Showing papers on "Center manifold published in 2017"
TL;DR: This paper focuses on the Turing-Hopf (TH) bifurcation and obtains the explicit dynamical classification in its neighborhood by calculating and investigating the normal form on the center manifold and demonstrates that this TH interaction would significantly enhance the diversity of spatial patterns and trigger the alternative paths for the pattern development.
Abstract: Intertidal mussels can self-organize into periodic spot, stripe, labyrinth, and gap patterns ranging from centimeter to meter scales. The leading mathematical explanations for these phenomena are the reaction-diffusion-advection model and the phase separation model. This paper continues the series studies on analytically understanding the existence of pattern solutions in the reaction-diffusion mussel-algae model. The stability of the positive constant steady state and the existence of Hopf and steady-state bifurcations are studied by analyzing the corresponding characteristic equation. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the explicit dynamical classification in its neighborhood by calculating and investigating the normal form on the center manifold. Using theoretical and numerical simulations, we demonstrates that this TH interaction would significantly enhance the diversity of spatial patterns and trigger the alternative paths for the pattern development.
97 citations
TL;DR: In this paper, the authors implemented the autonomous stabilization of an encoding manifold spanned by Schroedinger cat states in a superconducting cavity, and showed coherent oscillations between these states analogous to the Rabi rotation of a qubit protected against phase-flips.
Abstract: Manipulating the state of a logical quantum bit usually comes at the expense of exposing it to decoherence. Fault-tolerant quantum computing tackles this problem by manipulating quantum information within a stable manifold of a larger Hilbert space, whose symmetries restrict the number of independent errors. The remaining errors do not affect the quantum computation and are correctable after the fact. Here we implement the autonomous stabilization of an encoding manifold spanned by Schroedinger cat states in a superconducting cavity. We show Zeno-driven coherent oscillations between these states analogous to the Rabi rotation of a qubit protected against phase-flips. Such gates are compatible with quantum error correction and hence are crucial for fault-tolerant logical qubits.
94 citations
TL;DR: In this paper, the authors describe the use of a quadratic manifold for the model order reduction of structural dynamics problems featuring geometric nonlinearities, where the manifold is tangent to a subspace spanned by the most relevant vibration modes, and its curvature is provided by modal derivatives obtained by sensitivity analysis.
76 citations
TL;DR: The potential of the quadratic manifold approach is investigated in a numerical study, where several variations of the approach are compared on different examples, giving a clear indication of where the proposed approach is applicable.
52 citations
TL;DR: In this article, a branched center manifold is constructed in a neighborhood of a singular point of a 2-dimensional integral current which is almost minimizing in a suitable sense, and the construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of two-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3-dimensional area minimizing cones.
Abstract: We construct a branched center manifold in a neighborhood of a singular point of a 2-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of 2-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3-dimensional area minimizing cones.
41 citations
TL;DR: By constructing a suitable LyapunovPerron operator via giving asymptotic behavior of MittagLeffler function, an interesting center stable manifold theorem is obtained.
40 citations
TL;DR: In this paper, the stability and Hopf bifurcation of the positive steady state to a general scalar reaction diffusion equation with distributed delay and Dirichlet boundary condition are investigated.
32 citations
TL;DR: In this paper, a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state was studied and its bifurcation diagram in the plane of the two feedback strengths was presented.
Abstract: We study a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state-dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three contain only constant delays. We implemented this expansion and the computation of the normal form coef...
31 citations
TL;DR: In this article, a physiologically based corticothalamic model of large-scale brain activity is used to analyze critical dynamics of transitions from normal arousal states to epileptic seizures, which correspond to Hopf bifurcations.
