Center Manifold Approach Research Articles

Wave dynamics of a viscoelastic liquid

A falling film of a weakly viscoelastic liquid (Walters’ liquid B′′) is studied in detail for low to moderate values of the Reynolds number. A linear stability analysis within the framework of the Orr–Sommerfeld-type boundary value problem reveals a destabilizing effect of the viscoelastic parameter near the threshold of instability but a stabilizing effect of the viscoelastic parameter on the primary instability far away from the threshold of instability. The surface equation is derived to decipher the existence of nonlinear waves based on the centre manifold approach. However, it renders an unbounded solution in the short-wave regime. In order to avoid singular behaviour found in the surface equation, a regularized equation, a simplified second-order model, and a full second-order model are proposed. In the linear regime, both the regularized equation and the model equations show excellent agreement with the results of the Orr–Sommerfeld-type boundary value problem, i.e., the viscoelastic parameter demonstrates a dual role in the primary instability. In the nonlinear regime, the results of the regularized equation are verified with those of the simplified second-order model. Both the regularized equation and the simplified second-order model show qualitatively similar characteristics of the nonlinear waves. The speed and maximum amplitude corresponding to the steady-state travelling waves attain fixed values in the drag-inertia regime. Moreover, the speed and maximum amplitude acquired from the regularized equation underestimate the results of the simplified second-order model when the Reynolds number lies in the drag-inertia regime. The time-dependent simulation of the regularized equation displays a train of solitary waves downstream to a monochromatic time periodic forcing at the inlet. The separation distance between the first two solitary pulses is computed, and it seems to approach an asymptotic value when the time is very large. This fact assures the possibility of a bound-state formation among the solitary pulses downstream for the weakly viscoelastic liquid.

Stability and Hopf Bifurcation Analysis of a Stage-Structured Predator–Prey Model with Delay

In this work, a Lotka–Volterra type predator–prey system with time delay and stage structure for the predators is proposed and analyzed. By using the permanence theory for infinite dimensional system, we get that the system is permanent if some conditions are satisfied. The local and global stability of the positive equilibrium is presented. The existence of Hopf bifurcation around the positive equilibrium is observed. Further, by using the normal form theory and center manifold approach, we derive the explicit formulas determining the stability of bifurcating periodic solutions and the direction of Hopf bifurcation. Numerical simulations are carried out by Matlab software to explain the theoretical results. We find that combined time delay and stage structure can affect the dynamical behavior of the system.

Center Manifold Theory for the Motions of Camphor Boats with Delta Function

Various collective motions of camphor boats have been studied. Camphor boats are self-driven particles that interact with each other through the change of surface tension on water surface by camphor molecules. Consequently, even in a one-dimensional circuit, a congested state or jamming can be observed (Suematsu et al. in Phys Rev E 81:056210, 2010). In this phenomenon, the unidirectional motion of each particle is considered a traveling wave, and the concentration of camphor molecules forms a pulse shape. Hence, each pair of particles interacts with each other like a pulse–pulse interaction. Thus, we expect that the center manifold theories proposed in Ei (J Dyn Differ Equ 14:85–137, 2002) and Ei et al. (Physica D 165:176–198 2002) are applicable for the analysis of the collective motion of camphor boats. However, spatial discontinuity in our model, in particular the existence of Dirac delta functions in a linearized operator, does not fulfill the requirement in the reduction process because the authors developed their theories in $$L^2$$ -framework and for smooth nonlinearity in Ei (2002) and Ei et al. (2002). In this article, we extend the results obtained in Ei (2002) and Ei et al. (2002) and establish a new center manifold approach in $$(H^1)^*$$ -framework. Finally, we succeed to rigorously reduce a mathematical model (Nagayama et al. in Physica D: Nonlinear Phenom 194:151–165, 2004) coupled with a Newtonian equation and a reaction–diffusion equation including delta functions to an ordinary differential system that represents the motions of camphor boats.

