Journal articles: 'Centre manifold theory'

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Rendall, Alan D. "Cosmological Models and Centre Manifold Theory." General Relativity and Gravitation 34, no. 8 (August 2002): 1277–94. http://dx.doi.org/10.1023/a:1019734703162.

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ZHANG, CHUNRUI, and BAODONG ZHENG. "CODIMENSION ONE BIFURCATION OF EQUIVARIANT NEURAL NETWORK MODEL WITH DELAY." International Journal of Bifurcation and Chaos 20, no. 04 (April 2010): 1255–59. http://dx.doi.org/10.1142/s0218127410026459.

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In this paper, we consider double zero singularity of a symmetric BAM neural network model with a time delay. Based on the normal form approach and the center manifold theory, we obtain the normal form on the centre manifold with double zero singularity. Some numerical simulations support our analysis results.

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Lin, Xiaodong, Joseph W. H. So, and Jianhong Wu. "Centre manifolds for partial differential equations with delays." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 122, no. 3-4 (1992): 237–54. http://dx.doi.org/10.1017/s0308210500021090.

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SynopsisA centre manifold theory for reaction-diffusion equations with temporal delays is developed. Besides an existence proof, we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order. As an illustration, this reduction procedure is applied to the Hutchinson equation with diffusion.

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LIU, L., Y. S. WONG, and B. H. K. LEE. "APPLICATION OF THE CENTRE MANIFOLD THEORY IN NON-LINEAR AEROELASTICITY." Journal of Sound and Vibration 234, no. 4 (July 2000): 641–59. http://dx.doi.org/10.1006/jsvi.1999.2895.

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WANG KAI-GE, WANG YU-LONG, and SUN YIN-GUAN. "APPLICATION OF CENTRE MANIFOLD THEORY IN GENERALIZED MAXWELL-BLOCH LASER EQUATIONS." Acta Physica Sinica 45, no. 1 (1996): 46. http://dx.doi.org/10.7498/aps.45.46.

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Psarros, N., G. Papaschinopoulos, and C. J. Schinas. "Semistability of two systems of difference equations using centre manifold theory." Mathematical Methods in the Applied Sciences 39, no. 18 (March 6, 2016): 5216–22. http://dx.doi.org/10.1002/mma.3904.

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Liu, Wei, and Yaolin Jiang. "Dynamics of a Modified Predator-Prey System to allow for a Functional Response and Time Delay." East Asian Journal on Applied Mathematics 6, no. 4 (October 19, 2016): 384–99. http://dx.doi.org/10.4208/eajam.141214.050616a.

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AbstractA modified predator-prey system described by two differential equations and an algebraic equation is discussed. Formulae for determining the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions are derived differential-algebraic system theory, bifurcation theory and centre manifold theory. Numerical simulations illustrate the results, which includes quite complex dynamical behaviour.

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Valls, Claudia. "Stability of some solutions for elliptic equations on a cylindrical domain." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2109 (June 10, 2009): 2647–62. http://dx.doi.org/10.1098/rspa.2009.0110.

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We study analytically a class of solutions for the elliptic equation where α >0 and ε is a small parameter. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every α >0, it contains solutions that are defined for large values of time and they are very close (of order O ( ε )) to a linear torus for long times (of order O ( ε −1 )). The proof uses the fact that the equation leaves invariant a smooth centre manifold and, for the restriction of the system to the centre manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

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JI, J. C., X. Y. LI, Z. LUO, and N. ZHANG. "TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY." International Journal of Bifurcation and Chaos 22, no. 03 (March 2012): 1250060. http://dx.doi.org/10.1142/s0218127412500605.

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The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.

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Mielke, Alexander. "On Saint-Venant's problem for an elastic strip." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 110, no. 1-2 (1988): 161–81. http://dx.doi.org/10.1017/s0308210500024938.

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SynopsisThe equilibrium equations for elastic deformations of an infinite strip are considered. Under the assumption of sufficiently small strains along the whole body, it is shown that all solutions lie on a six-dimensional manifold. This is achieved by rewriting the field equations as a differential equation in a function spaceover the cross-section, the axial variable taken as time. Then the theory of centre manifolds for elliptic systems applies. Thus the local Saint-Venant's problem is solved. Moreover, the structure of the finite-dimensional solution space is analysed to reveal exactly the two-dimensional rod equations of Kirchhoff. The constitutive relations for this rod model are calculated in a mathematically rigorous way out of the constitutive law of the material forming the strip.

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Groves, M. D. "An Existence Theory for Gravity–Capillary Solitary Water Waves." Water Waves 3, no. 1 (February 16, 2021): 213–50. http://dx.doi.org/10.1007/s42286-020-00045-7.

