The dispersion properties of drift waves in a low /? weakly collisional plasma, in the presence o... more The dispersion properties of drift waves in a low /? weakly collisional plasma, in the presence of ion acoustic or Langmuir waves parallel to magnetic lines, are investigated. It is shown that the non-linear interaction with an ion acoustic wave mainly leads to a shift of the drift wave spectrum towards lower frequencies (and only under very particular conditions affects the stability of the drift wave). The interaction with a Langmuir wave, on the contrary, yields a contribution to the imaginary part of the frequency. The conditions under which this stabilizes unstable drift modes are derived and discussed.
Annali di Matematica Pura ed Applicata, Series 4, 1976
(***)-P. I~EGRII~I (Roma) (***) ]J. SAIXA])O~I (Roma)(***)-M. SCALIA (Roma)(***) A D~IO GRAFFI ne... more (***)-P. I~EGRII~I (Roma) (***) ]J. SAIXA])O~I (Roma)(***)-M. SCALIA (Roma)(***) A D~IO GRAFFI nel suo 70 ° compleanno Summary.-A concept o] total stability/or continuous or discrete dynamical systems and a generalized de]inition o] bi/ureation are given: 4t is possible to show the link between an abrupt change o/the asymptotic behaviour o] a ]amily o] ]lows and the arising o] new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour o] ]lows near an invariant compact set. They are obtained by means o/an extension o/ Liapunov's direct method. Introduction.
The subject of the present lecture is part of a work [1] devoted to some questions related to the... more The subject of the present lecture is part of a work [1] devoted to some questions related to the problem of turbulence as proposed by the Landau model [2]. We give a very brief survey of the problems which are dealt with. Let μ, be a real parameter, Fμ a vector field family de-fined on a suitable vectorial space E and consider the corresponding evo-lution equation $$\begin{array}{*{20}c}{\frac{{{\rm{dx}}}}{{{\rm{dt}}}}{\rm{=F}}_{\rm{\mu}}\left({\rm{x}}\right){\rm{,}}}&{{\rm{F}}_{\rm{\mu}}\left({\rm{0}}\right){\rm{=0}}}\\\end{array}$$ (1)
ABSTRACT An analysis of the relationship between asymptotic stability and total stability for abs... more ABSTRACT An analysis of the relationship between asymptotic stability and total stability for abstract dynamic systems is presented. The measure of perturbations on initial data as well as the measure of the distance at any given time between two motions are evaluated in terms of a measure h of stability, introduced as a function defined on the domain of the dynamic system. It is demonstrated that uniform asymptotic stability implies uniform total stability. This result is obtained by means of a characterization of h-uniform asymptotic stability in terms of a one-parameter family of Liapunov functions.
Publisher Summary This chapter discusses the stability problems for Hopf bifurcation. It explores... more Publisher Summary This chapter discusses the stability problems for Hopf bifurcation. It explores the general problem of asymptotic stability of the periodic orbits arising in the Hopf bifurcation. The bifurcating periodic orbits are found to be attracting under the general assumption that 0 is asymptotically stable for μ = 0, which is the critical value of the parameter, and there exists an odd integer h ∈{3, …,k}, such that the character of 0 is recognizable in a suitable sense by the terms of f(0,·) of degree ≤ h. The occurrence of P(h) can be recognized by using a classical procedure of Poincare, and this procedure is reduced to the analysis of linear algebraic systems. With respect to the problem of asymptotic stability of bifurcating periodic orbits, P(h) is not only sufficient but also necessary for the bifurcating periodic orbits to be attracting and for the attractivity to have an appropriate structural character.
02 c (ordl) end for each nelghborhood N of fOr !{ c l{*, thete exLsts f€Nsuchthat(1.2}heeexactlyJ... more 02 c (ordl) end for each nelghborhood N of fOr !{ c l{*, thete exLsts f€Nsuchthat(1.2}heeexactlyJncntrlvlelperl"adleotbltslylng ln fntar) whose perlod tr 1n [2n-6"rgr+62]; (!il) fcr any E € (0,*r), 6 € (0,6.1) there of f0, F c ltJ* euch that lf f € N and tf exlsts a nelghborhood N Y le a periodlc orblt of
Journal of Mathematical Analysis and Applications, 1980
Submirred bx V. Lakshmikanrham Explicit criteria for the asymptotic stability (or instability) of... more Submirred bx V. Lakshmikanrham Explicit criteria for the asymptotic stability (or instability) of bifurcating closed orbits are given for a class of abstract evolution equations. ' For the extensive literature on this subject. see 18. 14. I9 I.
Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli st... more Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli studenti della Laurea Triennale in Matematica dell’Università Sapienza di Roma
We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Ne... more We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian self-attraction. Under the assumption of infinitesimal tickness of the stratum we show the existence of stationary motions during which the stratum is approximatly a round torus (with radii r, R and R>>r) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helix-like trajectory around the circumference of radius R that connects the centers of the cross sections of the torus.
We rigorously prove the existence of directed transport for a certain class of ac-driven nonlinea... more We rigorously prove the existence of directed transport for a certain class of ac-driven nonlinear one dimensional systems, namely the generation of transport with a preferred direction in the absence of a net driving force.
We consider a Lagrangian differential system. The celebrated theorem of Lagrange-Dirichlet ensure... more We consider a Lagrangian differential system. The celebrated theorem of Lagrange-Dirichlet ensures that a stationary solution of this system is stable, provided that the corresponding critical point of the potential function is a proper {local} maximum.
We consider the problem of the planar motion of four point vortices with intensities (1, 1, 1, e)... more We consider the problem of the planar motion of four point vortices with intensities (1, 1, 1, e), in a Eulerian incompressible fluid, as a perturbation of the problem of three unit vortices. The unperturbed problem is reduced to a planar autonomous Hamiltonian system which admits saddle connections. For e > 0 and sufficiently small, we also reduce, in a neighborhood of the above saddle connections, the problem to a planar Hamiltonian system, which is no longer autonomous but periodically time dependent. The Poincare-map of the perturbed problem presents transversal intersections between stable and unstable manifolds of two hyperbolic points; this implies that there are new regions of chaotic behavior, different from the ones previously found by Ziglin
The dispersion properties of drift waves in a low /? weakly collisional plasma, in the presence o... more The dispersion properties of drift waves in a low /? weakly collisional plasma, in the presence of ion acoustic or Langmuir waves parallel to magnetic lines, are investigated. It is shown that the non-linear interaction with an ion acoustic wave mainly leads to a shift of the drift wave spectrum towards lower frequencies (and only under very particular conditions affects the stability of the drift wave). The interaction with a Langmuir wave, on the contrary, yields a contribution to the imaginary part of the frequency. The conditions under which this stabilizes unstable drift modes are derived and discussed.
Annali di Matematica Pura ed Applicata, Series 4, 1976
(***)-P. I~EGRII~I (Roma) (***) ]J. SAIXA])O~I (Roma)(***)-M. SCALIA (Roma)(***) A D~IO GRAFFI ne... more (***)-P. I~EGRII~I (Roma) (***) ]J. SAIXA])O~I (Roma)(***)-M. SCALIA (Roma)(***) A D~IO GRAFFI nel suo 70 ° compleanno Summary.-A concept o] total stability/or continuous or discrete dynamical systems and a generalized de]inition o] bi/ureation are given: 4t is possible to show the link between an abrupt change o/the asymptotic behaviour o] a ]amily o] ]lows and the arising o] new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour o] ]lows near an invariant compact set. They are obtained by means o/an extension o/ Liapunov's direct method. Introduction.
The subject of the present lecture is part of a work [1] devoted to some questions related to the... more The subject of the present lecture is part of a work [1] devoted to some questions related to the problem of turbulence as proposed by the Landau model [2]. We give a very brief survey of the problems which are dealt with. Let μ, be a real parameter, Fμ a vector field family de-fined on a suitable vectorial space E and consider the corresponding evo-lution equation $$\begin{array}{*{20}c}{\frac{{{\rm{dx}}}}{{{\rm{dt}}}}{\rm{=F}}_{\rm{\mu}}\left({\rm{x}}\right){\rm{,}}}&{{\rm{F}}_{\rm{\mu}}\left({\rm{0}}\right){\rm{=0}}}\\\end{array}$$ (1)
ABSTRACT An analysis of the relationship between asymptotic stability and total stability for abs... more ABSTRACT An analysis of the relationship between asymptotic stability and total stability for abstract dynamic systems is presented. The measure of perturbations on initial data as well as the measure of the distance at any given time between two motions are evaluated in terms of a measure h of stability, introduced as a function defined on the domain of the dynamic system. It is demonstrated that uniform asymptotic stability implies uniform total stability. This result is obtained by means of a characterization of h-uniform asymptotic stability in terms of a one-parameter family of Liapunov functions.
