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Janusz Mierczyński

Publications and source records attributed to Janusz Mierczyński.

At least 19 recordsLinked to original sources
arXiv2026

Time-periodic carrying simplex for a competitive system of Carathéodory ODEs

We consider time-periodic competitive systems of ordinary differential equations of Kolmogorov type. However, compared with standard assumptions, we relax the regularity of the time-dependent per-capita growth rates by imposing much weaker regularity, namely Carathéodory conditions. An important tool in investigating such systems is the concept of carrying simplex, that is, of an unordered invariant manifold of codimension one that attracts all nonzero orbits. We define the carrying simplex via the compact attractor of compact sets of an extended flow, and that attractor can be obtained as the limit of the actions of the solution operator on some set. Compared with previous papers, our approach has more dynamical flavour, and, further, provides a method of numerical approximation of the carrying simplex. Another feature of our paper is that we prove that the system restricted to the extended carrying simplex is topologically conjugate to a system of one dimension less. This property, appearing in the path-breaking paper by Morris W. Hirsch, has been almost universally neglected in the later papers.

math.DS↗
arXiv2024

Systems of parabolic equations with delays: Continuous dependence on parameters

Results on continuous dependence on parameters, as well as on regularization, of solutions to linear systems of parabolic partial differential equations of second order with delay are given. One of the main features is that the topology on the zero order and delay coefficients is the weak-* topology on an $L_{\infty}$ space, whereas the assumptions on the second- and first-order coefficients are slightly stronger. This is important in the context of generation of linear skew-product dynamical systems by such equations.

math.AP↗
arXiv2023

Existence of the carrying simplex for a retrotone map

We present a dynamical approach to the study of unordered, attracting manifolds of retrotone maps commonly known as carrying simplices. Our approach is novel in that it uses the radial representation of unordered manifolds over the probability simplex coupled with distances between these manifolds measured by way of the Harnack and Hausdorff metrics. We establish Kuratowski convergence of radial representations of unordered manifolds to a unique function which then provides the locally Lipschitz radial representation of the carrying simplex.

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arXiv2022

Parabolic Differential Equations with Bounded Delay

We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.

math.AP↗
arXiv2019

Linearization and invariant manifolds on the carrying simplex for competitive maps

A folklore result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In particular, the classical Grobman-Hartman theorem can not be used on the carrying simplex. In this paper, we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. On the other hand, our results also give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics. Although we focus on monotone maps, many results of the paper can be applied to maps that admit a non-smooth center-manifold.

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arXiv2018

Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as~well as on a space of $p$-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.

math.DS↗
arXiv2018

The $C^1$ property of convex carrying simplices for competitive maps

For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In the present paper it is shown that its convexity implies that it is a $C^1$ submanifold-with-corners, neatly embedded in the non-negative orthant. The proof uses the characterization of neat embedding in terms of inequalities between Lyapunov exponents for ergodic invariant measures supported on the boundary of the carrying simplex.

math.DS↗
arXiv2017

Instability in linear cooperative systems of ordinary differential equations

It is well known that, contrary to the autonomous case, the stability/instability of solutions of nonautonomous linear ordinary differential equations $x' = A(t) x$ is in no relation to the sign of the real parts of the eigenvalues of $A(t)$. In particular, the real parts of all eigenvalues can be negative and bounded away from zero, nonetheless there is a solution of magnitude growing to infinity. In this paper we present a method of constructing examples of such systems when the matrices $A(t)$ have positive off-diagonal entries (strongly cooperative systems). We illustrate those examples both with interactive animations and analytically. The paper is written in such a way that it can be accessible to students with diverse mathematical backgrounds/skills.

math.CA↗
arXiv2017

Time averaging for nonautonomous/random linear parabolic equations

Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.

math.AP↗
arXiv2017

Lower estimates of top Lyapunov exponent for cooperative random systems of linear ODEs

For cooperative random linear systems of ordinary differential equations a method is presented of obtaining lower estimates of the top Lyapunov exponent. The proofs are based on applying some polynomial Lyapunov-like function. Known estimates for the dominant eigenvalue of a nonnegative matrix due to G. Frobenius and L. Yu. Kolotilina are shown to be specializations of our results.

math.DS↗
arXiv2017

On smoothness of carrying simplices

We consider dissipative strongly competitive systems $\dot{x}_{i}=x_{i}f_{i}(x)$ of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface $Σ$, called the carrying simplex. In this note we give an amenable condition for $Σ$ to be a $C^{1}$ submanifold-with-corners. We also provide conditions, based on a recent work of M. Bena\"ım "On invariant hypersurfaces of strongly monotone maps", J. Differential Equations 137 (1997), 302-319, guaranteeing that $Σ$ is of class $C^{k+1}$.

math.DS↗
arXiv2017

Averaging in random systems of nonnegative matrices

It is proved that for the top Lyapunov exponent of a random matrix system of the form $\{A D(ω)\}$, where $A$ is a nonnegative matrix and $D(ω)$ is a diagonal matrix with positive diagonal entries, is bounded from below by the top Lyapunov exponent of the averaged system. This is in contrast to what one should expect of systems describing biological metapopulations.

math.DS↗
arXiv2017

Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.

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arXiv2016

Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory," Trans. Amer. Math. Soc. 365 (2013), pp. 5329-5365, to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.

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