Hopf bifurcation in an activator–inhibitor system with network

A network is introduced to describe the activator–inhibitor system, where the network structure represents the movement directions of molecule random walk. We show that a Hopf bifurcation occurs in the activator–inhibitor system by the linear stability analysis. By an extension of the center manifold approach, we also prove that the Hopf bifurcation is stable and its direction is backward.

Dynamical system analysis of a three fluid cosmological model: an invariant manifold approach

The present paper considers a three-fluid cosmological model consisting of noninteracting dark matter, dark energy and baryonic matter in the background of the Friedman–Robertson–Walker–Lemaître flat spacetime. It has been assumed that the dark matter takes the form of dust whereas the dark energy is a quintessence (real) scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. The field equations for this model takes the form of an autonomous dynamical system after some suitable changes of variables. Then a complete stability analysis is done considering all possible parameter (the adiabatic index of the baryonic matter and the parameter arising from the dark energy potential) values and for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperbolic critical points, the invariant manifold theory (center manifold approach) is applied. Finally various topologically different phase planes and vector field diagrams are produced and the cosmological interpretation of this model is presented.

Asymptotic integration of functional differential systems with oscillatory decreasing coefficients: a center manifold approach

Asymptotic integration of functional differential systems with oscillatory decreasing coefficients: a center manifold approach

Robust Position Stabilization of Underactuated Surface Vessels With Unknown Modeling Parameters Via Simple P/D‐Like Feedback: The Center Manifold Approach

AbstractIn this work, two smooth time‐invariant control laws are proposed to achieve asymptotic position stabilization of underactuated surface vessels despite modeling parameter uncertainties. The proposed control laws take a strikingly simple linear proportional derivative (PD)‐like feedback form or just a simplest proportional (P)‐like one, where the former relies on measuring the relative position error and surge/yaw velocity and the latter relies on measuring only the relative position error, decreasing or minimizing the sensing cost. The stabilities of the closed‐loop systems for the both control laws are strictly proved via the center manifold approach despite the unknown model parameters. The effectiveness of the proposed control laws is verified by simulation examples.

On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems

Numerical computation of nonlinear normal modes (NNMs) for conservative vibratory systems is addressed, with the aim of deriving accurate reduced-order models up to large amplitudes. A numerical method is developed, based on the center manifold approach for NNMs, which uses an interpretation of the equations as a transport problem, coupled to a periodicity condition for ensuring manifold's continuity. Systematic comparisons are drawn with other numerical methods, and especially with continuation of periodic orbits, taken as reference solutions. Three different mechanical systems, displaying peculiar characteristics allowing for a general view of the performance of the methods for vibratory systems, are selected. Numerical results show that invariant manifolds encounter folding points at large amplitude, generically (but not only) due to internal resonances. These folding points involve an intrinsic limitation to reduced-order models based on the centre manifold and on the idea of a functional relationship between slave and master coordinates. Below that amplitude limit, numerical methods are able to produce reduced-order models allowing for a precise prediction of the backbone curve.

Invariant manifolds for nonsmooth systems

For piecewise smooth systems we describe mechanisms to obtain a similar reduction to a lower dimensional system as has been achieved for smooth systems via the center manifold approach. It turns out that for nonsmooth systems there are invariant quantities as well which can be used for a bifurcation analysis but the form of the quantities is more complicated. The approximation by piecewise linear systems (PWLS) provides a useful concept. In the case of PWLS, the invariant sets are given as invariant cones. For nonlinear perturbations of PWLS the invariant sets are deformations of those cones. The generation of invariant manifolds and a bifurcation analysis establishing periodic orbits are demonstrated; also an example for which multiple cones exist is provided.

Stabilization of nonlinear systems via the center manifold approach

This paper considers the problem of the stabilization of affine nonlinear control systems. First, we assume that the systems under investigation are of the generalized Byrnes–Isidori normal form. A new way to approximate the center manifold is proposed, which can reduce the error degree of the center manifold approximation. A new matrix product, called the semi-tensor product, is introduced to obtain the approximation of the center manifold. Then the Lyapunov function with homogeneous derivative (LFHD) is used to design a stable center manifold by state feedback control. Finally, the method is applied to general affine nonlinear control systems.

Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift–Hohenberg equation

This work studies the spatio-temporal dynamics of a generic integral–differential equation subject to additive random fluctuations. It introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for Fokker–Planck equations, and studies analytically the systems’ stability near Turing bifurcations. In addition two types of fluctuation are studied, namely fluctuations uncorrelated in space and time, and global fluctuations, which are constant in space but uncorrelated in time. We show that the global fluctuations shift the Turing bifurcation threshold. This shift is proportional to the fluctuation variance. Applications to a neural field equation and the Swift–Hohenberg equation reveal the shift of the bifurcation to larger control parameters, which represents a stabilization of the system. All analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral–differential equation. To gain some insight into experimental manifestations, the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively.

Subgrid and interelement interactions affect discretisations of stochastically forced diffusion

Constructing discrete models of stochastic partial differential equations is very delicate. I apply dynamical systems theory to support spatial discretisations of the stochastically forced diffusion equation. To apply stochastic centre manifold theory, divide the physical domain into finite sized elements by introducing insulating internal boundaries which are later removed to fully couple the dynamical interactions between neighbouring elements. The approach automatically parametrises the stochastically forced microscale, subgrid structures within each element. The crucial aspect of this work is that we explore how a multitude of noise processes interact within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in a spatial discretisation. References Jinqiao Duan, Kening Lu, and Bjorn Schmalfuss. Invariant manifolds for stochastic partial differential equations. The Annals of Probability , 31:2109--2135, 2003. doi:10.1214/aop/1068646380 . E. Knobloch and K. A. Wiesenfeld. Bifurcations in fluctuating systems: The centre manifold approach. J. Stat Phys , 33:611--637, 1983. T. MacKenzie and A. J. Roberts. Holistic discretisation of shear dispersion in a two-dimensional channel. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001 , volume 44, pages C512--C530, March 2003. http://anziamj.austms.org.au/V44/CTAC2001/Mack . A. J. Roberts. Low-dimensional modelling of dynamics via computer algebra. Computer Phys. Comm. , 100:215--230, 1997. A. J. Roberts. Holistic discretisation ensures fidelity to {Burgers'} equation. Applied Numerical Modelling , 37:371--396, 2001. A. J. Roberts. A holistic finite difference approach models linear dynamics consistently. Mathematics of Computation , 72:247--262, 2002. http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01448-5 . A. J. Roberts. Derive boundary conditions for holistic discretisations of {Burgers'} equation. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001 , volume 44, pages C664--C686, March 2003. http://anziamj.austms.org.au/V44/CTAC2001/Robe . A. J. Roberts. A step towards holistic discretisation of stochastic partial differential equations. In Jagoda Crawford and A. J. Roberts, editors, Proc. of 11th Computational Techniques and Applications Conference CTAC-2003 , volume 45, pages C1--C15, December 2003. [Online] http://anziamj.austms.org.au/V45/CTAC2003/Robe [December 14, 2003]. J. C. Robinson. The asymptotic completeness of inertial manifolds. Nonlinearity , 9:1325--1340, 1996. http://www.iop.org/EJ/abstract/0951-7715/9/5/013 . N. {Sri Namachchivaya} and Y. K. Lin. Method of stochastic normal forms. Int. J. Nonlinear Mechanics , 26:931--943, 1991. D. Blomker, M. Hairer, and G. A. Pavliotis. Modulation equations: stochastic bifurcation in large domains. Communications in Mathematical Physics , 258:479--512, 2005. doi:10.1007/s00220-005-1368-8 . P. Boxler. A stochastic version of the centre manifold theorem. Probab. Th. Rel. Fields , 83:509--545, 1989. Xu Chao and A. J. Roberts. On the low-dimensional modelling of {Stratonovich} stochastic differential equations. Physica A , 225:62--80, 1996. doi:10.1016/0378-4371(95)00387-8 . P. H. Coullet, C. Elphick, and E. Tirapegui. Normal form of a {Hopf} bifurcation with noise. Physics Letts , 111A(6):277--282, 1985.