Abstract: A physiologically based corticothalamic model of large-scale brain activity is used to analyze critical dynamics of transitions from normal arousal states to epileptic seizures, which correspond to Hopf bifurcations. This relates an abstract normal form quantitatively to underlying physiology that includes neural dynamics, axonal propagation, and time delays. Thus, a bridge is constructed that enables normal forms to be used to interpret quantitative data. The normal form of the Hopf bifurcations with delays is derived using Hale's theory, the center manifold theorem, and normal form analysis, and it is found to be explicitly expressed in terms of transfer functions and the sensitivity matrix of a reduced open-loop system. It can be applied to understand the effect of each physiological parameter on the critical dynamics and determine whether the Hopf bifurcation is supercritical or subcritical in instabilities that lead to absence and tonic-clonic seizures. Furthermore, the effects of thalamic and cortical nonlinearities on the bifurcation type are investigated, with implications for the roles of underlying physiology. The theoretical predictions about the bifurcation type and the onset dynamics are confirmed by numerical simulations and provide physiologically based criteria for determining bifurcation types from first principles. The results are consistent with experimental data from previous studies, imply that new regimes of seizure transitions may exist in clinical settings, and provide a simplified basis for control-systems interventions. Using the normal form, and the full equations from which it is derived, more complex dynamics, such as quasiperiodic cycles and saddle cycles, are discovered near the critical points of the subcritical Hopf bifurcations.
27 citations
TL;DR: The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point and bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium.
Abstract: The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium. Taking into account the non-smoothness caused by loss of contact between the tool and the workpiece, the boundary of the bistable region is also derived analytically. The results are verified by numerical continuation. The possibility of (transient) chaotic motions in the global non-smooth dynamics is shown.
26 citations
TL;DR: In this article, the dynamic behaviors of coupled reaction-diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay under the Neumann boundary conditions are investigated.
Abstract: In this paper, the dynamic behaviors of coupled reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay under the Neumann boundary conditions are investigated. By constructing a basis of phase space based on the eigenvectors of Laplace operator, the characteristic equation of this system is obtained. Then, the local stability of zero solution and the occurrence of Hopf bifurcation are established by regarding the time delay as the bifurcation parameter. In particular, by using the normal form theory and center manifold theorem of the partial differential equation, the normal forms are obtained, which determine the bifurcation direction and the stability of the periodic solutions. Finally, two examples are given to verify the theoretical results.
TL;DR: In this paper, the stability of a delayed phytoplankton-zooplankton system with Crowley-Martin functional response is investigated analytically, and the permanence and stability of both boundary and positive equilibrium points for the system with delay as well as the system without delay are analyzed.
Abstract: In this paper, a delayed phytoplankton-zooplankton system with Crowley-Martin functional response is investigated analytically. We study the permanence and analyze the stability of the both boundary and positive equilibrium points for the system with delay as well as the system without delay. The global asymptotic stability is discussed by constructing a suitable Lyapunov functional. Numerical analysis indicates that the delay does not change the stability of the positive equilibrium point. Furthermore, we also show that due to the increase of the delay there occurs a Hopf bifurcation of periodic solutions. It is found that population fluctuations will not appear under the condition of certain parameters. In addition, we determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions by applying a normal form method and center manifold theory. Finally, some numerical simulations are carried out to support our theoretical analysis results.
TL;DR: In this article, the stability of steady state and the existence of Hopf bifurcation near positive steady state are investigated by analyzing the distribution of eigenvalues of a spruce budworm model with diffusion and physiological structures.
TL;DR: In this article, the dynamics of a diffusive predator-prey model with time delay and Michaelis-Menten-type harvesting subject to Neumann boundary condition is considered.
Abstract: The dynamics of a diffusive predator–prey model with time delay and Michaelis–Menten-type harvesting subject to Neumann boundary condition is considered. Turing instability and Hopf bifurcation at positive equilibrium for the system without delay are investigated. Time delay-induced instability and Hopf bifurcation are also discussed. By the theory of normal form and center manifold, conditions for determining the bifurcation direction and the stability of bifurcating periodic solution are derived. Some numerical simulations are carried out for illustrating the theoretical results.
TL;DR: A class of time fractional Cohen–Grossberg neural networks with time delays in leakage terms and diffusion under homogeneous Neumann boundary conditions is investigated and the local stability of the trivial uniform steady state and the existence of Hopf bifurcation are established.
Abstract: In this paper, a class of time fractional Cohen---Grossberg neural networks with time delays in leakage terms and diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation are established. By using the normal form theory and the center manifold reduction of partial functional differential equations, explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main theoretical results.
