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.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Model ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
Approximate System ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Delays

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
The Stability ◽  
The Galerkin Method ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

In this paper, we develop Galerkin approximations for determining the stability of delay differential equations (DDEs) with time periodic coefficients and time periodic delays. Using a transformation, we convert the DDE into a partial differential equation (PDE) along with a boundary condition (BC). The PDE and BC we obtain have time periodic coefficients. The PDE is discretized into a system of ordinary differential equations (ODEs) using the Galerkin method with Legendre polynomials as the basis functions. The BC is imposed using the tau method. The resulting ODEs are time periodic in nature; thus, we resort to Floquet theory to determine the stability of the ODEs. We show through several numerical examples that the stability charts obtained from the Galerkin method agree closely with those obtained from direct numerical simulations.

Period Doubling Bifurcation and Center Manifold Reduction in a Time-periodic and Time-delayed Model of Machining

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1169-1187 ◽  
Author(s):  
Róbert Szalai ◽  
Gábor Stépán
Keyword(s):  
Center Manifold ◽  
Period Doubling ◽  
Characteristic Parameter ◽  
Period Doubling Bifurcation ◽  
Numerical Continuation ◽  
Machining Processes ◽  
Center Manifold Reduction ◽  
Periodic Delay ◽  
Delay Differential ◽  
Time Periodic

A closed-form calculation is presented for the analysis of the period-doubling bifurcation in the time-periodic delay-differential equation model of interrupted machining processes such as milling where the nonlinearity is essentially nonsymmetric. We prove the subcritical sense of this period-doubling bifurcation and approximate the emerging period-two oscillations by the Lyapunov—Perron method for computing the center manifold and by calculating the Poincaré—Lyapunov constant of the bifurcation analytically at certain characteristic parameter values. The existence of the unstable period-two oscillations around the stable stationary cutting is confirmed using a numerical continuation algorithm developed for time-periodic delay-differential equations.

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.

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