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AbstractIn the applied mathematics literature solitary gravity–capillary water waves are modelled by approximating the standard governing equations for water waves by a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). These formal arguments have been justified by sophisticated techniques such as spatial dynamics and centre-manifold reduction methods on the one hand and variational methods on the other. This article presents a complete, self-contained account of an alternative, simpler approach in which one works directly with the Zakharov–Craig–Sulem formulation of the water-wave problem and uses only rudimentary fixed-point arguments and Fourier analysis.

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Rocha, Filipe, Maíra Aguiar, Max Souza, and Nico Stollenwerk. "Time-scale separation and centre manifold analysis describing vector-borne disease dynamics." International Journal of Computer Mathematics 90, no. 10 (October 2013): 2105–25. http://dx.doi.org/10.1080/00207160.2013.783208.

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Roberts, A. J. "Boundary conditions for approximate differential equations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 34, no. 1 (July 1992): 54–80. http://dx.doi.org/10.1017/s0334270000007384.

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AbstractA large number of mathematical models are expressed as differential equations. Such models are often derived through a slowly-varying approximation under the assumption that the domain of interest is arbitrarily large; however, typical solutions and the physical problem of interest possess finite domains. The issue is: what are the correct boundary conditions to be used at the edge of the domain for such model equations? Centre manifold theory [24] and its generalisations may be used to derive these sorts of approximations, and higher-order refinements, in an appealing and systematic fashion. Furthermore, the centre manifold approach permits the derivation of appropriate initial conditions and forcing for the models [25, 7]. Here I show how to derive asymptotically-correct boundary conditions for models which are based on the slowly-varying approximation. The dominant terms in the boundary conditions typically agree with those obtained through physical arguments. However, refined models of higher order require subtle corrections to the previously-deduced boundary conditions, and also require the provision of additional boundary conditions to form a complete model.

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Roberts, A. J. "The application of centre-manifold theory to the evolution of system which vary slowly in space." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 29, no. 4 (April 1988): 480–500. http://dx.doi.org/10.1017/s0334270000005968.

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AbstractIn many physical problems, the system tends quickly to a particular structure, which then evolves relatively slowly in space and time. Various methods exist to derive equations describing the slow evolution of the particular structure; for example, the method of multiple scales. However, the resulting equations are typically valid only for a limited range of the parameters. In order to extend the range of validity and to improve the accuracy, correction terms must be found for the equations. Here we describe a procedure, inspired by centre-manifold theory, which provides a systematic approach to calculating a sequence of successively more accurate approximations to the evolution of the principal structure in space and time.The formal procedure described here raises a number of questions for future research. For example: what sort of error bounds can be obtained, do the approximations converge or are they strictly asymptotic, and what sort of boundary conditions are appropriate in a given problem?

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Psarros, N., G. Papaschinopoulos, and C. J. Schinas. "Study of the stability of a 3×3 system of difference equations using Centre Manifold Theory." Applied Mathematics Letters 64 (February 2017): 185–92. http://dx.doi.org/10.1016/j.aml.2016.09.002.

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Xue, Wei, and Yan Ling Guo. "Chaos Analysis and Control of Permanent Magnet Synchronous Motors." Advanced Materials Research 219-220 (March 2011): 88–92. http://dx.doi.org/10.4028/www.scientific.net/amr.219-220.88.

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As a nonlinear dynamic system, the permanent magnet synchronous motor (PMSM) can exhibit prominent chaotic characteristics under some choices of system parameters. The existence of chaotic attractor of the PMSM is verified through the centre manifold theory and Poincaré section. Chaotic phenomenon affects the normal operation of motor. In this paper, it makes the PMSM in a stable state to control chaos of the PMSM with a control strategy of delay feedback, which can eliminate chaos well.

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ÇELİK, CANAN, and GÖKÇEN ÇEKİÇ. "BIFURCATION ANALYSIS OF A LOGISTIC PREDATOR–PREY SYSTEM WITH DELAY." ANZIAM Journal 57, no. 4 (April 2016): 445–60. http://dx.doi.org/10.1017/s1446181116000055.

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We consider a coupled, logistic predator–prey system with delay. Mainly, by choosing the delay time${\it\tau}$as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time${\it\tau}$passes some critical values. Based on the normal-form theory and the centre manifold theorem, we also derive formulae to obtain the direction, stability and the period of the bifurcating periodic solution at critical values of ${\it\tau}$. Finally, numerical simulations are investigated to support our theoretical results.

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Sinou, J. J., F. Thouverez, and L. Jezequel. "Analysis of friction and instability by the centre manifold theory for a non-linear sprag-slip model." Journal of Sound and Vibration 265, no. 3 (August 2003): 527–59. http://dx.doi.org/10.1016/s0022-460x(02)01453-0.