TL;DR: The paper is formulated with the Holling–Tanner prey–predator model with Beddington–DeAngelis functional response including prey harvesting with the help of normal theory and center manifold theorem to verify the validity of analytic results of the proposed model.
TL;DR: In this article, the dynamics of a discrete-time predator-prey system is investigated in detail, and it is shown that the system undergoes flip bifurcation and Hopf bifurbation by using center manifold theorem and bifurlcation theory.
Abstract: The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto's chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.
TL;DR: In this paper, a delayed computer virus spreading model in the network with limited anti-virus ability is proposed, where local stability and the existence of a Hopf bifurcation are proved by taking the time delay as the bifurlcation parameter and analyzing the distribution of the roots of the corresponding characteristic equation.
Abstract: A delayed computer virus spreading model in the network with limited anti-virus ability is proposed in the present paper. Local stability and the existence of a Hopf bifurcation are proved by taking the time delay as the bifurcation parameter and analyzing the distribution of the roots of the corresponding characteristic equation. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, a numerical example is presented to demonstrate our obtained results.
TL;DR: In this paper, the delays due to gestation of two kinds of zooplankton as parameters, the dynamics of a two-Zoobankton-phytop-layer model is studied, and the Hopf bifurcation about condition (5) should be studied by center manifold theorem and normal form.
Abstract: We take the delays due to gestation of two kinds of zooplankton as parameters, the dynamics of a two zooplankton-phytoplankton model is studied, we discussed the dynamics under six conditions: (1) τ 1 = τ 2 = 0 , (2) τ 1 > 0 , τ 2 = 0 , (3) τ 1 = 0 , τ 2 > 0 , (4) τ 1 = τ 2 > 0 , (5) τ1 ∈ (0, τ10), τ2 > 0, (6) τ2 ∈ (0, τ20), τ1 > 0, the Hopf bifurcation about condition (5) should be studied by center manifold theorem and normal form. At last, some simulations are given to support our results
TL;DR: In this paper, the authors set up a framework for center manifold reduction in fundamental networks and their quotients, and used this machinery to explain the difference in generic bifurcations between three example networks with identical spectral properties and identical robust synchrony spaces.
Abstract: Dynamical systems with a network structure can display anomalous bifurcations as a generic phenomenon. As an explanation for this it has been noted that homogeneous networks can be realized as quotient networks of so-called fundamental networks. The class of admissible vector fields for these fundamental networks is equal to the class of equivariant vector fields of the regular representation of a monoid. Using this insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to explain the difference in generic bifurcations between three example networks with identical spectral properties and identical robust synchrony spaces.
Abstract: In this paper, a delayed-diffusive predator–prey model with hyperbolic mortality and nonlinear prey harvesting subject to the homogeneous Neumann boundary conditions is investigated. Firstly, the global asymptotic stability of the unique positive constant equilibrium is obtained by an iteration technique. Secondly, regarding time delay as a bifurcation parameter and using the normal form theory and center manifold theorem, the existence, stability and direction of bifurcating periodic solutions are demonstrated, respectively. Finally, numerical simulations are conducted to illustrate the theoretical analysis.
TL;DR: In this paper, the authors extend the classical Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems subjected to a time-periodic perturbation.
Abstract: In this paper, we extend the classical Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth system subjected to a time-periodic perturbation. In this class, we suppose there exists a switching manifold with a more general form such that the plane is divided into two zones, and the dynamics in each zone is governed by a smooth system. Furthermore, we assume that the unperturbed system is a general planar piecewise-smooth system with non-zero trace and possesses a piecewise-smooth homoclinic orbit transversally crossing the switching manifold. We also define a reset map to describe the instantaneous impact rule on the switching manifold when a trajectory arrives at the switching manifold. Through a series of geometrical analysis and perturbation techniques, we obtain a Melnikov-type function to measure the separation of the unstable manifold and stable manifold under the effect of the time-periodic perturbations and the reset map. Finally, we use the presented Melnikov function to study global bifurcations and chaotic dynamics for a concrete planar piecewise-linear oscillator.
TL;DR: This paper deals with a competitor-competitor-mutualist Lotka-Volterra model and several concrete formulae determine the properties of bifurcating periodic solutions by applying the normal form theory and the center manifold principle.
