Mathematics › Geometry and Topology

Advanced Differential Equations and Dynamical Systems

Works Count: 45633

Wikipedia

Description

This cluster of papers focuses on the study of bifurcations in planar polynomial systems, particularly emphasizing piecewise linear structures, limit cycles, Hopf bifurcations, discontinuous systems, nilpotent singularities, Darboux integrability, and Abelian integrals.

Keywords

Bifurcations; Planar Systems; Piecewise Linear; Limit Cycles; Polynomial Vector Fields; Hopf Bifurcation; Discontinuous Systems; Nilpotent Singularity; Darboux Integrability; Abelian Integrals

Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires

1981-01-01

Jean-Michel Bony

chier doit contenir la présente mention de copyright. chier doit contenir la présente mention de copyright.

View PDF Chat

On the dynamics of polynomial-like mappings

1985-01-01

Adrien Douady , John H. Hubbard

Applications de type polynomial. Familles analytiques de telles applications. Resultats negatifs. Familles a un parametre d'applications de degre 2. Petites copies de M dans M. Carrottes pour le dessert Applications de type polynomial. Familles analytiques de telles applications. Resultats negatifs. Familles a un parametre d'applications de degre 2. Petites copies de M dans M. Carrottes pour le dessert

View PDF Chat

Potential theory in modern function theory

1959-01-01

正次辻

Qualitative Theory of Planar Differential Systems

2006-01-01

Freddy Dumortier , Joan Carles Artés Ferragud , Jaume Llibre

Representation theory and automorphic functions

1969-01-01

М. И. Граев , Ilya Piatetski-Shapiro , I. M. Gel'fand +1 more

Nonlinear Control Systems

1989-01-01

Alberto Isidori

Singularities and Groups in Bifurcation Theory

1988-01-01

Martin Golubitsky , Ian Stewart , David G. Schaeffer

Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

1964-01-01

Heisuke Hironaka

Entropy for group endomorphisms and homogeneous spaces

1971-01-01

Rufus Bowen

Topological entropy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript d Baseline left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation … Topological entropy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript d Baseline left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{h_d}(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript d Baseline left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{h_d}(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with measure theoretic entropy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; in particular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis upper T right-parenthesis equals h Subscript d Baseline left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h(T) = {h_d}(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for Haar measure and affine maps <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on compact metrizable groups. A particular case of this yields the well-known formula for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a toral automorphism.

View PDF Chat

A General Theorem on Local Controllability

1987-01-01

H.J. Sussmann

We prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point. Earlier results of Brunovsky, Hermes, Jurdjevic, Crouch and Byrnes, Sussmann and Grossmann, are … We prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point. Earlier results of Brunovsky, Hermes, Jurdjevic, Crouch and Byrnes, Sussmann and Grossmann, are shown to be particular cases of this result. Also, a number of new sufficient conditions are obtained. All these results follow from one simple general principle, namely, that local controllability follows whenever brackets with certain symmetries can be “neutralized,” in a suitable way, by writing them as linear combinations of brackets of a lower degree. Both the class of symmetries and the definition of “degree” can be chosen to suit the problem.

Transformation Groups in Differential Geometry

1972-01-01

Shôshichi Kobayashi

HIDDEN ATTRACTORS IN DYNAMICAL SYSTEMS. FROM HIDDEN OSCILLATIONS IN HILBERT–KOLMOGOROV, AIZERMAN, AND KALMAN PROBLEMS TO HIDDEN CHAOTIC ATTRACTOR IN CHUA CIRCUITS

2013-01-01

Г. А. Леонов , Н. В. Кузнецов

From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, … From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors. At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50–60s of the last century, the investigations of widely known Markus–Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes. Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit. This survey is dedicated to efficient analytical–numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods.

XVI. Functions of positive and negative type, and their connection the theory of integral equations

1909-01-01

James W. Mercer

The present memoir is the outcome of an attempt to obtain the conditions under which a given symmetric and continuous function k ( s, t ) is definite, in the … The present memoir is the outcome of an attempt to obtain the conditions under which a given symmetric and continuous function k ( s, t ) is definite, in the sense of Hilbert. At an early stage, however, it was found that the class of definite functions was too restricted to allow the determination of necessary and sufficient conditions in terms of the determinants of § 10. The discovery that this could be done for functions of positive or negative type, and the fact that almost all the theorems which are true of definite functions are, with slight modification, true of these, led finally to the abandonment of the original plan in favour of a discussion of the properties of functions belonging to the wider classes. The first part of the memoir is devoted to the definition of various terms employed, and to the re-statement of the consequences which follow from Hilbert’s theorem.