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KARAOGLU, E., and H. MERDAN. "HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS." ANZIAM Journal 55, no. 3 (January 2014): 214–31. http://dx.doi.org/10.1017/s1446181114000054.

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AbstractThe aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

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Zhou, Yi, Zheng Fei Wu, and Yu Hong Huo. "Dynamical Analysis of a Calcium Oscillation Model in Non-Excitable Cells." Applied Mechanics and Materials 226-228 (November 2012): 521–25. http://dx.doi.org/10.4028/www.scientific.net/amm.226-228.521.

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The Borghans-Dupont model of calcium oscillations based on both the calcium-induced calcium release and calcium-activated inositol trisphosphate concentration degradation is considered. Dynamical effect of the stimulation level on the calcium oscillation behavior is studied. The qualitative theory of differential equations is used to explain the mechanism of these oscillations. We investigate the existence, types, stability and bifurcations of the equilibria by applying the centre manifold theorem, stability theory and bifurcation theory and prove that oscillations are due to supercritical Hopf bifurcation. Finally, we perform numerical simulations, including time courses, phase portraits and bifurcation diagram, to validate the correctness and the effectiveness of our theoretical analysis. These results may be instructive for understanding the role of the stimulation level played in complex dynamics in this model.

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Wei, Junjie, and Chunbo Yu. "Hopf bifurcation analysis in a model of oscillatory gene expression with delay." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 4 (July 8, 2009): 879–95. http://dx.doi.org/10.1017/s0308210507000091.

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The dynamics of a gene expression model with time delay are investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions has been derived by using the theory of the centre manifold and the normal forms method. The global existence of periodic solutions has been established using a global Hopf bifurcation result by Wu and a Bendixson criterion for higher-dimensional ordinary differential equations due to Li and Muldowney.

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Yamgoué, Serge Bruno, and Timoléon Crépin Kofané. "Transient Chaos in Coupled Oscillators with Shape Deformable Potential." International Journal of Bifurcation and Chaos 13, no. 06 (June 2003): 1459–74. http://dx.doi.org/10.1142/s0218127403007333.

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The dynamics of a perturbed system consisting of a particle embedded in a strongly nonlinear potential V(ϕ; r, η) whose shape can be varied continuously as a function of r in the range -1 < r < 1, and coupled to an harmonic oscillator is analyzed. The perturbations are made of the coupling and damping forces. When they are removed, the subsystem formed by the anharmonic oscillator contains saddle points connected by homoclinic loops. Thus, the whole unperturbed system has saddle-centre points whose stable and unstable manifolds coincide in three-dimensional manifold. We concentrate our analysis of the perturbed system in the regions near these saddle-centre points. First, the Melnikov theory is used to investigate the presence of horseshoes chaos in the dynamics. Due to some discrepancies of the Melnikov theory, we next rederive the boundary between the regions of regular and irregular motions in the parameters space by using regular perturbation expansion. It is found here that for small values of the natural pulsation ω of the linear oscillator, complicated behavior intensifies in the system as the absolute value of the shape parameter r approaches zero. On the contrary, chaotic motion intensifies according as r decreases from values close to 1 to values close to -1, for large ω. The numerical analysis, which includes computation of maximal Lyapunov exponent, bifurcation diagrams and Poincaré sections, shows that the system exhibits transient stochastic behavior, but ultimately settles down on a simple set which is either a fixed point or a limit cycle. The dependence of this transient behavior on the system parameters agrees qualitatively well with the analytical predictions.

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HU, RUI, and YUAN YUAN. "Stability and Hopf bifurcation analysis for Nicholson's blowflies equation with non-local delay." European Journal of Applied Mathematics 23, no. 6 (August 10, 2012): 777–96. http://dx.doi.org/10.1017/s0956792512000265.

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We consider a diffusive Nicholson's blowflies equation with non-local delay and study the stability of the uniform steady states and the possible Hopf bifurcation. By using the upper- and lower solutions method, the global stability of constant steady states is obtained. We also discuss the local stability via analysis of the characteristic equation. Moreover, for a special kernel, the occurrence of Hopf bifurcation near the steady state solution and the stability of bifurcated periodic solutions are given via the centre manifold theory. Based on laboratory data and our theoretical results, we address the influence of various types of vaccinations in controlling the outbreak of blowflies.

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Zhang, Wenqi, Dan Jin, and Ruizhi Yang. "Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey." Mathematics 11, no. 3 (January 20, 2023): 556. http://dx.doi.org/10.3390/math11030556.

