Inertia sets allowed by matrix patterns

2018 ◽  
Vol 34 ◽  
pp. 343-355 ◽  
Author(s):  
Adam Berliner ◽  
Dale Olesky ◽  
Pauline Van den Driessche
Keyword(s):  
Dynamical Systems ◽  
Sufficient Condition ◽  
Sign Pattern ◽  
Zero Eigenvalue ◽  
Sign Patterns

Motivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias $\sn_n$ and $\SN{n}$ for sign and zero-nonzero patterns, respectively, are introduced. For an $n\times n$ sign pattern $\mc{A}$ that allows inertia $(0,n-1,1)$, a sufficient condition is given for $\mc{A}$ and every superpattern of $\mc{A}$ to allow $\sn_n$, and a family of such irreducible sign patterns for all $n\geq 3$ is specified. All zero-nonzero patterns (up to equivalence) that allow $\SN{3}$ and $\SN{4}$ are determined, and are described by their associated digraphs.

INTERTWINED BASINS OF ATTRACTION

2012 ◽  
Vol 22 (06) ◽  
pp. 1250130
Author(s):  
CHANGMING DING
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Basins Of Attraction ◽  
General Definition ◽  
Sufficient Condition ◽  
Topological Equivalence ◽  
Definition Of

This paper deals with intertwined basins of attraction for dynamical systems in a metric space. After giving a general definition of intertwining property, which is preserved by a topological equivalence between dynamical systems, we present a sufficient condition to guarantee the existence of intertwined basins for dynamical systems in ℝn.

scholarly journals Sign patterns that require eventual exponential nonnegativity

2015 ◽  
Vol 30 ◽  
Author(s):  
Craig Erickson
Keyword(s):  
Sign Pattern ◽  
Sign Patterns ◽  
Necessary And Sufficient

Sign patterns that require exponential nonnegativity are characterized. A set of conditions necessary for a sign pattern to require eventual exponential nonnegativity are established. It is shown that these conditions are also sufficient for an upper triangular sign pattern to require eventual exponential nonnegativity and it is conjectured that these conditions are both necessary and sufficient for any sign pattern to require eventual exponential nonnegativity. It is also shown that the maximum number of negative entries in a sign pattern that requires eventual exponential nonnegativity is (n−1)(n−2)/2 + 2

Potentially eventually positive star sign patterns

2016 ◽  
Vol 31 ◽  
pp. 541-548
Author(s):  
Yu Ber-Lin ◽  
Huang Ting-Zhu ◽  
Jie Cui ◽  
Deng Chunhua
Keyword(s):  
Positive Integer ◽  
Necessary Conditions ◽  
Positive Sign ◽  
Real Matrix ◽  
Sign Pattern ◽  
Positive Real ◽  
Sign Patterns

An $n$-by-$n$ real matrix $A$ is eventually positive if there exists a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is potentially eventually positive (PEP) if there exists an eventually positive real matrix $A$ with the same sign pattern as $\mathcal{A}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is a minimal potentially eventually positive sign pattern (MPEP sign pattern) if $\mathcal{A}$ is PEP and no proper subpattern of $\mathcal{A}$ is PEP. Berman, Catral, Dealba, et al. [Sign patterns that allow eventual positivity, {\it ELA}, 19(2010): 108-120] established some sufficient and some necessary conditions for an $n$-by-$n$ sign pattern to allow eventual positivity and classified the potentially eventually positive sign patterns of order $n\leq 3$. However, the identification and classification of PEP signpatterns of order $n\geq 4$ remain open. In this paper, all the $n$-by-$n$ PEP star sign patterns are classified by identifying all the MPEP star sign patterns.

scholarly journals Safe Reachability Verification of Nonlinear Switched Systems via a Barrier Density

2019 ◽  
Author(s):  
Aysegul Kıvılcım ◽  
Ozkan Karabacak ◽  
Rafael Wisniewski
Keyword(s):  
Dynamical Systems ◽  
Density Function ◽  
Switched Systems ◽  
Autonomous Systems ◽  
Sufficient Condition ◽  
Temporal Properties ◽  
Nonlinear Switched Systems ◽  
Arbitrary Switching ◽  
Initial States ◽  
Set Of Initial States

One of the notable temporal properties of dynamical systems is that a set of initial states leads the solutions to reach desired states avoiding a predetermined unsafe set.This property, that we call safe reachability has been studied in literature for autonomous systems using Barrier functionand Barrier densities [1]. In this paper, we generalize a sufficient condition for safe reachability of autonomous systemto switched systems under arbitrary switching signals. The condition relies upon the existence of a common Barrier density function for each subsystem. We apply the condition using the sum of squares method together with Putinar Positivstellensatz.

scholarly journals On the infinite Borwein product raised to a positive real power

2021 ◽  
Author(s):  
Michael J. Schlosser ◽  
Nian Hong Zhou
Keyword(s):  
Asymptotic Formula ◽  
Circle Method ◽  
Sign Pattern ◽  
Real Numbers ◽  
Positive Real ◽  
Infinite Products ◽  
Real Power ◽  
Sign Patterns ◽  
Divisibility Properties ◽  
Made In

AbstractIn this paper, we study properties of the coefficients appearing in the q-series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For $$p=3$$ p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $$\delta $$ δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $$p=3$$ p = 3 case.

scholarly journals Controllability of Nonlinear Fractional Dynamical Systems with a Mittag–Leffler Kernel

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2139
Author(s):  
Jiale Sheng ◽  
Wei Jiang ◽  
Denghao Pang ◽  
Sen Wang
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Fixed Point Theorem ◽  
Laplace Transform ◽  
Point Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

This paper is concerned with controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel. First, the solution of fractional dynamical systems with a Mittag–Leffler kernel is given by Laplace transform. In addition, one necessary and sufficient condition for controllability of linear fractional dynamical systems with Mittag–Leffler kernel is established. On this basis, we obtain one sufficient condition to guarantee controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel by fixed point theorem. Finally, an example is given to illustrate the applicability of our results.

scholarly journals A LIMIT THEOREM FOR OCCUPATION MEASURES OF LÉVY PROCESSES IN COMPACT GROUPS

2012 ◽  
Vol 13 (01) ◽  
pp. 1250008
Author(s):  
ARNO BERGER ◽  
STEVEN N. EVANS
Keyword(s):  
Limit Theorem ◽  
Dynamical Systems ◽  
Compact Group ◽  
Short Proof ◽  
Compact Groups ◽  
Sufficient Condition ◽  
Occupation Measure ◽  
Necessary And Sufficient Condition ◽  
Occupation Measures ◽  
Necessary And Sufficient

A short proof utilizing dynamical systems techniques is given of a necessary and sufficient condition for the normalized occupation measure of a Lévy process in a metrizable compact group to be asymptotically uniform with probability one.

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