% of MACCE in patients in whom no optimization technique was used (p=0.39). The type of implanted DES (second- vs. first-generation) had no influence on MACCE (4% vs 14%, respectively, p=0.12) and re-PCI (1.7% vs. 51% patients, respectively, p=0.28). Con-clusions: The treatment of bifurcation lesions with PCI represent high-risk procedures with high rate of MACCE. Stenting technique, optimization of PCI and the generation of implanted stent should be personalized for each case to balance risk of the procedure. In this setting, the operator experience might be the factor of better outcome, which should be further investigated.

Keywords: coronary bifurcation, drug eluting stents, long-term follow-up, percutaneous coronary interventions

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Authors: Artem Nuriev, Olga Zaitseva

Abstract:

This paper is devoted to the study of a viscous incompressible flow around a circular cylinder performing harmonic oscillations, especially the steady streaming phenomenon. The research methodology is based on the asymptotic explanation method combined with the computational bifurcation analysis. Present studies allow to identify several regimes of the secondary streaming with different flow structures. The results of the research are in good agreement with experimental and numerical simulation data.

Keywords: oscillating cylinder, secondary streaming, flow regimes, asymptotic and bifurcation analysis

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Authors: Doo Byong Bae, Jae Jun Yoo, Il Gyu Park, Choi Seowon, Oh Chang Kook

Abstract:

There are several wind load provisions to evaluate the wind response on tank structures such as API, Euro-code, etc. the assessment of wind action applying these provisions is made by performing the finite element analysis using both linear bifurcation analysis and geometrically nonlinear analysis. By comparing the pressure patterns obtained from the analysis with the results of wind tunnel test, most appropriate wind load criteria will be recommended.

Keywords: wind load, finite element analysis, linear bifurcation analysis, geometrically nonlinear analysis

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Authors: Wang Zhulin, Min Fuhong, Peng Guangya, Wang Yaoda, Cao Yi

Abstract:

In this paper, a magnetron memristor model based on hyperbolic sine function is presented and the correctness proved by studying the trajectory of its voltage and current phase, and then a memristor chaotic system with the memristor model is presented. The phase trajectories and the bifurcation diagrams and Lyapunov exponent spectrum of the magnetron memristor system are plotted by numerical simulation, and the chaotic evolution with changing the parameters of the system is also given. The paper includes numerical simulations and mathematical model, which confirming that the system, has a wealth of dynamic behavior.

Keywords: memristor, chaotic circuit, dynamical behavior, chaotic system

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