Introduction to Riemann Surfaces.

1959-01-01

T. K. Pan , George Ś. Springer

Introduction: 1-1 Algebraic functions and Riemann surfaces 1-2 Plane fluid flows 1-3 Fluid flows on surfaces 1-4 Regular potentials 1-5 Meromorphic functions 1-6 Function theory on a torus General Topology: … Introduction: 1-1 Algebraic functions and Riemann surfaces 1-2 Plane fluid flows 1-3 Fluid flows on surfaces 1-4 Regular potentials 1-5 Meromorphic functions 1-6 Function theory on a torus General Topology: 2-1 Topological spaces 2-2 Functions and mappings 2-3 Manifolds Riemann Surface of an Analytic Function: 3-1 The complete analytic function 3-2 The analytic configuration Covering Manifolds: 4-1 Covering manifolds 4-2 Monodromy theorem 4-3 Fundamental group 4-4 Covering transformations Combinatorial Topology: 5-1 Triangulation 5-2 Barycentric coordinates and subdivision 5-3 Orientability 5-4 Differentiable and analytic curves 5-5 Normal forms of compact orientable surfaces 5-6 Homology groups and Betti numbers 5-7 Invariance of the homology groups 5-8 Fundamental group and first homology group 5-9 Homology on compact surfaces Differentials and Integrals: 6-1 Second-order differentials and surface integrals 6-2 First-order differentials and line integrals 6-3 Stokes' theorem 6-4 The exterior differential calculus 6-5 Harmonic and analytic differentials The Hilbert Space of Differentials: 7-1 Definition and properties of Hilbert space 7-2 Smoothing operators 7-3 Weyl's lemma and orthogonal projections Existence of Harmonic and Analytic Differentials: 8-1 Existence theorems 8-2 Countability of a Riemann surface Uniformization: 9-1 Schlichtartig surfaces 9-2 Universal covering surfaces 9-3 Triangulation of a Riemann surface 9-4 Mappings of a Riemann surface onto itself Compact Riemann Surfaces: 10-1 Regular harmonic differentials 10-2 The bilinear relations of Riemann 10-3 Bilinear relations for differentials with singularities 10-4 Divisors 10-5 The Riemann-Roch theorem 10-6 Weierstrass points 10-7 Abel's theorem 10-8 Jacobi inversion problem 10-9 The field of algebraic functions 10-10 The hyperelliptic case References Index.

Balls and metrics defined by vector fields I: Basic properties

1985-01-01

Alexander Nagel , Elias M. Stein , Stephen Wainger

View PDF Chat

Singularities of vector fields

1974-12-01

Floris Takens

View PDF Chat

One-Parameter Bifurcations in Planar Filippov Systems

2003-08-01

Yuri A. Kuznetsov , S. Rinaldi , Alessandra Gragnani

We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological … We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

Global transformations of nonlinear systems

1983-01-01

L.R. Hunt , Renjeng Su , George E. Meyer

Recent results have established necessary and sufficient conditions for a nonlinear system of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}(t) = f(x(t))-u(t)g(x(t))</tex> . with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(0) = 0</tex> , to be … Recent results have established necessary and sufficient conditions for a nonlinear system of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}(t) = f(x(t))-u(t)g(x(t))</tex> . with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(0) = 0</tex> , to be locally equivalent in a neighborhood of the origin in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> to a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{g\ldot[f\dotg],...,(ad^{n-1}f\ldotg)}</tex> span an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional space and that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{g\ldot[f\ldot g],...,(ad^{n-2}f\ldotg)}</tex> is an involutive set.

