IMPROVED DETECTION OF BIFURCATIONS IN LARGE NONLINEAR SYSTEMS VIA THE CONTINUATION OF INVARIANT SUBSPACES ALGORITHM

2001 ◽  
Vol 11 (08) ◽  
pp. 2277-2285 ◽  
Author(s):  
M. J. FRIEDMAN
Keyword(s):  
Nonlinear Systems ◽  
Similarity Transformation ◽  
Invariant Subspaces ◽  
Iterative Refinement ◽  
Test Functions ◽  
Triangular Form ◽  
Reliable Detection ◽  
Parameter Dependent ◽  
Orthogonal Similarity Transformation

The Continuation of Invariant Subspaces (CIS) algorithm [Demmel et al., 2001; Dieci & Friedman, 2001] produces a smooth orthogonal similarity transformation to block triangular form of a parameter dependent matrix A(s). The CIS algorithm is an adaptation of an iterative refinement technique for improving the accuracy of computed invariant subspaces. On the basis of the CIS algorithm, new test functions are defined for an improved, more reliable detection of bifurcations in problems with simple bifurcations near multiple bifurcations, problems with multiple bifurcations, and problems with symmetries. Illustrative examples are given.

ON THE LOCATION AND CONTINUATION OF HOPF BIFURCATIONS IN LARGE-SCALE PROBLEMS

2008 ◽  
Vol 18 (05) ◽  
pp. 1589-1597 ◽  
Author(s):  
M. FRIEDMAN ◽  
W. QIU
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Orthonormal Basis ◽  
Invariant Subspaces ◽  
Hopf Bifurcations ◽  
Augmented System ◽  
Large Scale Problems ◽  
System Technique ◽  
Matlab Package ◽  
Parameter Dependent

CL_MATCONT is a MATLAB package for the study of dynamical systems and their bifurcations. It uses a minimally augmented system for continuation of the Hopf curve. The Continuation of Invariant Subspaces (CIS) algorithm produces a smooth orthonormal basis for an invariant subspace [Formula: see text] of a parameter-dependent matrix A(s). We extend a minimally augmented system technique for location and continuation of Hopf bifurcations to large-scale problems using the CIS algorithm, which has been incorporated into CL_MATCONT. We compare this approach with using a standard augmented system and show that a minimally augmented system technique is more suitable for large-scale problems. We also suggest an improvement of a minimally augmented system technique for the case of the torus continuation.

Uniform modeling of parameter dependent nonlinear systems

2012 ◽  
Vol 13 (11) ◽  
pp. 850-858 ◽  
Author(s):  
Najmeh Eghbal ◽  
Naser Pariz ◽  
Ali Karimpour
Keyword(s):  
Nonlinear Systems ◽  
Parameter Dependent

scholarly journals Some Theorems On Matrices With Real Quaternion Elements

1955 ◽  
Vol 7 ◽  
pp. 191-201 ◽  
Author(s):  
N. A. Wiegmann
Keyword(s):  
Similarity Transformation ◽  
Elementary Divisor ◽  
Quaternion Matrix ◽  
Unitary Similarity ◽  
Triangular Form ◽  
Quaternion Matrices ◽  
Divisor Theory ◽  
Real Quaternion ◽  
Unitary Similarity Transformation

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.

Triangular representations of linear algebras

1953 ◽  
Vol 49 (4) ◽  
pp. 595-600 ◽  
Author(s):  
M. P. Drazin
Keyword(s):  
Commutative Algebra ◽  
Similarity Transformation ◽  
Closed Field ◽  
Triangular Form ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Common Eigenvector ◽  
General Systems ◽  
The Matrix ◽  
Matrix Case

It is well known that the elements of any given commutative algebra (and hence of any commutative set) of n × n matrices, over an algebraically closed field K, have a common eigenvector over K; indeed, the elements of such an algebra can be simultaneously reduced to triangular form (by a suitable similarity transformation). McCoy (5) has shown that a triangular reduction is always possible even for matrix algebras satisfying a condition substantially weaker than commutativity. Our aim in this note is to extend these results to more general systems (our arguments being, incidentally, simpler than some used for the matrix case even by writers subsequent to McCoy).

Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Nonlinear Systems ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Design Procedure ◽  
Singular Matrix ◽  
Triangular Form ◽  
Nonlinear Observers ◽  
Numerical Example ◽  
Lower Triangular ◽  
Necessary And Sufficient

AbstractPerfect nonlinear fractional descriptor observers for fractional descriptor continuous-time nonlinear systems are proposed. Necessary and sufficient conditions for the existence of the observers are established. The design procedure of the nonlinear fractional observers is given. It is based on the elementary row (column) operations and reducing the singular matrix of the system to upper (lower) triangular form. The effectiveness of the procedure is demonstrated on a numerical example.

Adaptive Controller and Observer Design for a Class of Nonlinear Systems

2005 ◽  
Vol 128 (3) ◽  
pp. 712-717 ◽  
Author(s):  
Yongliang Zhu ◽  
Prabhakar R. Pagilla
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Lyapunov Function ◽  
Adaptive Controller ◽  
Observer Design ◽  
Dynamic Friction ◽  
Unknown Parameters ◽  
Single Link ◽  
Suitable Form ◽  
Parameter Dependent

Design of a stable adaptive controller and observer for a class of nonlinear systems that contain product of unmeasurable states and unknown parameters is considered. The nonlinear system is cast into a suitable form based on which a stable adaptive controller and observer are designed using a parameter dependent Lyapunov function. The class of nonlinear systems considered is practically relevant; mechanical systems with dynamic friction fall into this category. Experimental results on a single-link mechanical system with dynamic friction are shown for the proposed design.

Symbolic Bifurcation Boundaries and Time-Dependent Resonance Sets of Time-Periodic Nonlinear Systems

1999 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Parameter Space ◽  
Local Stability ◽  
Degree Of Freedom ◽  
Time Dependent ◽  
Periodic Systems ◽  
Stability Boundaries ◽  
System Parameters ◽  
Parameter Dependent ◽  
Time Periodic

Abstract A recent computational technique is utilized for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Because this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. In addition, the time-dependent normal forms and resonance sets for one and two degree-of-freedom time-periodic nonlinear systems are analyzed. For this purpose, the Liapunov-Floquet (L-F) transformation is employed which transforms the periodic variational equations into an equivalent form in which the linear system matrix is constant. Both quadratic and cubic nonlinearities are investigated, and all possible cases for the single degree-of-freedom case are studied. The above algorithm for computing stability boundaries may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

THE PARAMETER-DEPENDENT FERMION STATES FOR MOLECULES WITH SYMPLECTIC SYMMETRY

2006 ◽  
Vol 05 (04) ◽  
pp. 779-799
Author(s):  
C. C. SUN ◽  
B. F. LI ◽  
Z. S. LI ◽  
H. X. ZHANG ◽  
X. R. HUANG
Keyword(s):  
Quantum Chemistry ◽  
Explicit Form ◽  
Symplectic Group ◽  
Similarity Transformation ◽  
Optimization Procedure ◽  
Complete Classification ◽  
Branching Rule ◽  
Parameter Dependent

Under a certain kind of similarity transformation, a parameter-dependent (abbreviated as PD) symplectic group chain Sp(2M) ⊃ Sp(2M - 2) ⊃ ⋯ ⊃ Sp(2) that is characterized by a set of pairing parameters is introduced to build up the PD antisymmetrized fermion states for molecules with symplectic symmetry, and these states will be useful in carrying out the optimization procedure in quantum chemistry. In order to make a complete classification of the states, a generalized branching rule associated with the symplectic group chain is proposed. Further, we are led to the result that the explicit form of the PD antisymmetrized fermion states is obtained in terms of M one-particle operators and M geminal operators.

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