Abstract: This paper deals with a competitor-competitor-mutualist Lotka-Volterra model. A series of sufficient criteria guaranteeing the stability and the occurrence of Hopf bifurcation for the model are obtained. Several concrete formulae determine the properties of bifurcating periodic solutions by applying the normal form theory and the center manifold principle. Computer simulations are given to support the theoretical predictions. At last, biological meaning and a conclusion are presented.
TL;DR: In this article, the authors investigated the dynamics of a time-delay ratio-dependent predator-prey model with stage structure for the predator and showed that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions.
Abstract: In this paper, we investigate the dynamics of a time-delay ratio-dependent predator-prey model with stage structure for the predator. This predator-prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang.[26] We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.
TL;DR: In this article, an algorithm for determining the existence of Hopf bifurcation of a system of delayed reaction-diffusion equations with the Neumann boundary conditions is presented.
Abstract: We present an algorithm for determining the existence of a Hopf bifurcation of a system of delayed reaction–diffusion equations with the Neumann boundary conditions. The conditions on parameters of the system that a Hopf bifurcation occurs as the delay parameter passes through a critical value are determined. These conditions depend on the coefficients of the characteristic equation corresponding to linearization of the system. Furthermore, an algorithm to obtain the formulas for determining the direction of the Hopf bifurcation, the stability, and period of the periodic solution is given by using the Poincare normal form and the center manifold theorem. Finally, we give several examples and some numerical simulations to show the effectiveness of the algorithm proposed.
TL;DR: In this article, the existence of quasiperiodic solutions of the KAM problem on R N + 1 of the form (1) Δ x u + u y y + g ( x, u ) = 0, ( x, y ) ∈ R N × R, where g (x, u ) is a sufficiently regular function with g ( ⋅, 0 ) ≡ 0.
TL;DR: In this paper, a discrete stage-structured and harvested predator-prey model is established, which is based on a predator-predator model with Type III functional response, and theoretical methods are used to investigate existence of equilibria and local properties.
Abstract: First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of R+3, by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically compute...
TL;DR: In this article, the authors considered the dynamics of a delayed diffusive predator-prey model with herd behavior and hyperbolic mortality under Neumann boundary conditions and derived the stability of the positive equilibria and the existence of Hopf bifurcations induced by delay.
Abstract: In this paper, we consider the dynamics of a delayed diffusive predator-prey model with herd behavior and hyperbolic mortality under Neumann boundary conditions. Firstly, by analyzing the characteristic equations in detail and taking the delay as a bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations induced by delay are investigated. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Finally, some numerical simulations are also carried out and we obtain the unstable spatial periodic solutions, which are induced by the subcritical Hopf bifurcation.
TL;DR: A locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits a positive automorphic Kahler potential and can be embedded into a Hopf manifold if its dimension is at least 3 as discussed by the authors.
Abstract: A locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits a positive automorphic Kahler potential. A compact LCK manifold with potential can be embedded into a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. We provide an alternative argument for compact complex surfaces, deducing the embedding theorem from the Spherical Shell Conjecture.
TL;DR: In this article, the authors consider a one-dimensional half-space problem for a system of viscous conservation laws which is deduced to a symmetric hyperbolic-parabolic system under assuming that the system has a strictly convex entropy function.
Abstract: In this paper, we consider a one-dimensional half-space problem for a system of viscous conservation laws which is deduced to a symmetric hyperbolic–parabolic system under assuming that the system has a strictly convex entropy function. We firstly prove existence of a stationary solution by assuming that a boundary strength is sufficiently small. The existence of the stationary solution is characterized by the number of negative characteristics. In the case where one characteristic speed is zero at spatial asymptotic state x →∞, we assume that the characteristic field corresponding to the characteristic speed 0 is genuinely nonlinear in order to show existence of a degenerate stationary solution with the aid of a center manifold theory. We next prove that the stationary solution is time asymptotically stable under a smallness assumption on an initial perturbation in the Sobolev space. The key to proof is to derive the uniform a priori estimates by using the energy method, where the stability condition of ...