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The transition from regular to irregular motions, explained as travel on Riemann surfaces
Full text linkPeriodic Solutions of a System of Complex ODEs. II. Higher Periods
No full textIn a previous paper the real evolution of the system of ODEs ¨zn + zn = N m=1, m=n gnm(zn - zm) -3 , zn zn(t), zn dzn(t) dt , n = 1, . . . , N is discussed in CN , namely the N dependent variables zn, as well as the N(N - 1) (arbitrary!) "coupling constants" gnm, are considered to be complex numbers, while the independent variable t ("time") is real. In that context it was proven that there exists, in the phase space of the initial data zn(0), zn(0), an open domain having infinite measure, such that all trajectories emerging from it are completely periodic with period 2, zn(t + 2) = zn(t). In this paper we investigate, both by analytcal techniques and via the display of numerical simulations, the remaining solutions, and in particular we show that there exist many -- emerging out of sets of initial data having nonvanishing measures in the phase space of such data -- that are also completely periodic but with periods which are integer multiples of 2. We also elcidate the mechanism that yields nonperiodic solutions, including those characterized by a "chaotic" behavior, namely those associated, in the context of the initial-value problem, with a sensitive dependence on the initial data
Helical configurations of elastic rods in the presence of a long-range interaction potential
No full textRecently, the integrability of the stationary Kirchhoff equations describing an elastic rod folded in the shape of a circular helix was proven. In this paper we explicitly work out the solutions to the stationary Kirchhoff equations in the presence of a long-range potential which describes the average constant force due to a Morse-type interaction acting among the points of the rod. The average constant force results to be parallel to the normal vector to the central line of the folded rod; this condition remarkably permits to preserve the integrability (indeed the solvability) of the corresponding Kirchhoff equations if the elastic rod features constant or periodic stiffnesses and vanishing intrinsic twist. Furthermore, we discuss the elastic energy density with respect to the radius and pitch of the helix, showing the existence of stationary points, namely stable and unstable configurations, for plausible choices of the featured parameters corresponding to a real bio-polymer
Solvable Nonlinear Evolution PDEs in Multidimensional Space
No full textA class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schrödinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type. Isochronous variants of these evolution PDEs are also considered
Solvable nonlinear evolution PDEs in multidimensional space involving elliptic functions
No full textSolvable nonlinear evolution PDEs in multidimensional space involving trigonometric functions
No full textIsochronous rate equations describing chemical reactions
No full textWe consider systems of ordinary differential equations in the plane featuring at most quadratic nonlinearities. It is known that, up to linear transformations of the variables, there are only four systems for which the origin is an isochronous center, that is, for which all orbits in the vicinity of the origin are periodic with the same, fixed period. On the other hand, if, after an affine transformation, the system's coefficients satisfy certain positivity requirements, these systems can be interpreted as kinetic equations for chemical reactions. Here we show that, for two of these four isochronous systems, it is possible to find an affine transformation such that the transformed system obeys all these positivity conditions. For the third we can show that this is not possible, whereas for the fourth the issue remains to some extent open. Hence for the two cases mentioned above these systems may be interpreted as kinetic equations describing isochronous chemical reactions
Traveling Waves in Elastic Rods with Arbitrary Curvature and Torsion
No full textThe dynamic Kirchhoff equations, describing a thin elastic rod of infinite length, are considered in connection with the study of the conformations of polymeric chains. A novel special traveling wave solution that can be interpreted as a conformational soliton propagating at constant speed is obtained, featuring arbitrary non-constant curvature and torsion of the rod, in the simple case of constant cross-section, homogeneous density and elastic isotropy. This traveling wave corresponds to a specific constraint on the twist-to-bend ratio of the constant stiffness parameters, which in turn appears to be compatible with the experimental evidence for the mechanical properties of real polymeric chains. Due to such a constraint, the square of the velocity of the solitary wave is directly proportional to the bending stiffness and inversely proportional to the density and to the principal momentum of inertia of the rod. Several applications to the study of conformational changes in polymeric chains are given
The transition from regular to irregular motions, explained as travel on Riemann surfaces
Full text linkWe introduce and discuss a simple Hamiltonian dynamical system, interpretable as a three-body problem in the (complex) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology—illustrating the onset in a deterministic context of irregular motions—is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere
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