Friction induced vibration for an aircraft brake system—Part 2: Non-linear dynamics

Non-linear dynamics due to friction induced vibrations in a complex aircraft brake model are investigated. This paper outlines a non-linear strategy, based on the center manifold concept and the rational in order to evaluate the non-linear dynamical behaviour of a system in the neighbourhood of a critical steady-state equilibrium point. In order to obtain time–history responses, the complete set of non-linear dynamic equations may be integrated numerically. But this procedure is both time consuming and costly to perform when parametric design studies are needed. So it is necessary to use non-linear analysis: the center manifold approach and the rational approximants are used to obtain the limit cycle of the non-linear system and to study the behaviour of the system in the unstable region. Results from these non-linear methods are compared with results obtained by integrating the full original system. These non-linear methods appear very interesting in regard to computational time and also necessitate very few computer resources.

Accurately Model the Kuramoto--Sivashinsky Dynamics with Holistic Discretization

We analyze the nonlinear Kuramoto--Sivashinsky equation to develop accurate discretizations modeling its dynamics on coarse grids. The analysis is based upon center manifold theory, so we are assured that the discretization accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing isolating internal boundaries which are later removed. Comprehensive numerical solutions and simulations show that the holistic discretizations excellently reproduce the steady states and the dynamics of the Kuramoto--Sivashinsky equation. The Kuramoto--Sivashinsky equation is used as an example to show how holistic discretization may be successfully applied to fourth-order, nonlinear, spatio-temporal dynamical systems. This novel center manifold approach is holistic in the sense that it treats the dynamical equations as a whole, not just as the sum of separate terms.

On the dynamics of free and excited oscillations of a simple portal frame foundation

In this paper we search for the dynamics of a simple portal structure in the free and in the periodic excitation cases. By using the Center Manifold approach and Averaging Method, we obtain results on both stability and bifurcation of equilibrium points and periodic orbits.

Rotating spirals of curvature flows: a center manifold approach

Rigidly rotating spiral wave patterns of Archimedean shape are a prominent phenomenon in the spatiotemporal evolution of planar reaction diffusion systems. A first analysis by Wiener and Rosenblueth [27] was motivated by electrical waves in heart tissue. For experimental evidence of Archimedean spirals in Belousov– Zhabotinsky systems and other excitable chemical reaction media see, for example, [24–26,5,22,20,21]. See also the recent survey [17] and references therein. Fluid convection experiments, although modeled by the Navier–Stokes system rather than by reaction and diffusion, also exhibit dynamically intricate spiral wave patterns; see [19]. For a survey of mathematical issues related to spiral waves in reaction diffusion systems, see [10]. One mathematical approach, initiated already by Wiener, aims at a geometric description of the spatiotemporal dynamics of the sharp wave fronts that seem to emanate from a highly focused “core” region or “tip” and that take the form of an Archimedeam spiral, far away from the tip. This approach, where the wave front is represented simply by a planar curve z = z(s) ∈ R2, parametrized over arclength s ≥ 0, is sometimes called kinematic theory of spiral waves and has been developed on a mostly formal level. See, for example, [15,14,18,17] and references therein. Typically the dynamics of the curve z(s) is then described by a curvature driven flow

