Showing papers on "Center manifold published in 2013"
TL;DR: A general delayed bidirectional associative memory neural network model with n+1 neurons is considered and it is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons.
Abstract: Recent studies on Hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two, three, four, five, or six neurons. It is well known that neural networks are complex and large-scale nonlinear dynamical systems, so the dynamics of the delayed neural networks are very rich and complicated. Although discussing the dynamics of networks with a few neurons may help us to understand large-scale networks, there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large-scale networks. In this paper, a general delayed bidirectional associative memory neural network model with n+1 neurons is considered. By analyzing the associated characteristic equation, the local stability of the trivial steady state is examined, and then the existence of the Hopf bifurcation at the trivial steady state is established. By applying the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution. Furthermore, the paper highlights situations where the Hopf bifurcations are particularly critical, in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections. It is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons. Numerical simulations are carried out to illustrate the main results.
86 citations
TL;DR: A general framework for solving partial differential equations on manifolds represented by meshless points, i.e., point clouds, without parameterization or connection information is presented.
Abstract: In this paper we present a general framework for solving partial differential equations on manifolds represented by meshless points, i.e., point clouds, without parameterization or connection information. Our method is based on a local approximation of the manifold as well as functions defined on the manifold, such as using least squares, simultaneously in a local intrinsic coordinate system constructed by local principal component analysis using $K$ nearest neighbors. Once the local reconstruction is available, differential operators on the manifold can be approximated discretely. The framework extends to manifolds of any dimension. The complexity of our method scales well with the total number of points and the true dimension of the manifold (not the embedded dimension). The numerical algorithms, error analysis, and test examples are presented.
85 citations
Posted Content•
TL;DR: In this paper, a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure was presented, and it was shown that the eigenvectors and eigenvalues of these LaplACians converge in the limit of infinitely many independent random samples.
Abstract: Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Vector Diffusion Maps and showed that the connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples. We generalize the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.
77 citations
TL;DR: In this article, a three dimensional eco-epidemiological model consisting of susceptible prey, infected prey and predator is proposed and analyzed, where the parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected.
Abstract: A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.
57 citations
TL;DR: An algorithm is given to show that chaotic tri-neuron networks can be used for encrypting a color image and the model can exhibit pitchfork, Hopf, homoclinic, and double-limit cycles bifurcations.
Abstract: This brief reports a retarded functional differential equation modeling tri-neuron network with time delay. The Bogdanov-Takens (B-T) bifurcation is investigated by using the center manifold reduction and the normal form method. We get the versal unfolding of the norm forms at the B-T singularity and show that the model can exhibit pitchfork, Hopf, homoclinic, and double-limit cycles bifurcations. Some numerical simulations are given to support the analytic results and explore chaotic dynamics. Finally, an algorithm is given to show that chaotic tri-neuron networks can be used for encrypting a color image.
50 citations
TL;DR: In this article, the authors generalized the economic growth model with endogenous labor shift under a dual economy proposed by Cai (Applied Mathematics Letters 21, 774-779 (2008)) by introducing a time delay in the physical capital.
Abstract: The economic growth model with endogenous labor shift under a dual economy proposed by Cai (Applied Mathematics Letters 21, 774-779 (2008)) is generalized in this paper by introducing a time delay in the physical capital. By choosing the delay as a bifurcation parameter, it is proved that the delayed model has unique nonzero equilibrium and a Hopf bifurcation is proven to exist as the delay crosses a critical value. Moreover the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated in this paper by applying the center manifold theorem and the normal form theory.
49 citations
TL;DR: In this paper, the authors considered the differential-algebraic predator-prey model with predator harvesting and two delays and analyzed the stability and Hopf bifurcation of the proposed system.
Abstract: In this paper, we consider the differential-algebraic predator–prey model with predator harvesting and two delays. By using the new normal form of differential-algebraic systems, center manifold theorem and bifurcation theory, we analyze the stability and the Hopf bifurcation of the proposed system. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of the delayed differential-algebraic systems. Finally, numerical simulations are given to show the consistency with theoretical analysis obtained here.
