A dynamical system method for solving nonlinear ill-posed problems

2008 ◽  
Vol 197 (1) ◽  
pp. 399-406 ◽  
Author(s):  
Li Li ◽  
Bo Han
Keyword(s):  
Dynamical System ◽  
System Method ◽  
Ill Posed ◽  
Dynamical System Method

scholarly journals Separation Method of Semi-Fixed Variables Together with Dynamical System Method for Solving Nonlinear Time-Fractional PDEs with Higher-Order Terms

2021 ◽  
Author(s):  
Weiguo Rui
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
Exact Solutions ◽  
Fractional Order ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
System Method ◽  
Higher Order Terms ◽  
Fractional Models ◽  
Dynamical System Method

Abstract It is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. In this paper, a new approach which combined with the separation method of semi-fixed variables and dynamical system method is introduced. As example, a time-fractional reaction-diffusion equation with higher-order terms is studied under the different kinds of fractional-order differential operators. In different parametric regions, phase portraits of systems which derived from the reaction-diffusion equation are presented. Existence and dynamic properties of solutions of this nonlinear time-fractional models are investigated. In some special parametric conditions, some exact solutions of this time-fractional models are obtained. The dynamical properties of some exact solutions are discussed and the graphs of them are illustrated.PACS: 02.30.Jr; 02.30.Oz; 02.70.-c; 02.70.Mv; 02.90.+p; 04.20.Jb; 05.10.-a

scholarly journals Anisotropic power-law inflation for a model of two scalar and two vector fields

2021 ◽  
Vol 81 (6) ◽  
Author(s):  
Tuan Q. Do ◽  
W. F. Kao
Keyword(s):  
Dynamical System ◽  
Power Law ◽  
Vector Fields ◽  
Bianchi Type ◽  
Scalar Fields ◽  
Field Extension ◽  
Type I ◽  
Anisotropic Solution ◽  
System Method ◽  
Dynamical System Method

AbstractInspired by an interesting counterexample to the cosmic no-hair conjecture found in a supergravity-motivated model recently, we propose a multi-field extension, in which two scalar fields are allowed to non-minimally couple to two vector fields, respectively. This model is shown to admit an exact Bianchi type I power-law solution. Furthermore, stability analysis based on the dynamical system method is performed to show that this anisotropic solution is indeed stable and attractive if both scalar fields are canonical. Nevertheless, if one of the two scalar fields is phantom then the corresponding anisotropic power-law inflation turns unstable as expected.

The dynamic system method and the traps

1998 ◽  
Vol 30 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Odile Brandière
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Dynamic System ◽  
Stochastic Algorithms ◽  
System Method ◽  
Dynamical System Method ◽  
Unstable Direction ◽  
Differential Equation Method

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.

scholarly journals No-go theorem for inflation in Ricci-inverse gravity

2021 ◽  
Vol 81 (5) ◽  
Author(s):  
Tuan Q. Do
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Fourth Order ◽  
Singularity Problem ◽  
System Method ◽  
Dynamical System Method

AbstractIn this paper, we study the so-called Ricci-inverse gravity, which is a very novel type of fourth-order gravity proposed recently. In particular, we are able to figure out both isotropically and anisotropically inflating universes to this model. More interestingly, these solutions are shown to be free from a singularity problem. However, stability analysis based on the dynamical system method shows that both isotropic and anisotropic inflation of this model turn out to be unstable against field perturbations. This result implies a no-go theorem for both isotropic and anisotropic inflation in the Ricci-inverse gravity.

scholarly journals Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hang Zheng ◽  
Yonghui Xia ◽  
Yuzhen Bai ◽  
Guo Lei
Keyword(s):  
Dynamical System ◽  
Equilibrium Points ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
System Method ◽  
Dynamical System Method ◽  
Nonzero Equilibrium ◽  
Derivative Order

In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order α. Phase portraits and simulations are given to show the validity of the obtained solutions.

scholarly journals No-go theorem for inflation in an extended Ricci-inverse gravity model

2022 ◽  
Vol 82 (1) ◽  
Author(s):  
Tuan Q. Do
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Gravity Model ◽  
Fourth Order ◽  
Order Term ◽  
Second Order ◽  
System Method ◽  
Walker Metric ◽  
Dynamical System Method

AbstractIn this paper, we propose an extension of the Ricci-inverse gravity, which has been proposed recently as a very novel type of fourth-order gravity, by introducing a second order term of the so-called anticurvature scalar as a correction. The main purpose of this paper is that we would like to see whether the extended Ricci-inverse gravity model admits the homogeneous and isotropic Friedmann–Lemaitre–Robertson–Walker metric as its stable inflationary solution. However, a no-go theorem for inflation in this extended Ricci-inverse gravity is shown to appear through a stability analysis based on the dynamical system method. As a result, this no-go theorem implies that it is impossible to have such stable inflation in this extended Ricci-inverse gravity model.

scholarly journals Stable small spatial hairs in a power-law k-inflation model

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
Tuan Q. Do
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Electromagnetic Fields ◽  
Power Law ◽  
Bianchi Type ◽  
Type I ◽  
Inflation Model ◽  
System Method ◽  
Inflationary Phase ◽  
Dynamical System Method

AbstractIn this paper, we extend our investigation of the validity of the cosmic no-hair conjecture within non-canonical anisotropic inflation. As a result, we are able to figure out an exact Bianchi type I solution to a power-law k-inflation model in the presence of unusual coupling between scalar and electromagnetic fields as $$-f^2(\phi )F_{\mu \nu }F^{\mu \nu }/4$$ - f 2 ( ϕ ) F μ ν F μ ν / 4 . Furthermore, stability analysis based on the dynamical system method indicates that the obtained solution does admit stable and attractive hairs during an inflationary phase and therefore violates the cosmic no-hair conjecture. Finally, we show that the corresponding tensor-to-scalar ratio of this model turns out to be highly consistent with the observational data of the Planck 2018.

The dynamic system method and the traps

1998 ◽  
Vol 30 (01) ◽  
pp. 137-151
Author(s):  
Odile Brandière
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Dynamic System ◽  
Stochastic Algorithms ◽  
System Method ◽  
Dynamical System Method ◽  
Unstable Direction ◽  
Differential Equation Method

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.

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