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In this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property of Hopf bifurcation is investigated by the theory of the centre manifold and normal form method. Through the numerical simulations, it is observed that the anti-predator behaviour parameter η, the memory-based diffusion coefficient parameter d, and memory delay τ can affect the stability of the coexisting equilibrium under some parameters and cause the spatially inhomogeneous oscillation of prey and predator’s densities.

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Daogao, Wei, Jiang Tong, Chen Changhe, Jiang Yibin, Pan Ning, and Pan Zhijie. "Hopf bifurcation character of an interactive vehicle–road shimmy system under bisectional road conditions." Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 231, no. 3 (August 5, 2016): 405–17. http://dx.doi.org/10.1177/0954407016640874.

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The adhesion coefficients of a bisectional road have significant coupled influences on the shimmy characteristics of the front wheels of a vehicle. A four-degree-of-freedom model for a representative sport utility vehicle was established. This model considered the adhesion coefficients of a bisectional road and the friction of the steering system of the suspension. The existence and stability of the system’s limit cycles were qualitatively determined using the Hopf bifurcation theorem and the centre manifold theory based on the model. The influences of the adhesion coefficients on the Hopf bifurcation characteristics of the system were calculated using a numerical method. The results showed that the road adhesion coefficient μ1 of the left front wheel and the road coefficient μ2 of the right front wheel significantly affected the vehicle shimmy and coupling relationship when different. Keeping μ1 at a certain value, the swing angles and the angular velocities of the two front wheels consistently decreased when μ2 decreased. The phenomenon repeatedly occurred when the difference Δ μ between the adhesion coefficients increased. Moreover, the discrepancy between the amplitude of the left front wheel and the amplitude of the right front wheel is much more apparent when both the adhesion coefficients are larger. Good agreement between the shimmy characteristics of the two wheels was also found when comparing the results using the centre manifold reduced-dimensions method with the numerical method. Furthermore, a higher reduced order caused the reduced system to be closer to the original system.

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Smoller, Joel, Blake Temple, and Zeke Vogler. "An instability of the standard model of cosmology creates the anomalous acceleration without dark energy." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2207 (November 2017): 20160887. http://dx.doi.org/10.1098/rspa.2016.0887.

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We identify the condition for smoothness at the centre of spherically symmetric solutions of Einstein’s original equations without the cosmological constant or dark energy. We use this to derive a universal phase portrait which describes general, smooth, spherically symmetric solutions near the centre of symmetry when the pressure p =0. In this phase portrait, the critical k =0 Friedmann space–time appears as a saddle rest point which is unstable to spherical perturbations. This raises the question as to whether the Friedmann space–time is observable by redshift versus luminosity measurements looking outwards from any point. The unstable manifold of the saddle rest point corresponding to Friedmann describes the evolution of local uniformly expanding space–times whose accelerations closely mimic the effects of dark energy. A unique simple wave perturbation from the radiation epoch is shown to trigger the instability, match the accelerations of dark energy up to second order and distinguish the theory from dark energy at third order. In this sense, anomalous accelerations are not only consistent with Einstein’s original theory of general relativity, but are a prediction of it without the cosmological constant or dark energy.

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Roberts, A. J. "Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations." LMS Journal of Computation and Mathematics 9 (2006): 193–221. http://dx.doi.org/10.1112/s146115700000125x.

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AbstractConstructing numerical models of noisy partial differential equations is a very delicate task. Our long-term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a second step, we consider here a small domain, representing a finite element, and derive a one-degree-of-freedom model for the dynamics in the element; stochastic centre manifold theory supports the model. The approach automatically parametrises the microscale structures induced by spatially varying stochastic noise within the element. The crucial aspect of this work is that we explore how a multitude of microscale noise processes may interact in nonlinear dynamical systems. The analysis finds that noise processes with coarse structure across a finite element are the significant noises for the modelling. Further, the nonlinear dynamics abstracts effectively new noise sources over the macroscale time-scales resolved by the model.

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Zuo, Hong Kun, Quan Bao Ji, and Yi Zhou. "Hopf Bifurcation and Numerical Simulation in a Calcium Oscillation Model." Applied Mechanics and Materials 226-228 (November 2012): 510–15. http://dx.doi.org/10.4028/www.scientific.net/amm.226-228.510.

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Calcium oscillations play a very important role in providing the intracellular signaling, and many mathematical models have been proposed to describe calcium oscillations. The Shen-Larter model presented here is based on calcium-induced calcium release (CICR) and the inositol trisphosphate cross-coupling (ICC). Nonlinear dynamics of this model is investigated by using the centre manifold theorem and bifurcation theory, including the variation in classification and stability of equilibria with different parameter values. The results show that the appearance and disappearance of calcium oscillations are due to subcritical Hopf bifurcation of equilibria. The numerical simulations are performed in order to illustrate the correctness of our theoretical analysis, including the bifurcation diagram of fixed points, the phase diagram of the system in two dimensional space and time series.