45 citations
TL;DR: In this article, a theory of invariant manifolds for the steady Boltzmann equation was developed for boundary layers and nonlinear waves, and the analysis of these wave patterns allowed us to understand the Sone Diagram for the study of the complete condensation boundary value problem.
Abstract: We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green’s function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of −1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne’s problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves.
42 citations
TL;DR: In this article, the authors developed a methodology to establish stability and bifurcation dynamics generated by a class of delayed economic model, whose state variable is described by the scalar delay differential equation of the form.
Abstract: This investigation aims at developing a methodology to establish stability and bifurcation dynamics generated by a class of delayed economic model, whose state variable is described by the scalar delay differential equation of the form . At appropriate parameter values, linear stability and Hopf bifurcation including its direction and stability of the economic model are obtained. The main tools to obtain our results are the normal form method and the center manifold theory introduced by Hassard. Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior. Our results extend and complement some earlier publications.
40 citations
TL;DR: In this article, inertia is added to a simplified neuron system with time delay and the stability of the trivial equilibrium of the network is analyzed and the condition for the existence of Hopf bifurcation is obtained by discussing the associated characteristic equation.
Abstract: In this paper, inertia is added to a simplified neuron system with time delay. The stability of the trivial equilibrium of the network is analyzed and the condition for the existence of Hopf bifurcation is obtained by discussing the associated characteristic equation. Hopf bifurcation is investigated by using the perturbation scheme without the norm form theory and the center manifold theorem. Numerical simulations are performed to validate the theoretical results and chaotic behaviors are observed. Phase plots, time history plots, power spectra, and Poincare section are presented to confirm the chaoticity. To the best of our knowledge, the chaotic behavior in this paper is new to the previously published works.
39 citations
Book•
24 Mar 2013
TL;DR: In this paper, a perturbation theory for singularly perturbed dynamical systems is proposed, where the perturbations of the Euler top are assumed to break symmetry in the dynamical system.
Abstract: Symmetries in dynamical systems.- Symplccticity, reversibility and elliptic operators.- The Rolling Disc.- Testing for Sn-Symmetry with a Recursive Detective.- Normal forms of vector fields satisfying certain geometric conditions.- On symmetric ?-limit sets in reversible flows.- Symmetry Breaking in Dynamical Systems.- Invariant Cj functions and center manifold reduction.- Hopf bifurcation at k-fold resonances in conservative systems.- KAM theory and other perturbation theories.- Families of Quasi-Periodic Motions in Dynamical Systems Depending on Parameters.- Towards a Global Theory of Singularly Perturbed Dynamical Systems.- Equivariant Perturbations of the Euler Top.- On stability loss delay for a periodic trajectory.- Parametric and autoparametric resonance.- Global attractors and bifurcations.- Infinite dimensional systems.- Modulated waves in a perturbed Korteweg-de Vries equation.- Hamiltonian Perturbation Theory for Concentrated Structures in Inhomogeneous Media.- On instability of minimal foliations for a variational problem on T2.- Local and Global Existence of Multiple Waves Near Formal Approximations.- Time series analysis.- Estimation of dimension and order of time series.- Numerical continuation and bifurcation analysis.- On the computation of normally hyperbolic invariant manifolds.- The Computation of Unstable Manifolds Using Subdivision and Continuation.
TL;DR: In this article, a numerical approach based on the center manifold approach for nonlinear normal modes (NNMs) for conservative vibratory systems is presented, with the aim of deriving accurate reduced-order models up to large amplitudes.
TL;DR: An explicit, computationally efficient, recursive formula is presented for computing the normal form and center manifold of general n-dimensional systems associated with Hopf bifurcation.
Abstract: An explicit, computationally efficient, recursive formula is presented for computing the normal form and center manifold of general n-dimensional systems associated with Hopf bifurcation. Maple program is developed based on the analytical formulas, and shown to be computationally efficient, using two examples.