The Jacobian matrix and global univalence of mappings

1965-04-01

David Gale , Hukukane Nikaidō

The topology of normal singularities of an algebraic surface and a criterion for simplicity

1961-12-01

David Mumford

View PDF Chat

Homogeneous Lyapunov function for homogeneous continuous vector field

1992-12-01

Lionel Rosier

The Jacobian conjecture: Reduction of degree and formal expansion of the inverse

1982-01-01

Hyman Bass , E. H. Connell , David L. Wright

Introduction I. The Jacobian Conjecture 1. Statement of the Jacobian Problem; first observations 2. Some history of the Jacobian Conjecture 3. Faulty proofs 4. The use of stabilization and of … Introduction I. The Jacobian Conjecture 1. Statement of the Jacobian Problem; first observations 2. Some history of the Jacobian Conjecture 3. Faulty proofs 4. The use of stabilization and of formal methods II. The Reduction Theorem 1. Notation 2. Statement of the Reduction Theorem 3. Reduction to degree 3 4. Proof of the Reduction Theorem 5. r-linearization and unipotent reduction 6. Nilpotent rank 1 Jacobians III. The Formal Inverse 1. Notation 2. Abhyankar's Inversion Formula 3. The terms Gj 4. The tree expansion G = lT(\/a(T))lfPTf 5. Calculations References

View PDF Chat

On the non-vanishing of the Jacobian in certain one-to-one mappings

1936-01-01

Hans Lewy

View PDF Chat

Classification of solutions for an integral equation

2005-12-19

Wenxiong Chen , Congming Li , Biao Ou

Abstract Let n be a positive integer and let 0 < α < n . Consider the integral equation We prove that every positive regular solution u ( x ) … Abstract Let n be a positive integer and let 0 < α < n . Consider the integral equation We prove that every positive regular solution u ( x ) is radially symmetric and monotone about some point and therefore assumes the form with some constant c = c ( n , α) and for some t > 0 and x 0 ϵ ℝ n . This solves an open problem posed by Lieb 12 . The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper. Moreover, we show that the family of well‐known semilinear partial differential equations is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.

A Lie-Backlund approach to equivalence and flatness of nonlinear systems

1999-05-01

Michel Fliess , Jean Lévine , Philippe Martin +1 more

A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent … A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie-Backlund isomorphism. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie-Backlund isomorphism can be realized by an endogenous feedback. The differentially flat nonlinear systems introduced by the authors (1992) via differential algebraic techniques, are generalized and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability.

View PDF Chat

A Geometric Approach to Global-Stability Problems

1996-07-01

Michael Y. Li , James S. Muldowney

A new criterion for the global stability of equilibria is derived for nonlinear autonomous ordinary differential equations in any finite dimension based on recent developments in higher-dimensional generalizations of the … A new criterion for the global stability of equilibria is derived for nonlinear autonomous ordinary differential equations in any finite dimension based on recent developments in higher-dimensional generalizations of the criteria of Bendixson and Dulac for planar systems and on a local version of the $C^1 $ closing lemma of Pugh. The classical result of Lyapunov is obtained as a special case.

Asymptotic controllability implies feedback stabilization

1997-01-01

Francis Clarke , Yuri S. Ledyaev , Eduardo D. Sontag +1 more

It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed … It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a control-Lyapunov function, iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium. A major technical problem, and one of the contributions of the present paper, concerns the precise meaning of "solution" when using a discontinuous controller.

Orbits of families of vector fields and integrability of distributions

1973-01-01

Héctor J. Sussmann

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an arbitrary set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:mrow … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an arbitrary set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vector fields on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that the orbits of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> submanifolds of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and that, moreover, they are the maximal integral submanifolds of a certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distribution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{P_D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (In general, the dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript upper D Baseline left-parenthesis m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{P_D}(m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will not be the same for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m element-of upper M"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m \in M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow’s theorem to the maximal integral submanifolds of the smallest distribution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that every vector field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Lie algebra generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belongs to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (i.e. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis m right-parenthesis element-of normal upper Delta left-parenthesis m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X(m) \in \Delta (m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m element-of upper M"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m \in M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Their work therefore requires the additional assumption that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not assumed in proving the first main result. It turns out that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is integrable if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta equals upper P Subscript upper D"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Delta = {P_D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and this fact makes it possible to derive a characterization of integrability and Chow’s theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.

View PDF Chat

Geometric homogeneity with applications to finite-time stability

2005-05-16

Sanjay P. Bhat , Dennis S. Bernstein

View PDF Chat

I. Liquid crystals. On the theory of liquid crystals

1958-01-01

F. C. Frank

The first page of this article is displayed as the abstract. The first page of this article is displayed as the abstract.

Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere

1985-05-01

Morris W. Hirsch

A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The main … A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The main results in this article are the following. A cooperative system cannot have nonconstant attracting periodic solutions. In a cooperative system whose Jacobian matrices are irreducible the forward orbit converges for almost every point having compact forward orbit closure. In a cooperative system in 2 dimensions, every solution is eventually monotone. Applications are made to generalizations of positive feedback loops.

