Page 633 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 633
DYNAMICAL SYSTEMS AND ORDINARY DIFFERENTIAL EQUATIONS AND
APPLICATIONS
Multigrid Fast Sweep Method For Computation of Isostables and
Isochrons
Bojan Crnkovic´, bojan.crnkovic@uniri.hr
University of Rijeka, Croatia
Coauthor: Jerko Skific´
We propose a fast iterative multigrid algorithm for the computation of isostables and isochrons
for dynamical systems with stable limit cycles or fixed points in high dimensions. We solve a
first-order static Hamilton–Jacobi equation with a constant source term using a Eulerian Fast
Sweeping Method developed for this type of problem. We reduce the number of iteration of
the standard Fast Sweeping Method using nested grids and demonstrate the speed up on several
illustrative examples.
Keywords: isostables, isochrons, Hamilton–Jacobi, multigrid
Conditions of global solvability, Lyapunov stability, Lagrange stability
and dissipativity for time-varying semilinear differential-algebraic
equations, and applications
Maria Filipkovska, filipkovskaya@ilt.kharkov.ua
B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine, Ukraine
Theorems on the existence and uniqueness of global solutions, Lagrange stability and insta-
bility, dissipativity (ultimate boundedness), Lyapunov stability and instability, and asymptotic
stability for time-varying semilinear DAEs (nonautonomous degenerate ordinary differential
equations) will be presented, and mathematical models of nonlinear time-varying electrical cir-
cuits will be considered in order to demonstrate the application of the presented theorems. The
features and advantages of the obtained theorems will also be discussed. The talk is based on
the results published in the journals “Differential Equations”, “Global and Stochastic Analysis”,
and “Proceedings of the Institute of Mathematics and Mechanics”.
Non-uniformly hyperbolic dynamics for some classes of piecewise smooth
systems
Sergey Kryzhevich, kryzhevicz@gmail.com
Saint Petersburg State University, Russian Federation
We consider piecewise smooth systems (PWS) of ordinary differential equations. The phase
space is supposed to be split into two (or more) subsets with a threshold being a peicewise
smooth surface. For such systems, smoothly depending on a parameter, the so-called grazing
bifurcation is considered. Roughly speaking, this bifurcation corresponds to the existence of a
periodic solution tangent to the threshold.
Basing on the approaches of previous author’s works and some new ideas on estimating
Lyapunov exponents for near-grazing periodic solutions, we describe non-uniformly hyperbolic
invariant sets for some classes of PWS. In other words, local coexistence of infinitely many
periodic solutions with distinct dimensions of stable/unstable manifolds is proved for those
631
APPLICATIONS
Multigrid Fast Sweep Method For Computation of Isostables and
Isochrons
Bojan Crnkovic´, bojan.crnkovic@uniri.hr
University of Rijeka, Croatia
Coauthor: Jerko Skific´
We propose a fast iterative multigrid algorithm for the computation of isostables and isochrons
for dynamical systems with stable limit cycles or fixed points in high dimensions. We solve a
first-order static Hamilton–Jacobi equation with a constant source term using a Eulerian Fast
Sweeping Method developed for this type of problem. We reduce the number of iteration of
the standard Fast Sweeping Method using nested grids and demonstrate the speed up on several
illustrative examples.
Keywords: isostables, isochrons, Hamilton–Jacobi, multigrid
Conditions of global solvability, Lyapunov stability, Lagrange stability
and dissipativity for time-varying semilinear differential-algebraic
equations, and applications
Maria Filipkovska, filipkovskaya@ilt.kharkov.ua
B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine, Ukraine
Theorems on the existence and uniqueness of global solutions, Lagrange stability and insta-
bility, dissipativity (ultimate boundedness), Lyapunov stability and instability, and asymptotic
stability for time-varying semilinear DAEs (nonautonomous degenerate ordinary differential
equations) will be presented, and mathematical models of nonlinear time-varying electrical cir-
cuits will be considered in order to demonstrate the application of the presented theorems. The
features and advantages of the obtained theorems will also be discussed. The talk is based on
the results published in the journals “Differential Equations”, “Global and Stochastic Analysis”,
and “Proceedings of the Institute of Mathematics and Mechanics”.
Non-uniformly hyperbolic dynamics for some classes of piecewise smooth
systems
Sergey Kryzhevich, kryzhevicz@gmail.com
Saint Petersburg State University, Russian Federation
We consider piecewise smooth systems (PWS) of ordinary differential equations. The phase
space is supposed to be split into two (or more) subsets with a threshold being a peicewise
smooth surface. For such systems, smoothly depending on a parameter, the so-called grazing
bifurcation is considered. Roughly speaking, this bifurcation corresponds to the existence of a
periodic solution tangent to the threshold.
Basing on the approaches of previous author’s works and some new ideas on estimating
Lyapunov exponents for near-grazing periodic solutions, we describe non-uniformly hyperbolic
invariant sets for some classes of PWS. In other words, local coexistence of infinitely many
periodic solutions with distinct dimensions of stable/unstable manifolds is proved for those
631