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Sandstede, B., and A. Scheel. "Essential instability of pulses and bifurcations to modulated travelling waves." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 6 (1999): 1263–90. http://dx.doi.org/10.1017/s0308210500019387.

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Reaction-diffusion systems on the real line are considered. Localized travelling waves become unstable when the essential spectrum of the linearization about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a family of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcating patterns can be parametrized by the wavelength of the steady patterns; they are time-periodic in a moving frame. A major difficulty in analysing this bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov–Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touching the imaginary axis.

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BHUNU, C. P., J. M. TCHUENCHE, W. GARIRA, G. MAGOMBEDZE, and S. MUSHAYABASA. "MODELING THE EFFECTS OF SCHISTOSOMIASIS ON THE TRANSMISSION DYNAMICS OF HIV/AIDS." Journal of Biological Systems 18, no. 02 (June 2010): 277–97. http://dx.doi.org/10.1142/s0218339010003196.

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A schistosomiasis and HIV/AIDS co-infection model is presented as a system of nonlinear ordinary differential equations. Qualitative analysis (properties) of the model are presented. The disease-free equilibrium is shown to be locally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The Centre Manifold theory is used to show that the schistosomiasis only and HIV/AIDS only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. The model is numerically analyzed to assess the effects of schistosomiasis on the dynamics of HIV/AIDS. Analysis of the reproduction numbers and numerical simulations show that an increase of schistosomiasis cases result in an increase of HIV/AIDS cases, suggesting that schistosomiasis control have a positive impact in controlling the transmission dynamics of HIV/AIDS.

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Magal, Pierre, and Shigui Ruan. "Sustained oscillations in an evolutionary epidemiological model of influenza A drift." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2116 (December 3, 2009): 965–92. http://dx.doi.org/10.1098/rspa.2009.0435.

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Understanding the seasonal/periodic reoccurrence of influenza will be very helpful in designing successful vaccine programmes and introducing public-health interventions. However, the reasons for seasonal/periodic influenza epidemics are still not clear, even though various explanations have been proposed. In this paper, we study an age-structured type evolutionary epidemiological model of influenza A drift, in which the susceptible class is continually replenished because the pathogen changes genetically and immunologically from one epidemic to the next, causing previously immune hosts to become susceptible. Applying our recently established centre manifold theory for semi-linear equations with non-dense domain, we show that Hopf bifurcation occurs in the model. This demonstrates that the age-structured type evolutionary epidemiological model of influenza A drift has an intrinsic tendency to oscillate owing to the evolutionary and/or immunological changes of the influenza viruses.

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Groves, M. D., and A. Mielke. "A spatial dynamics approach to three-dimensional gravity-capillary steady water waves." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 1 (February 2001): 83–136. http://dx.doi.org/10.1017/s0308210500000809.

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This paper contains a rigorous existence theory for three-dimensional steady gravity-capillary finite-depth water waves which are uniformly translating in one horizontal spatial direction x and periodic in the transverse direction z. Physically motivated arguments are used to find a formulation of the problem as an infinite-dimensional Hamiltonian system in which x is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system. General statements concerning the existence of waves which are periodic or quasiperiodic in x (and periodic in z) are made by applying standard tools in Hamiltonian-systems theory to the reduced equations.A critical curve in Bond number–Froude number parameter space is identified which is associated with bifurcations of generalized solitary waves. These waves are three dimensional but decay to two-dimensional periodic waves (small-amplitude Stokes waves) far upstream and downstream. Their existence as solutions of the water-wave problem confirms previous predictions made on the basis of model equations.

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Mushayabasa, S., and C. P. Bhunu. "Modeling Schistosomiasis and HIV/AIDS Codynamics." Computational and Mathematical Methods in Medicine 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/846174.

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We formulate a mathematical model for the cointeraction of schistosomiasis and HIV/AIDS in order to assess their synergistic relationship in the presence of therapeutic measures. Comprehensive mathematical techniques are used to analyze the model steady states. The disease-free equilibrium is shown to be locally asymptotically stable when the associated disease threshold parameter known as the basic reproduction number for the model is less than unity. Centre manifold theory is used to show that the schistosomiasis-only and HIV/AIDS-only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. The impact of schistosomiasis and its treatment on the dynamics of HIV/AIDS is also investigated. To illustrate the analytical results, numerical simulations using a set of reasonable parameter values are provided, and the results suggest that schistosomiasis treatment will always have a positive impact on the control of HIV/AIDS.