View PDF Chat

Quantitative universality for a class of nonlinear transformations

1978-07-01

Mitchell J. Feigenbaum

The double scroll family

1986-11-01

Leon O. Chua , M. Komuro , Takashi Matsumoto

This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equivalent class of piecewise-linear differential equations which includes the … This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equivalent class of piecewise-linear differential equations which includes the double scroll as a special case. A necessary and sufficient condition for two such piecewise-linear vector fields to be linearly equivalent is that their respective eigenvalues be a scaled version of each other. In the special case where they are identical, we have exact equivalence in the sense of linear conjugacy. An explicit normalform equation in the context of global bifurcation is derived and parametrized by their eigenvalues. Analytical expressions for various Poincaré maps are then derived and used to characterize the birth and the death of the double scroll, as well as to derive an approximate one-dimensional map in analytic form which is useful for further bifurcation analysis. In particular, the analytical expressions characterizing various half-return maps associated with the Poincaré map are used in a crucial way to prove the existence of a Shilnikov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and death (extinction of chaos) of the double scroll. The unifying theme throughout this paper is to analyze the double scroll system as an unfolding of a large family of piecewise-linear vector fields in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^3</tex> . Using this approach, we were able to prove that the chaotic dynamics of the double scroll is quite common, and is robust because the associated horseshoes predicted from Shilnikov's theorem are structurally stable. In fact, it is exhibited by a large family (in fact, infinitely many linearlyequivalent circuits) of vector fields whose associated piecewise-linear differential equations bear no resemblance to each other. It is therefore remarkable that the normalized eigenvalues, which is a local concept, completely determine the system's global qualitative behavior.

Introduction to the Arithmetic Theory of Automorphic Functions

1972-10-01

Marvin Tretkoff , Goro Shimura

Structural controllability

1974-06-01

Ching-Tai Lin

The new concepts of "structure" and "structural controllability" for a linear time-invariant control system (described by a pair ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,b</tex> )) are defined and studied. The physical justification … The new concepts of "structure" and "structural controllability" for a linear time-invariant control system (described by a pair ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,b</tex> )) are defined and studied. The physical justification of these concepts and examples are also given. The graph of a pair ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,b</tex> ) is also defined. This gives another way of describing the structure of this pair. The property of structural controllability is reduced to a property of the graph of the pair ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,b</tex> ). To do this, the basic concept of a "cactus" and the related concept of a "precactus" are introduced. The main result of this paper states that the pair ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,b</tex> ) is structurally controllable if an only if the graph of ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,b</tex> ) is "spanned by a cactus." The result is also expressed in a more conventional way, in terms of some properties of the pair ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,b</tex> ).

The Intermediate Jacobian of the Cubic Threefold

1972-03-01

C. Herbert Clemens , Phillip Griffiths

Intermediate Jacobians of threefolds 1. Algebraic correspondences and homology relations.2. Families of algebraic curves on a threefold.3. The intermediate Jacobian and its polarizing class.4. The Abel-Jacobi mapping.Part Two.Geometry of cubic … Intermediate Jacobians of threefolds 1. Algebraic correspondences and homology relations.2. Families of algebraic curves on a threefold.3. The intermediate Jacobian and its polarizing class.4. The Abel-Jacobi mapping.Part Two.Geometry of cubic hypersurfaces 5.The dual mapping, Lefschetz hypersurfaces.6. Cubic hypersurfaces.7. The variety of lines on a cubic hypersurface.Part Three.The cubic threefold 8.The Fano surface of lines on a cubic threefold, the double point case.9. A topological model for the Fano surface, the non-singular case.10. Distinguished divisors on the Fano surface.Part Four.The intermediate Jacobian of the cubic threefold 11.The Gherardelli-Todd isomorphism.12.The Gauss map and the tangent bundle theorem.13.The "double-six", Torelli, and irrationality theorems.Appendices A. Equivalence relations on the algebraic one-cycles lying on a cubic threefold.B. Unirationality.C. Mumford's theory of Prym varieties and a comment on moduli.

View PDF Chat

Homogeneous Vector Bundles

1957-09-01

Raoul Bott

Switched Linear Systems

2005-01-01

Zhendong Sun , Shuzhi Sam Ge

Nonlinear oscillations, dynamical systems, and bifurcations of vector fields

1986-06-01

Jack D. Cowan

Qualitative theory of differential equations

1961-01-01

Analytic Functions of Several Complex Variables.