Methods to reduce non-linear mechanical systems for instability computation

Non-linear dynamical structures depending on control parameters are encountered in many areas of science and engineering. In the study of non-linear dynamical systems depending on a given control parameter, the stability analysis and the associated non-linear behaviour in a near-critical steady-state equilibrium point are two of the most important points; they make it possible to validate and characterize the non-linear structures. Stability is investigated by determining eigenvalues of the linearized perturbation equations about each steady-state operating point, or by calculating the Jacobian of the system at the equilibrium points. While the conditions and the values of the parameters which cause instability can be investigated by using linearized equations of motion studies of the non-linear behaviour of vibration problems, on the other hand, require the complete non-linear expressions of systems. Due to the complexity of non-linear systems and to save time, simplifications and reductions in the mathematical complexity of the non-linear equations are usually required. The principal idea for these non-linear methods is to reduce the order of the system and eliminate as many non-linearities as possible in the system of equations. In this paper, a study devoted to evaluating the instability phenomena in non-linear models is presented. It outlines stability analysis and gives a non-linear strategy by constructing a reduced order model and simplifying the non-linearities, based on three non-linear methods: the centre manifold concept, the rational approximants and the Alternating Frequency/Time domain method. The computational procedures to determine the reduced and simplified system via the centre manifold approach and the fractional approximants, as well as the approximation of the responses as a Fourier series via the harmonic balance method, are presented and discussed. These non-linear methods for calculating the dynamical behaviour of non-linear systems with several degrees-of-freedom and non-linearities are tested in the case of mechanical systems with many degrees-of-freedom possessing polynomial non-linearities. Results obtained are compared with those estimated by a classical Runge-Kutta integration procedure. Moreover, an extension of the centre manifold approach using rational approximants is proposed and used to explore the dynamics of non-linear systems, by extending the domain of convergence of the non-linear reduced system and evaluating its performance and suitability.

Stabilization of General Nonlinear Control Systems via Center Manifold and Approximation Techniques

This paper considers the stabilization of a class of general nonlinear control systems. Under some mild conditions, a general nonlinear control system can be transformed into a normal form, which is suitable for the center manifold approach. A kind of controller in the polynomial form is proposed to stabilize the systems. First, the type and degree of controllers are chosen to assure the approximation degree of the center manifold. Then the coefficients are chosen to make the dynamics of the center manifold of the closed-loop systems stable. To test the approximate stability of the dynamics on the center manifold, the Lyapunov function with homogeneous derivative proposed in l6r is used.

Analysis of friction and instability by the centre manifold theory for a non-linear sprag-slip model

This paper presents the research devoted to the study of instability phenomena in non-linear model with a constant brake friction coefficient. Indeed, the impact of unstable oscillations can be catastrophic. It can cause vehicle control problems and component degradation. Accordingly, complex stability analysis is required. This paper outlines stability analysis and centre manifold approach for studying instability problems. To put it more precisely, one considers brake vibrations and more specifically heavy trucks judder where the dynamic characteristics of the whole front axle assembly is concerned, even if the source of judder is located in the brake system. The modelling introduces the sprag-slip mechanism based on dynamic coupling due to buttressing. The non-linearity is expressed as a polynomial with quadratic and cubic terms. This model does not require the use of brake negative coefficient, in order to predict the instability phenomena. Finally, the centre manifold approach is used to obtain equations for the limit cycle amplitudes. The centre manifold theory allows the reduction of the number of equations of the original system in order to obtain a simplified system, without loosing the dynamics of the original system as well as the contributions of non-linear terms. The goal is the study of the stability analysis and the validation of the centre manifold approach for a complex non-linear model by comparing results obtained by solving the full system and by using the centre manifold approach. The brake friction coefficient is used as an unfolding parameter of the fundamental Hopf bifurcation point.

Generalized normal form and stabilization of non-linear systems

This paper considers the normal form of non-linear control systems. First we propose a generalized relative degree (relative degree vector) for non-linear single (respectively, multiple) input control system, which is called the point relative degree (respectively, point relative degree vector). For the systems without output, the concepts of essential relative degree (respectively, essential relative degree vector) and the essential point relative degree (respectively, essential point relative degree vector) are defined. Unlike the classical definition which requires regularity, the point relative degree (vector) is always well defined. Using these new concepts the generalized normal form is obtained. Its relationship with the Jacobian linearization is investigated. Using it, a straightforward computation algorithm is provided to achieve the generalized normal form. Based on the generalized normal form we prove that with an additional condition, if the zero-dynamics is stable the overall system is stabilizable by using pseudo-linear state feedback control. For the systems under generalized normal form with unstable zero dynamics, the centre manifold approach is applied. It is shown that the stabilization technique via a designed centre manifold is still applicable to this kind of general non-linear control system.