Publisher Summary This chapter discusses the stability problems for Hopf bifurcation. It explores... more Publisher Summary This chapter discusses the stability problems for Hopf bifurcation. It explores the general problem of asymptotic stability of the periodic orbits arising in the Hopf bifurcation. The bifurcating periodic orbits are found to be attracting under the general assumption that 0 is asymptotically stable for μ = 0, which is the critical value of the parameter, and there exists an odd integer h ∈{3, …,k}, such that the character of 0 is recognizable in a suitable sense by the terms of f(0,·) of degree ≤ h. The occurrence of P(h) can be recognized by using a classical procedure of Poincare, and this procedure is reduced to the analysis of linear algebraic systems. With respect to the problem of asymptotic stability of bifurcating periodic orbits, P(h) is not only sufficient but also necessary for the bifurcating periodic orbits to be attracting and for the attractivity to have an appropriate structural character.
02 c (ordl) end for each nelghborhood N of fOr !{ c l{*, thete exLsts f€Nsuchthat(1.2}heeexactlyJ... more 02 c (ordl) end for each nelghborhood N of fOr !{ c l{*, thete exLsts f€Nsuchthat(1.2}heeexactlyJncntrlvlelperl"adleotbltslylng ln fntar) whose perlod tr 1n [2n-6"rgr+62]; (!il) fcr any E € (0,*r), 6 € (0,6.1) there of f0, F c ltJ* euch that lf f € N and tf exlsts a nelghborhood N Y le a periodlc orblt of
Journal of Mathematical Analysis and Applications, 1980
Submirred bx V. Lakshmikanrham Explicit criteria for the asymptotic stability (or instability) of... more Submirred bx V. Lakshmikanrham Explicit criteria for the asymptotic stability (or instability) of bifurcating closed orbits are given for a class of abstract evolution equations. ' For the extensive literature on this subject. see 18. 14. I9 I.
Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli st... more Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli studenti della Laurea Triennale in Matematica dell’Università Sapienza di Roma
We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Ne... more We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian self-attraction. Under the assumption of infinitesimal tickness of the stratum we show the existence of stationary motions during which the stratum is approximatly a round torus (with radii r, R and R>>r) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helix-like trajectory around the circumference of radius R that connects the centers of the cross sections of the torus.
We rigorously prove the existence of directed transport for a certain class of ac-driven nonlinea... more We rigorously prove the existence of directed transport for a certain class of ac-driven nonlinear one dimensional systems, namely the generation of transport with a preferred direction in the absence of a net driving force.
We consider a Lagrangian differential system. The celebrated theorem of Lagrange-Dirichlet ensure... more We consider a Lagrangian differential system. The celebrated theorem of Lagrange-Dirichlet ensures that a stationary solution of this system is stable, provided that the corresponding critical point of the potential function is a proper {local} maximum.
We consider the problem of the planar motion of four point vortices with intensities (1, 1, 1, e)... more We consider the problem of the planar motion of four point vortices with intensities (1, 1, 1, e), in a Eulerian incompressible fluid, as a perturbation of the problem of three unit vortices. The unperturbed problem is reduced to a planar autonomous Hamiltonian system which admits saddle connections. For e > 0 and sufficiently small, we also reduce, in a neighborhood of the above saddle connections, the problem to a planar Hamiltonian system, which is no longer autonomous but periodically time dependent. The Poincare-map of the perturbed problem presents transversal intersections between stable and unstable manifolds of two hyperbolic points; this implies that there are new regions of chaotic behavior, different from the ones previously found by Ziglin
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Papers by Piero Negrini