1968-04-01

J. W. Gray , Robert C. Gunning , Hugo Rossi

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others … The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.

Ergodic Theorems

1985-12-31

Ulrich Krengel

Stable Mappings and Their Singularities

1973-01-01

Martin Golubitsky , Victor Guillemin

Stable and Unitary Vector Bundles on a Compact Riemann Surface

1965-11-01

M. S. Narasimhan , C. S. Seshadri

Nonlinear Control Systems

1995-01-01

Alberto Isidori

The purpose of this book is to present a self-contained description of the fun damentals of the theory of nonlinear control systems, with special emphasis on the differential geometric approach. … The purpose of this book is to present a self-contained description of the fun damentals of the theory of nonlinear control systems, with special emphasis on the differential geometric approach. The b

Liapunov Functions and Stability in Control Theory

2005-01-01

Andrea Bacciotti , Lionel Rosier

View PDF Chat

Polynomial Automorphisms and the Jacobian Conjecture

2021-01-01

Arno van den Essen , Shigeru Kuroda , Anthony J. Crachiola

This book deals with the development on polynomial rings in the last two decades since the publication of "Polynomial Automorphisms" by Arno van den Essen. It is written by the … This book deals with the development on polynomial rings in the last two decades since the publication of "Polynomial Automorphisms" by Arno van den Essen. It is written by the experts who themselves have been able to make a significant contribution on these problems.

View PDF Chat

Global center and limit cycles in generalized piecewise cubic Liénard systems

2025-06-25

Ting Chen , Jian-Wen Peng | Nonlinear Analysis Real World Applications

Modules of Derivations, Logarithmic Ideals and Singularities of Maps on Analytic Varieties

2025-06-24

Carles Bivià-Ausina , Konstantinos Kourliouros , María Aparecida Soares Ruas | Results in Mathematics

Monotonous Period Function for Equivariant Differential Equations with Homogeneous Nonlinearities

2025-06-24

Armengol Gasull , David Rojas | Mediterranean Journal of Mathematics

Extraspecial normalizers and local systems in characteristic 2

2025-06-24

| Princeton University Press eBooks

Two further kinds of local system in characteristic 2

2025-06-24

| Princeton University Press eBooks

Sufficient Conditions to have Global Centers via Branching Theory

2025-06-23

Jaume Giné , Xiaoqi Yang | Qualitative Theory of Dynamical Systems

Abstract It has always been challenging to identify global centers in a planar differential system, despite the recent proposal of a new algorithm to determine such centers. In this paper, … Abstract It has always been challenging to identify global centers in a planar differential system, despite the recent proposal of a new algorithm to determine such centers. In this paper, we present a novel method for establishing sufficient conditions to achieve a global center. This method is based on determining all the branches that pass through the point at infinity. We apply the method to a specific example to demonstrate how it circumvents the traditional blow-up procedure.

None

2025-06-23

Fan Kang | Comptes Rendus Mathématique

Berger’s isoperimetric problem asks if the flat equilateral torus is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub></mml:math>-maximal. In 1996, Nadirashvili first gave a positive answer. In this paper, we use El Soufi–Ilias–Ros’s method … Berger’s isoperimetric problem asks if the flat equilateral torus is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub></mml:math>-maximal. In 1996, Nadirashvili first gave a positive answer. In this paper, we use El Soufi–Ilias–Ros’s method in [8] and Bryant’s result in [3] to give a new proof.

Dynamics of Degenerate Quadratic Lotka-Volterra Mappings Acting in a Four-Dimensional Simplex and Bigraphs

2025-06-21

| Deleted Journal

Investigating Monogenity in a Family of Cyclic Sextic Fields

2025-06-18

István Gaál | Mathematics

Jones characterized, among others, monogenity of a family of cyclic sextic polynomials. Our purpose is to study monogenity of the family of corresponding sextic number fields. We show that several … Jones characterized, among others, monogenity of a family of cyclic sextic polynomials. Our purpose is to study monogenity of the family of corresponding sextic number fields. We show that several of these number fields are monogenic, despite the defining polynomial of their generating element being non-monogenic. In the monogenic fields, there are several inequivalent generators of power integral bases. Our calculation also provides the first non-trivial application of the method described earlier to study monogenity in totally real extensions of imaginary quadratic fields, emphasizing the efficiency of that algorithm.