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Bhunu, C. P., and S. Mushayabasa. "Impact of Intravenous Drug Use on HIV/AIDS among Women Prisoners: A Mathematical Modelling Approach." ISRN Computational Biology 2013 (December 12, 2013): 1–8. http://dx.doi.org/10.1155/2013/718039.

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Intravenous drug use and tattooing remain one of the major routes of HIV/AIDS transmission among prisoners. We formulate and analyze a deterministic model for the role of intravenous drug use in HIV/AIDS transmission among women prisoners. With the aid of the Centre Manifold theory, the endemic equilibrium is shown to be locally asymptotically stable when the corresponding reproduction number is greater than unity. Analysis of the reproduction number and numerical simulations suggest that an increase in intravenous drug use among women prisoners as they fail to cope with prison settings fuels the HIV/AIDS epidemic in women prisoners. Failure to control HIV/AIDS among female prisoners may be a time bomb to their communities upon their release. Thus, it may be best to consider free needle/syringe exchange and drug substitution treatment programmes in women prisons as well as considering open prison systems for less serious crimes.

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MOUALEU, D. P., J. MBANG, R. NDOUNDAM, and S. BOWONG. "MODELING AND ANALYSIS OF HIV AND HEPATITIS C CO-INFECTIONS." Journal of Biological Systems 19, no. 04 (December 2011): 683–723. http://dx.doi.org/10.1142/s0218339011004159.

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Infection with the hepatitis C virus (HCV) is the most common coinfection in people with the human immunodeficiency virus (HIV), and hepatitis C is categorized as an HIV-related illness. The study of the joint dynamics of HIV and HCV present formidable mathematical challenges in spite the fact that they share similar routes of transmission. A deterministic model for the co-interaction of HCV and HIV in a community is presented and rigorously analyzed. The disease-free equilibrium is shown to be locally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than the unity. The Centre Manifold theory is used to show that the HCV only and HIV/AIDS only endemic equilibria are locally asymptotically stable when their associated reproduction numbers are greater than the unity. We compute two coexistence thresholds for the stability of boundary equilibria. Numerical results are presented to validate analytical results.

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MUSHAYABASA, S., and C. P. BHUNU. "MODELING THE IMPACT OF VOLUNTARY TESTING AND TREATMENT ON TUBERCULOSIS TRANSMISSION DYNAMICS." International Journal of Biomathematics 05, no. 04 (May 16, 2012): 1250029. http://dx.doi.org/10.1142/s1793524511001726.

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A deterministic model for evaluating the impact of voluntary testing and treatment on the transmission dynamics of tuberculosis is formulated and analyzed. The epidemiological threshold, known as the reproduction number is derived and qualitatively used to investigate the existence and stability of the associated equilibrium of the model system. The disease-free equilibrium is shown to be locally-asymptotically stable when the reproductive number is less than unity, and unstable if this threshold parameter exceeds unity. It is shown, using the Centre Manifold theory, that the model undergoes the phenomenon of backward bifurcation where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. The analysis of the reproduction number suggests that voluntary tuberculosis testing and treatment may lead to effective control of tuberculosis. Furthermore, numerical simulations support the fact that an increase voluntary tuberculosis testing and treatment have a positive impact in controlling the spread of tuberculosis in the community.

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Bona, J. L., F. Demengel, and K. Promislow. "Fourier splitting and dissipation of nonlinear dispersive waves." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 3 (1999): 477–502. http://dx.doi.org/10.1017/s0308210500021478.

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Presented herein is a new method for analysing the long-time behaviour of solutions of nonlinear, dispersive, dissipative wave equations. The method is applied to the generalized Korteweg–de Vries equation posed on the entire real axis, with a homogeneous dissipative mechanism included. Solutions of such equations that commence with finite energy decay to zero as time becomes unboundedly large. In circumstances to be spelled out presently, we establish the existence of a universal asymptotic structure that governs the final stages of decay of solutions. The method entails a splitting of Fourier modes into long and short wavelengths which permits the exploitation of the Hamiltonian structure of the equation obtained by ignoring dissipation. We also develop a helpful enhancement of Schwartz's inequality. This approach applies particularly well to cases where the damping increases in strength sublinearly with wavenumber. Thus the present theory complements earlier work using centre-manifold and group-renormalization ideas to tackle the situation wherein the nonlinearity is quasilinear with regard to the dissipative mechanism.

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Lichau, Karsten. "Soundproof Silences? Towards a Sound History of Silence." International Journal for History, Culture and Modernity 7, no. 1 (November 2, 2019): 840–67. http://dx.doi.org/10.18352/hcm.586.