01 Jan 2013
TL;DR: In this article, the generalized reduced basis method is used for the uniform approximation of the solution manifold of a linear PDE with smooth coefficients, where the set of solutions of a parameter-dependent linear partial differential equation with smooth functions typically form a compact manifold in a Hilbert space.
Abstract: The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold.
TL;DR: Based on the availability of prey and a simple predator-prey model, Liu et al. as discussed by the authors proposed a delay-dependent delay-sensitive model with predator migration to describe biological control and derived the direction and stability of Hopf bifurcation.
01 Jan 2013
TL;DR: In this article, Wu et al. investigated the dynamics of a TCP system described by a first-order nonlinear delay differential equation and showed that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values.
Abstract: This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with mem- ory. Transactions of the American Mathematical Society 350(12), 4799-4838 (1998)).
TL;DR: In this article, the authors compared the van der Pol system with a reduced version of the original system and showed that it is possible to predict the approaching explosive phase and the associated degeneracy of the manifold only through the dynamics of the reduced system.
TL;DR: Flexible guideway system for maglev train with time-delayed velocity feedback control gain with stability and direction of the bifurcation periodic solution analyzed by applying the normal form theory and the center manifold theorem.
TL;DR: The dynamics of a diffusive predator-prey system with the Beddington-DeAngelis functional response and two delays are considered and the sufficient conditions of the global stability of the positive equilibrium are given by the upper-lower solutions method.
TL;DR: In this paper, a delay induced prey-predator system with stage structure for prey is described and the dynamical characteristics of the system are rigorously studied using mathematical tools, and sufficient conditions are derived for the global stability.
Abstract: This paper describes a delay induced prey–predator system with stage structure for prey. The dynamical characteristics of the system are rigorously studied using mathematical tools. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Sufficient conditions are derived for the global stability of the system. The optimal harvesting problem is formulated and solved in order to achieve the sustainability of the system, keeping the ecological balance, and maximize the monetary social benefit. Maturation time delay of prey is incorporated and the existence of Hopf bifurcation phenomenon is examined at the coexistence equilibria. It is shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Moreover, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.
TL;DR: This work completes a previous author's paper which presents the description of the qualitative dynamical response of a rotary drilling system with a drag bit, using a model that takes into consideration the axial and the torsional vibration modes of the bit.
TL;DR: In this article, the effects of two time delays on the asymptotic dynamics of the Solow model were analyzed and the existence of Hopf bifurcation was proved.
Abstract: A further generalization of an economic growth model is the main topic of this paper. The paper specifically analyzes the effects on the asymptotic dynamics of the Solow model when two time delays are inserted: the time employed in order that the capital is used for production and the necessary time so that the capital is depreciated. The existence of a unique nontrivial positive steady state of the generalized model is proved and sufficient conditions for the asymptotic stability are established. Moreover, the existence of a Hopf bifurcation is proved and, by using the normal form theory and center manifold argument, the explicit formulas which determine the stability, direction, and period of bifurcating periodic solutions are obtained. Finally, numerical simulations are performed for supporting the analytical results.
TL;DR: In this paper, the authors give a complete stability analysis of the flow restricted to a neighborhood of the origin in any center manifold of the Moon-Rand systems, solve the center problem on the center manifold, and find sharp bounds on the number of limit cycles that can be made to bifurcate from the singularity when it is a focus.
Abstract: The Moon–Rand systems, developed to model control of flexible space structures, are systems of differential equations on R 3 with polynomial or rational right hand sides that have an isolated singularity at the origin at which the linear part has one negative and one pair of purely imaginary eigenvalues for all choices of the parameters. We give a complete stability analysis of the flow restricted to a neighborhood of the origin in any center manifold of the Moon–Rand systems, solve the center problem on the center manifold, and find sharp bounds on the number of limit cycles that can be made to bifurcate from the singularity when it is a focus. We generalize the Moon–Rand systems in a natural way, solve the center problem in several cases, and provide sufficient conditions for the existence of a center, which we conjecture to be necessary.