Inversion of adjunction for quotient singularities II: Non-linear actions

2025-06-18

Yusuke Nakamura , Kohsuke Shibata | Algebraic geometry

View PDF Chat

The period of the limit cycle bifurcating from a persistent polycycle

2025-06-17

David Marín , Lucas Queiroz , Jordi Villadelprat | Publicacions Matemàtiques

Ergodic optimization theory for a class of typical maps

2025-06-17

Wen Huang , Zeng Lian , Xiao Ma +2 more | Journal of the European Mathematical Society

We study the ergodic optimization problem for a class of typical maps including Axiom A attractors, Anosov diffeomorphisms, subshifts of finite type and uniformly expanding systems. In connection with the … We study the ergodic optimization problem for a class of typical maps including Axiom A attractors, Anosov diffeomorphisms, subshifts of finite type and uniformly expanding systems. In connection with the conjecture proposed by Yuan and Hunt in 1999, we prove that when the space of observables is C^{0, \alpha} with \alpha\in(0,1] or C^{1} (if well defined), the optimal (minimizing or maximizing) orbits are generically periodic, thus confirming the conjecture in those cases.

Study of Poincaré Map and limit cycles for non-smooth Welander’s Ocean Convection Model

2025-06-17

Yagor Romano Carvalho , Luiz F.S. Gouveia , Richard McGehee | Physica D Nonlinear Phenomena

The Number of Isolated Periodic Traveling Waves in a Class of Single Species Model

2025-06-16

Gang Kou , Yangjian Sun , Qing Ge | Qualitative Theory of Dynamical Systems

Bifurcation of Limit Cycles in a Class of Piecewise Smooth Cubic Rayleigh-Liénard Systems

2025-06-16

Fang Wu , Ting Chen , Xin Li | Qualitative Theory of Dynamical Systems

Some Aspects of Thermodynamic Formalism of Piecewise Smooth Vector Fields

2025-06-16

Marco Florentino , Tiago Carvalho , Jeferson Cassiano | Journal of Dynamics and Differential Equations

Higher-order exceptional points in composite non-Hermitan systems

2025-06-16

NULL AUTHOR_ID | Physical Review Research

First person – Nikhil Dev Narendradev

2025-06-15

| Journal of Cell Science

ABSTRACT First Person is a series of interviews with the first authors of a selection of papers published in Journal of Cell Science, helping researchers promote themselves alongside their papers. … ABSTRACT First Person is a series of interviews with the first authors of a selection of papers published in Journal of Cell Science, helping researchers promote themselves alongside their papers. Nikhil Dev Narendradev is first author on ‘ Endosomal RFFL ubiquitin ligase regulates mitochondrial morphology by targeting mitofusin 2’, published in JCS. Nikhil Dev is a PhD student in the lab of Srinivasa Murty Srinivasula at Indian Institute of Science Education and Research Thiruvananthapuram, India, working on understanding the crosstalk between different organelles and their regulation via post translational modifications mediated by ubiquitin protein ligases.

Theoretical and experimental observation of border-collision bifurcations, coexisting attractors, and complex Arnol’d tongues in a driven piecewise-constant oscillator

2025-06-15

Hirosi Kuriyama , Tadashi Tsubone , Munehisa Sekikawa +2 more | Chaos Solitons & Fractals

Book Review: Meeting the challenge of mathematical challenge in mathematics education. Roza Leikin (Ed.) (2023) Mathematical Challenges for All

2025-06-14

Edward A. Silver | Educational Studies in Mathematics

Reversible nilpotent centers with cubic nonlinearities

2025-06-13

Leandro Bacelar , Jaume Llibre | Rendiconti del Circolo Matematico di Palermo Series 2

Direct method of solution for a class of singular integral equations based on meromorphism of solution

2025-06-12

Jinyuan Du , Kit Ian Kou | Complex Variables and Elliptic Equations

Global dynamics analysis of a class of quadratic polynomial differential systems with an invariant plane in $$\mathbb {R}^3$$

2025-06-12

Ali Bakhshalizadeh , Jaume Llibre , Alex C. Rezende | Nonlinear Differential Equations and Applications NoDEA

New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory

2025-06-12

Joan C. Artés , Laurent Cairó , Jaume Llibre | AppliedMath

After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been … After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal.