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This article calls for a sound history of silence. Widely neglected within sound-historical research, exploring the manifold sounds of silence not only fills a lacuna in scholarship, but also poses critical challenges to current discussions in the flourishing field of sound history. This theoretical claim is based on empirical case studies from another still unwritten history: the political and cultural history of the minute’s silence, a political commemoration ceremony established in the aftermath of World War I. A practice theory approach makes it possible to understand how silence was produced in specific historical contexts through a complex set of cognitive, emotional, logistical, media, physiological, sensorial and kinesthetic practices that engage (or not) with the official call for silence and make it into success or failure. Conceiving of silence as a complex acoustical practice, the article aims to establish silence as a full-fledged topic of research at the centre of sound history and to inspire research on the historical and contemporary interplay between political structures and sensory or bodily practices.

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Din, Qamar, and Muhammad Asad Iqbal. "Bifurcation Analysis and Chaos Control for a Discrete-Time Enzyme Model." Zeitschrift für Naturforschung A 74, no. 1 (December 19, 2018): 1–14. http://dx.doi.org/10.1515/zna-2018-0254.

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AbstractBasically enzymes are biological catalysts that increase the speed of a chemical reaction without undergoing any permanent chemical change. With the application of Euler’s forward scheme, a discrete-time enzyme model is presented. Further investigation related to its qualitative behaviour revealed that discrete-time model shows rich dynamics as compared to its continuous counterpart. It is investigated that discrete-time model has a unique trivial equilibrium point. The local asymptotic behaviour of equilibrium is discussed for discrete-time enzyme model. Furthermore, with the help of the bifurcation theory and centre manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two existing chaos control methods, that is, Ott, Grebogi and Yorke feedback control and hybrid control strategy, are implemented. In particular, a novel chaos control technique, based on state feedback control is introduced for controlling chaos under the influence of flip and Hopf bifurcations in discrete-time enzyme model. Numerical simulations are provided to illustrate theoretical discussion and effectiveness of newly introduced chaos control method.

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Kumar, Rakesh, Anuj K. Sharma, and Kulbhushan Agnihotri. "Dynamics of an Innovation Diffusion Model with Time Delay." East Asian Journal on Applied Mathematics 7, no. 3 (August 2017): 455–81. http://dx.doi.org/10.4208/eajam.201216.230317a.

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AbstractA nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

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Liu, Zhigang, V. Madhusudanan, M. N. Srinivas, ChukwuNonso H. Nwokoye, and Tadele Degefa Geleto. "An Epidemic Patch-Enabled Delayed Model for Virus Propagation: Towards Evaluating Bifurcation and White Noise." Mathematical Problems in Engineering 2022 (August 17, 2022): 1–19. http://dx.doi.org/10.1155/2022/3763858.

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The massive disruptions caused by malware, such as a virus in computer networks and other aspects of information and communication technology, have generated attention, making it a hot research topic. While antivirus and firewalls can be effective, there is also a need to understand the spread patterns of viral infection using epidemic models to curb its incidences. Many previous research attempts have produced analytical models for computer viruses under various infectiousness situations. As a result, we suggested the SLBS model, which considers infection latency and transient immunity in patched nodes. Under certain conditions, the local stability of all equilibrium points is investigated. By setting the delay parameter, we established the occurrence of a Hopf bifurcation (HB) as it crossed a crucial point by several analyses. We also used the centre manifold theorem and normal form theory to examine the attributes of the HB. While the former was used to study the time delay and direction of Hopf bifurcation, the latter was used to investigate external noise and its intensities. Finally, numerical simulations two dimensional and three-dimensional graphs were used to depict the perturbations of the model, thus bolstering the essentiality of the study.

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Djellit, Ilham, and Baya Laadjal. "Center manifold in continuous time systems and computation." Facta universitatis - series: Electronics and Energetics 15, no. 3 (2002): 429–49. http://dx.doi.org/10.2298/fuee0203429d.

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The objective in this paper is to give some results of bifurcation equations, is concerned with the bifurcation from an equilibrium point in the case when the linear approximation has eigenvalues with zero real parts As we know, there is an intimate relationship between changes of stability and bifurcation. We formulate the main theorems that allow one to reduce dimension of a given system near a local bifurcation. We treat only continuous case. Center manifold theory is a method which uses power series expansions in the neighborhood of an equilibrium point in order to reduce the dimension of a system of ordinary differential equation. We will discuss some aspects of the center manifold. In this paper we will be concerned with the question of how to reduce a system to its center manifold. The calculation of center manifolds involves the manipulation of truncated power series. Coefficients of the quadratic Taylor expansion representing the center manifold can be computed via a recursive procedure, each step of which involves solving a linear system of algebraic equations. We present programs by Maple to accomplish such computations.