TL;DR: In this article, the effect of time delay on conservation of forestry biomass was studied by proposing a non-linear mathematical model, in which it is assumed that the density of forest biomass depletes due to the presence of human population and it is being conserved by applying some technological efforts.
Abstract: In this paper, we have studied the effect of time delay on conservation of forestry biomass by proposing a non-linear mathematical model. In the modeling process, it is assumed that the density of forestry biomass depletes due to the presence of human population and it is being conserved by applying some technological efforts. The analysis of model shows that the density of forestry biomass may be conserved if the technological effort is applied within the appropriate time. A longer delay in applying technological effort for its conservation destabilizes the system. The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the mathematical results.
TL;DR: A diffusive predator-prey model with Holling type II functional response and the no-flux boundary condition incorporating a constant prey refuge and an algorithm for determining the properties of the Hopf bifurcation is derived.
Abstract: A diffusive predator-prey model with Holling type II functional response and the no-flux boundary condition incorporating a constant prey refuge is considered. Globally asymptotically stability of the positive equilibrium is obtained. Regarding the constant number of prey refuge m as a bifurcation parameter, by analyzing the distribution of the eigenvalues, the existence of Hopf bifurcation is given. Employing the center manifold theory and normal form method, an algorithm for determining the properties of the Hopf bifurcation is derived. Some numerical simulations for illustrating the analysis results are carried out.
TL;DR: In this paper, a nonlinear mathematical model for control of unemployment in the developing countries by incorporating time delay in creating new vacancies was proposed and analyzed, where three dynamic variables have been considered, namely, (i) unemployed persons, (ii) employed persons, and (iii) newly created vacancies The model is studied using stability theory of differential equations.
Abstract: In this paper, we have proposed and analyzed a nonlinear mathematical model for control of unemployment in the developing countries by incorporating time delay in creating new vacancies In the modeling process, three dynamic variables have been considered, namely, (i) number of unemployed persons, (ii) number of employed persons, and (iii) number of newly created vacancies The model is studied using stability theory of differential equations It is found that the model has only one equilibrium, which is stable in absence of delay It is further shown that this stable equilibrium becomes unstable as delay crosses some critical value This critical value of delay has been obtained analytically Further, direction of Hopf bifurcation and stability of the bifurcating periodic solutions are studied by applying the normal form theory and the center manifold theorem Numerical simulation of the model has been carried out to illustrate the analytical results
TL;DR: In this article, stability and bifurcations in a simply supported rotating shaft are studied, where the shaft is modeled as an inextensional spinning beam with large amplitude, including the effects of nonlinear curvature and inertia.
Abstract: In this paper, stability and bifurcations in a simply supported rotating shaft are studied. The shaft is modeled as an in-extensional spinning beam with large amplitude, which includes the effects of nonlinear curvature and inertia. To include the internal damping, it is assumed that the shaft is made of a viscoelastic material. In addition, the torsional stiffness and external damping of the shaft are considered. To find the boundaries of stability, the linearized shaft model is used. The bifurcations considered here are Hopf and double zero eigenvalues. Using center manifold theory and the method of normal form, analytical expressions are obtained, which describe the behavior of the rotating shaft in the neighborhood of the bifurcations.
TL;DR: In this paper, the authors investigated the Bogdanov-Takens bifurcation in an oscillator with negative damping and delayed position feedback by using center manifold reduction and normal form theory.
TL;DR: In this article, the dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail Conditions of the existence for flip bifurcation and Hopf bifurbation are derived by using the center manifold theorem and bifurlcation theory, and then further illustrated by numerical simulations.
Abstract: In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations Chaos in the sense of Marotto is proved by both analytical and numerical methods Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior The dynamical behavior can move from complex instable states to stable states as the Allee constant increases (within a limited value) Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given
TL;DR: In this article, the existence of a global invariant manifold of bubble states for the mass-conserving Allen-Cahn Equation in two space dimensions was established and the dynamics for the center of the bubble was given.
Abstract: We establish the existence of a global invariant manifold of bubble states for the mass-conserving Allen-Cahn Equation in two space dimensions and give the dynamics for the center of the bubble.