scholarly journals Reduction principle at work

2021 ◽  
Vol 8 (1) ◽  
pp. 46-74
Author(s):  
Christian Pötzsche ◽  
Evamaria Russ
Keyword(s):  
Center Manifold ◽  
Reduction Principle ◽  
Critical Stability ◽  
Center Manifold Reduction ◽  
Infinite Dimensional ◽  
Linearized Stability ◽  
Dichotomy Spectrum ◽  
Nonautonomous Case ◽  
Necessary And Sufficient ◽  
Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

scholarly journals Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jinbin Wang ◽  
Rui Zhang ◽  
Lifenq Ma
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Two Dimensional ◽  
Main Attention ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Double Hopf Bifurcation ◽  
Physical Phenomena ◽  
Manifold Reduction ◽  
Theoretical Results

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

2020 ◽  
Vol 30 (09) ◽  
pp. 2050130 ◽  
Author(s):  
Shangzhi Li ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Banach Spaces ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Delay Effect ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Equivariant Hopf Bifurcation ◽  
Manifold Reduction

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

Center-manifold reduction for laser equations with detuning

1989 ◽  
Vol 40 (3) ◽  
pp. 1422-1427 ◽  
Author(s):  
Gian-Luca Oppo ◽  
Antonio Politi
Keyword(s):  
Center Manifold ◽  
Center Manifold Reduction ◽  
Manifold Reduction

scholarly journals Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Massimiliano Ferrara ◽  
Luca Guerrini ◽  
Giovanni Molica Bisci
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Method ◽  
Production Function ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

scholarly journals Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

2014 ◽  
Vol 24 (04) ◽  
pp. 1450057 ◽  
Author(s):  
Yu. A. Kuznetsov ◽  
H. G. E. Meijer ◽  
B. Al Hdaibat ◽  
W. Govaerts
Keyword(s):  
Normal Form ◽  
Center Manifold ◽  
Blow Up ◽  
Center Manifold Reduction ◽  
Parameter Transformation ◽  
Homological Equation ◽  
Standard Parameter ◽  
Parameter Dependent ◽  
Manifold Reduction ◽  
Fredholm Solvability

An improved homoclinic predictor at a generic codim 2 Bogdanov–Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.

scholarly journals A center manifold application: existence of periodic travelling waves for the 2D abcd-Boussinesq system

2017 ◽  
Vol 11 (12) ◽  
pp. 641-667
Author(s):  
Jose R. Quintero ◽  
Alex M. Montes
Keyword(s):  
Center Manifold ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Type System ◽  
Boussinesq System ◽  
Wave Regime ◽  
Center Manifold Reduction ◽  
Small Initial Data ◽  
Weakly Nonlinear ◽  
Infinite Dimensional

In this paper we study the existence of periodic travelling waves for the 2D abcd Boussinesq type system related with the three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. Small solutions that are periodic in the direction of translation form an infinite-dimensional family, by characterizing them using a center manifold reduction of infinite dimension and codimension due to the fact that at the linear level we are dealing with an ill-posed mixed-type initial-value problem. As happens for the Benney-Luke model and the KP II model for wave speed large enough and large surface tension, we show that a unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). As a consequence of this fact, we show that the spatial evolution of bottom velocity is governed by a dispersive, nonlocal, nonlinear wave equation.

Center Manifold Reduction of Periodic Delay Differential Systems

2007 ◽  
Author(s):  
Eric A. Butcher ◽  
Venkatesh Deshmukh ◽  
Ed Bueler
Keyword(s):  
Differential Equations ◽  
Center Manifold ◽  
Perturbation Expansion ◽  
Flip Bifurcation ◽  
Restoring Force ◽  
Center Manifold Reduction ◽  
Periodic Coefficients ◽  
Delay Differential ◽  
Manifold Reduction ◽  
Time Periodic

A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.

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