Rigid centres on the center manifold of tridimensional differential systems

2021 ◽  
pp. 1-23
Author(s):  
Adam Mahdi ◽  
Claudio Pessoa ◽  
Jarne D. Ribeiro
Keyword(s):  
Center Manifold ◽  
Center Manifolds ◽  
Differential Systems ◽  
Definition Of

Motivated by the definition of rigid centres for planar differential systems, we introduce the study of rigid centres on the center manifolds of differential systems on $\mathbb {R}^{3}$ . On the plane, these centres have been extensively studied and several interesting results have been obtained. We present results that characterize the rigid systems on $\mathbb {R}^{3}$ and solve the centre-focus problem for several families of rigid systems.

PRACTICAL COMPUTATION OF NORMAL FORMS ON CENTER MANIFOLDS AT DEGENERATE BOGDANOV–TAKENS BIFURCATIONS

2005 ◽  
Vol 15 (11) ◽  
pp. 3535-3546 ◽  
Author(s):  
YU. A. KUZNETSOV
Keyword(s):  
Normal Form ◽  
Numerical Evaluation ◽  
Center Manifold ◽  
Normal Forms ◽  
Center Manifolds ◽  
Linear Transformations ◽  
Double Zero ◽  
Practical Computation ◽  
Generalized Eigenvectors

Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov–Takens (nonsemisimple double-zero) bifurcation for n-dimensional systems with arbitrary n ≥ 2. These formulas are equally suitable for both symbolic and numerical evaluation and allow one to classify all codim 3 Bogdanov–Takens bifurcations in generic multidimensional ODEs. They are also applicable to systems with symmetries. We perform no preliminary linear transformations but use only critical (generalized) eigenvectors of the linearization matrix and its transpose. The derivation combines the approximation of the center manifold with the normalization on it. Three known models are used as test examples to demonstrate advantages of the method.

The Attitude Stability for Longitudinal Motion of Underwater Vehicle

2001 ◽  
Author(s):  
Deshi Wang ◽  
Renbin Xiao ◽  
Ming Yang
Keyword(s):  
Bifurcation Theory ◽  
Center Manifold ◽  
Equilibrium Points ◽  
Autonomous Underwater Vehicles ◽  
Stability Loss ◽  
Underwater Vehicles ◽  
Longitudinal Motion ◽  
Center Manifolds ◽  
One Dimensional ◽  
Attitude Stability

Abstract Although the equations describing the longitudinal motions of underwater vehicles are typically nonlinear, the linearized equations are still employed to design the depth controller by the traditional analysis methods in engineering for the sake of simplicity. The reduction of the nonlinearity loses the dynamics near the singular points, which may be responsible for the sudden climb or dive. The nonlinear systems limited in the longitudinal plane of the underwater vehicles are analyzed on center manifold through the bifurcation theory. It focuses on the case that single zero root in Jacobi matrix occurs at equilibrium points corresponding to nominal trajectory with varied angles of the elevator or the direction change of the flows. The center manifolds are calculated and one-dimensional bifurcation equations on the center manifolds are obtained and analyzed. Based on the transcritical bifurcation diagram, we have found the mechanism of the attitude stability loss as well as the abnormal trajectory of autonomous underwater vehicles. It gives good explainations to the practical climbing jump and diving fall and delivers the theoretical tools to design the controller and to design dynamics. Numerical simulation verifies the results.

AN EXPLICIT RECURSIVE FORMULA FOR COMPUTING THE NORMAL FORM AND CENTER MANIFOLD OF GENERAL n-DIMENSIONAL DIFFERENTIAL SYSTEMS ASSOCIATED WITH HOPF BIFURCATION

2013 ◽  
Vol 23 (06) ◽  
pp. 1350104 ◽  
Author(s):  
YUN TIAN ◽  
PEI YU
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Recursive Formula ◽  
Computationally Efficient ◽  
Differential Systems ◽  
Analytical Formulas

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.

On the Stability and Boundedness of Differential Systems

1962 ◽  
Vol 58 (3) ◽  
pp. 492-496 ◽  
Author(s):  
V. Lakshmikantham
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Real Banach Space ◽  
Closed Linear Operator ◽  
Differential Systems ◽  
The Real ◽  
The Stability ◽  
Definition Of ◽  
Image Position ◽  
The Right

Consider the differential systemswhere A(t), g(t, y) and g(t, y) are operators acting in the real Banach space E, A(t) is an unbounded, closed, linear operator for each t in 0 ≤ t < ∞ and x0, y0 belong to the domain of definition of the operator A (t0). Let ‖x‖ denote the norm of an element x ε: E and R(λ, t) the resolvent of A(t). Here and in the following the prime denotes the right-hand derivative.

scholarly journals Tihonov theory and center manifolds for inhibitory mechanisms in enzyme kinetics

2017 ◽  
Vol 8 (1) ◽  
pp. 81-102
Author(s):  
A. M. Bersani ◽  
A. Borri ◽  
A. Milanesi ◽  
P. Vellucci
Keyword(s):  
Phase Space ◽  
Enzyme Kinetics ◽  
Chemical Reaction ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Numerical Results ◽  
Center Manifolds ◽  
Inhibitory Mechanisms

Abstract In this paper we study the chemical reaction of inhibition, determine the appropriate parameter ε for the application of Tihonov's Theorem, compute explicitly the equations of the center manifold of the system and find sufficient conditions to guarantee that in the phase space the curves which relate the behavior of the complexes to the substrates by means of the tQSSA are asymptotically equivalent to the center manifold of the system. Some numerical results are discussed.

scholarly journals Existence and controllability of higher-order nonlinear fractional integrodifferential systems via resolvent operator

2021 ◽  
Author(s):  
Abdul Haq ◽  
N Sukavanam
Keyword(s):  
Fixed Point ◽  
Mild Solution ◽  
Resolvent Operator ◽  
Approximate Controllability ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Existence Of Solution ◽  
Differential Systems ◽  
Fixed Point Approach ◽  
Definition Of

This work analyzes the existence of solution and approximate controllability for higher order non-linear fractional integro-differential systems with Riemann-Liouville derivatives in Banach spaces. Firstly, the definition of mild solution for the system is derived. Then a set of sufficient conditions for the existence of mild solution and approximate controllability of the system is obtained. The discussions are based on fixed point approach, and the theory of convolution and fractional resolvent. To illustrate the feasibility of developed theory, an example is given.

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