Jacobian Matrix and Singularity

2025-06-10

Hamid D. Taghirad | CRC Press eBooks

A Piecewise-Linear Fixed Point Theorem

2025-06-09

David J. W. Simpson | American Mathematical Monthly

[The] System of Mesnil-Durand Considered in Relation to Orders of Battle and to Battles

2025-06-06

Jonathan S. Abel | BRILL eBooks

Intersection of orbits for polynomials in characteristic p

2025-06-06

Simone Coccia , Dragos Ghioca , Jungin Lee +1 more | Journal of Number Theory

Inhomogeneous Khintchine–Groshev theorem without monotonicity

2025-06-04

Seongmin Kim

Abstract The Khintchine–Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of ‐approximable numbers, given a monotonic function . Allen and … Abstract The Khintchine–Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of ‐approximable numbers, given a monotonic function . Allen and Ramírez removed the monotonicity condition from the inhomogeneous Khintchine–Groshev theorem for cases with and conjectured that it also holds for . In this paper, we prove this conjecture in the case of . We also prove it for the case of with a rational inhomogeneous parameter.

Vanishing and nearby boundary cycles of complex non-isolated singularities

2025-06-03

Marcelo Aguilar , Aurélio Menegon , José Seade

Globally Generated Vector Bundles with c1 = 1 over Grassmannians

2025-06-02

Dazhi Zhang | Deleted Journal

Algebraic integrability with bounded genus

2025-06-02

Carlos Galindo , Francisco Monserrat , Elvira Pérez-Callejo

A new formal series method and its application to the center problem in Z2-equivariant nilpotent vector fields

2025-06-01

Feng Li , Yusen Wu , Ting Chen +1 more | Bulletin des Sciences Mathématiques

Classification of area‐strict limits of planar BV homeomorphisms

2025-06-01

Daniel Campbell , Aapo Kauranen , Emanuela Radici

Abstract We present a classification of area‐strict limits of planar homeomorphisms. This class of mappings allows for cavitations and fractures but fulfil a suitable generalisation of the INV condition of … Abstract We present a classification of area‐strict limits of planar homeomorphisms. This class of mappings allows for cavitations and fractures but fulfil a suitable generalisation of the INV condition of Müller and Spector (Arch. Rational Mech. Anal. 131 (1995), no. 1, 1–66). As pointed out by J. Ball, these features are expected in limit configurations of elastic deformations. De Philippis and Pratelli introduced the no‐crossing condition which characterises the closure of planar homeomorphisms. In the current paper, we show that a suitable version of this concept is equivalent with a map, , being the area‐strict limit of BV homeomorphisms. This extends our results from Campbell et al. (J. Funct. Anal. 285 (2023), no. 3, Paper No. 109953, 30), where we proved that the no‐crossing BV condition for a BV map was equivalent with the map being the m‐strict limit of homeomorphisms (i.e. and ). Further, we show that the no‐crossing BV condition is equivalent with a seemingly stronger version of the same condition.

On a minimal set of generators for the polynomial algebra of five variables in a generic degree

2025-05-30

Nguyễn Sum , Pham Do Tai

An invitation to some of K. R. Parthasarathy's writings: Part II

2025-05-30

Luigi Accardi , Yun Gang Lu , Amit Vutha

Homotopy homoclinic orbit and global dynamics analysis of the double-excited Duffing–van der Pol system

2025-05-30

Danjin Zhang , Meirong Ren , Youhua Qian

Jacobian Analysis

2025-05-30

Chao Chen , Wesley Au , Shao Liu

Local analytic integrability for a class of degenerate planar vector fields

2025-05-29

Antonio Algaba , M. Reyes , Jaume Giné

Abstract Using the normal form theory and the existence of an algebraic inverse integrating factor we characterize the local analytic integrability of the systems whose quasi-homogeneous leading term is <m:math … Abstract Using the normal form theory and the existence of an algebraic inverse integrating factor we characterize the local analytic integrability of the systems whose quasi-homogeneous leading term is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfenced> </m:math> $\left({a}_{1}{y}^{3}+{a}_{2}{x}^{3}y,{b}_{1}{x}^{5}+{b}_{2}{x}^{2}{y}^{2}\right)$ . More specifically we prove that the analytic integrable vector fields inside such family are orbitally equivalent to a semi-quasi-homogeneus system, that is, are not orbitally equivalent to its lowest-degree quasi-homogeneous term.