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Kusakabe, Yuta. "An implicit function theorem for sprays and applications to Oka theory." International Journal of Mathematics 31, no. 09 (July 17, 2020): 2050071. http://dx.doi.org/10.1142/s0129167x20500718.

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We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson’s elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov’s condition [Formula: see text], and the equivariant localization principle is also given.

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Jendoubi, Ch. "On the theory of integral manifolds for some delayed partial differential equations with nondense domain}." Ukrains’kyi Matematychnyi Zhurnal 72, no. 6 (June 17, 2020): 776–89. http://dx.doi.org/10.37863/umzh.v72i6.6020.

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UDC 517.9 Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation ⅆ u ⅆ t = ( A + B ( t ) ) u ( t ) + f ( t , u t ) , t ∈ R , ( 1 ) where ( A , D ( A ) ) satisfies the Hille – Yosida condition, ( B ( t ) ) t ∈ R is a family of operators in ℒ ( D ( A ) ¯ , X ) satisfying some measurability and boundedness conditions, and the nonlinear forcing term f satisfies ‖ f ( t , ϕ ) - f ( t , ψ ) ‖ ≤ φ ( t ) ‖ ϕ - ψ ‖ 𝒞 , here, φ belongs to some admissible spaces and ϕ , ψ ∈ 𝒞 : = C ( [ - r ,0 ] , X ) . We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for such solutions.Our main methods are invoked by the extrapolation theory and the Lyapunov – Perron method based on the admissible functions properties.

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TURAEV, D. "ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS." International Journal of Bifurcation and Chaos 06, no. 05 (May 1996): 919–48. http://dx.doi.org/10.1142/s0218127496000515.

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An analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension dc which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong-unstable foliations, and the Lyapunov dimension dL which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values dc and dL. A connection with the problem of hyperchaos is discussed.

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Burchard, Almut, Bo Deng, and Kening Lu. "Smooth conjugacy of centre manifolds." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 120, no. 1-2 (1992): 61–77. http://dx.doi.org/10.1017/s0308210500014980.

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SynopsisIn this paper, we prove that for a system of ordinary differential equations of class Cr+1,1, r≧0 and two arbitrary Cr+1, 1 local centre manifolds of a given equilibrium point, the equations when restricted to the centre manifolds are Cr conjugate. The same result is proved for similinear parabolic equations. The method is based on the geometric theory of invariant foliations for centre-stable and centre-unstable manifolds.

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Liu, Shuang, Shu-Ning Xia, Rui Yan, Zhen-Hua Wan, and De-Jun Sun. "Linear and weakly nonlinear analysis of Rayleigh–Bénard convection of perfect gas with non-Oberbeck–Boussinesq effects." Journal of Fluid Mechanics 845 (April 20, 2018): 141–69. http://dx.doi.org/10.1017/jfm.2018.225.

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The influences of non-Oberbeck–Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh–Bénard convection are examined. The working fluid is air with reference Prandtl number $Pr=0.71$ and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential $\unicode[STIX]{x1D716}$. Linear stability analysis of the thermal conduction state is performed. An $\unicode[STIX]{x1D716}^{2}$ scaling of the leading-order corrections of critical Rayleigh number $Ra_{cr}$ and disturbance growth rate $\unicode[STIX]{x1D70E}$ due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to $\unicode[STIX]{x1D716}$, and then the influence of aspect ratio $A$ is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.

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Boxler, Petra. "A stochastic version of center manifold theory." Probability Theory and Related Fields 83, no. 4 (December 1989): 509–45. http://dx.doi.org/10.1007/bf01845701.

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Ei, Shin-Ichiro, Kota Ikeda, Masaharu Nagayama, and Akiyasu Tomoeda. "Application of a center manifold theory to a reaction-diffusion system of collective motion of camphor disks and boats." Mathematica Bohemica 139, no. 2 (2014): 363–71. http://dx.doi.org/10.21136/mb.2014.143861.

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Bersani, A. M., A. Borri, A. Milanesi, and P. Vellucci. "Tihonov theory and center manifolds for inhibitory mechanisms in enzyme kinetics." Communications in Applied and Industrial Mathematics 8, no. 1 (March 28, 2017): 81–102. http://dx.doi.org/10.1515/caim-2017-0005.

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Abstract In this paper we study the chemical reaction of inhibition, determine the appropriate parameter ε for the application of Tihonov's Theorem, compute explicitly the equations of the center manifold of the system and find sufficient conditions to guarantee that in the phase space the curves which relate the behavior of the complexes to the substrates by means of the tQSSA are asymptotically equivalent to the center manifold of the system. Some numerical results are discussed.

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Journal articles on the topic 'Centre manifold theory'

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