Alexander P. Seyranian

Multiparameter stability theory with mechanical applications

Fundamentals of Stability Theory Bifurcation Analysis of Eigenvalues Stability Boundary of a General System Depending on Parameters Bifurcation Analysis of Roots and Stability of a Characteristic Polynomial Depending on Parameters Vibrations and Stability of a Conservative System Depending on Parameters Stability of a Linear Hamiltonian System Depending on Parameters Stability of Linear Gyroscopic Systems Depending on Parameters Mechanical Effects Related to Bifurcation of Eigenvalues and Singularities of the Stability Boundary Stability of Periodic Systems Depending on Parameters Stability Boundary of a General Periodic System Depending on Parameters Instability Domains of Oscillatory Systems with Small Parametric Excitation and Damping Stability Domains of Nonconservative Systems Under Small Parametric Excitation.

Sensitivity analysis for problems of dynamic stability

Abstract In mechanics, as well as in physics, the most general and important thing is to study the dependence of the characteristics of a physical process on problem parameters. Problems of dynamic stability for non-conservative systems involve determination of eigenvalues and eigenvectors. For these problems it is shown in general how the different sensitivity analyses can be performed without any new eigenvalue analyses. The main question relates to the change in flutter load as a function of change in stiffness, mass, boundary conditions, or load distribution. Discretized as well as non-discretized examples are presented in details.

Coupling of eigenvalues of complex matrices at diabolic and exceptional points

The paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.

Dynamics of the pendulum with periodically varying length

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of a childs swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with a numerical study. Chaotic motions of the pendulum depending on problem parameters are investigated numerically.

Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation

The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied.

Geometric phase around exceptional points

A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly {pi} for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to {pi} for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.

On gyroscopic stabilization

The mechanisms of transition between divergence, flutter, and stability for a class of conservative gyroscopic systems with parameters are studied. Two results are obtained which state sufficient conditions for gyroscopic stabilization of conservative systems with an even dimension and a negative definite stiffness matrix. A number of examples are given to demonstrate the feasibility of the results.

On Singularities of a Boundary of the Stability Domain

This paper deals with the study of generic singularities of a boundary of the stability domain in a parameter space for systems governed by autonomous linear differential equations

Collapse of the Keldysh Chains and Stability of Continuous Nonconservative Systems

\dot y=Ay

Sensitivity analysis and optimization of aeroelastic stability

or x(m) + a1x(m-1) + . . . + amx=0. It is assumed that elements of the matrix A and coefficients of the differential equation of mth order smoothly depend on one, two, or three real parameters. A constructive approach allowing the geometry of singularities (orientation in space, magnitudes of angles, etc.) to be determined with the use of tangent cones to the stability domain is suggested. The approach allows the geometry of singularities to be described using only first derivatives of the coefficients ai of the differential equation or first derivatives of the elements of the matrix A with respect to problem parameters with its eigenvectors and associated vectors calculated at the singular points of the boundary. Two methods of study of singularities are suggested. It is shown that they are constructive and can be applied to investigate more complicated singularities for multiparameter families of matrices or polynomials. Two physical examples are presented and discussed in detail.