Existence results on the one-dimensional Dirichlet problem suggested by the piecewise linear case

Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt’s and Zermélo’s paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the irreversible Second Law of Thermodynamics (Loschmidt’s) as well as periodic in the time (Zermélo’s, illustrating Poincaré recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals’ information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here.

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GENERALIZATIONS OF THE CHUA EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127492000513 ◽

1992 ◽

Vol 02 (04) ◽

pp. 889-909 ◽

Cited By ~ 6

Author(s):

RAY BROWN

Keyword(s):

Piecewise Linear ◽

Three Dimensional ◽

Type Ii ◽

Two Dimensional ◽

Chua’S Circuit ◽

Unimodal Maps ◽

One Dimensional ◽

Chua's Circuit ◽

The One ◽

Block Approach

In this paper we present two generalizations of the equations governing Chua’s circuit. In order to obtain the first generalization we simplify Chua’s equations by replacing the piecewise-linear term with a signum function. The resulting simplified system produces a double scroll similar to the one observed experimentally in Chua’s circuit. What is significant about this simplified system is that it can be reduced to what we shall call a two-dimensional single scroll, and from the two-dimensional single scroll we are able to derive a one-dimensional map. This entire derivation is carried out analytically, in contrast to the one-dimensional map analysis that has been carried out for the Lorenz equations which is based on axioms. After we have carried out our analysis for this simplified version of Chua’s equations, we use these equations as a guide to the construction of the first generalization to be presented in this paper. We call this a type-I generalization of Chua’s equations. The generalization consists in using a two-dimensional autonomous flow as a component in a three-dimensional autonomous flow in such a way that the resulting equations will have double scroll attractors similar to those observed experimentally in Chua’s circuit. The value of this generalization is that: (1) it provides a building block approach to the construction of chaotic circuits from simpler two-dimensional components which are not chaotic by themselves. In so doing it provides an insight into how chaotic systems can be built up from simple nonchaotic parts; (2) it illustrates a precise relationship between three-dimensional flows and one-dimensional maps. Of particular significance in this regard is a recent paper of Misiurewicz [1993], which analytically connects the two-dimensional single scroll to the class of unimodal maps, thus providing a framework within which a theory linking unimodal maps to three-dimensional flows may be possible. The second generalization is suggested by considering three-dimensional flows whose only nonlinearities are sigmoid, sgn, or piecewise-linear functions. Clearly, such flows are a generalization of the Chua equations. We call these equations type-II generalization Chua equations. The significance of this direction of investigation is that attractors similar to the Lorenz and Rössler attractors can be produced from type-II generalized Chua equations in a building block approach using only piecewise-linear vector fields. As a result we have a method of producing the Lorenz and Rössler dynamics in a circuit without the use of multipliers. This suggests that the type-II generalized Chua equations are in some sense fundamental in that the dynamics of the three most important autonomous three-dimensional differential equations producing chaos are seen as variations of a single class of equations whose nonlinearities are generalizations of the Chua diode.

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Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

Advanced Nonlinear Studies ◽

10.1515/ans-2012-0310 ◽

2012 ◽

Vol 12 (3) ◽

Cited By ~ 26

Author(s):

Isabel Coelho ◽

Chiara Corsato ◽

Franco Obersnel ◽

Pierpaolo Omari

Keyword(s):

Differential Equation ◽

Dirichlet Problem ◽

Positive Solutions ◽

Singular Problem ◽

Topological Methods ◽

One Dimensional ◽

Curvature Equation ◽

Existence And Multiplicity ◽

The One ◽

Multiplicity Of Positive Solutions

AbstractWe discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation.Depending on the behaviour of f = f (t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

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BIFURCATION ANALYSIS OF A PWL CHAOTIC CIRCUIT BASED ON HYSTERESIS THROUGH A ONE-DIMENSIONAL MAP

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401003085 ◽

2001 ◽

Vol 11 (07) ◽

pp. 1911-1927 ◽

Cited By ~ 6

Author(s):

FEDERICO BIZZARRI ◽

MARCO STORACE ◽

LAURA GARDINI ◽

RENZO LUPINI

Keyword(s):

Bifurcation Analysis ◽

Piecewise Linear ◽

Three Dimensional ◽

Chaotic Circuit ◽

One Dimensional ◽

Local And Global Bifurcations ◽

The One ◽

Planar Flows ◽

Analytical Expressions ◽

Linear Nature

Bifurcations in the dynamics of a chaotic circuit based on hysteresis are evaluated. Owing to the piecewise-linear nature of the nonlinear elements of the circuit, such bifurcations are discussed by resorting to a suitable one-dimensional map. As the ordinary differential equations governing the circuit are piecewise linear, the analytical expressions of their solutions can be derived in each linear region. Consequently, the proposed results have been obtained not by resorting to numerical integration, but by properly connecting pieces of planar flows. The bifurcation analysis is carried out by varying one of the three dimensionless parameters that the system of normalized circuit equations depends on. Local and global bifurcations, regular and chaotic asymptotic behaviors are pointed out by analyzing both the one-dimensional map and the three-dimensional flow induced by the circuit dynamics.

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THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

Journal of the London Mathematical Society ◽

10.1017/s0024610701002277 ◽

2001 ◽

Vol 64 (1) ◽

pp. 125-143 ◽

Cited By ~ 69

Author(s):

MEIRONG ZHANG

Keyword(s):

Dirichlet Problem ◽

Rotation Number ◽

Periodic Potential ◽

Schrödinger Operators ◽

Negative Integer ◽

Schrodinger Operators ◽

One Dimensional ◽

Higher Dimensional ◽

Periodic Potentials ◽

The One

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

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NUMERICAL SIMULATION OF CHARGED FULLERENE SPECTRUM

Mathematical Modelling and Analysis ◽

10.3846/mma.2019.017 ◽

2019 ◽

Vol 24 (2) ◽

pp. 263-275

Author(s):

Rafael Arutyunyan ◽

Yuri Obukhov ◽

Petr Vabishchevich

Keyword(s):

Spectral Problem ◽

Electron Spectrum ◽

Computational Algorithm ◽

Piecewise Linear ◽

Spherically Symmetric ◽

Charged Sphere ◽

One Dimensional ◽

Spectral Problems ◽

The One ◽

The Mathematical Model

The mathematical model of the electron spectrum of a charged fullerene is constructed on the basis of the potential of a charged sphere and the spherically symmetric potential of an uncharged fullerene. The electron spectrum is defined as the solution of the spectral problem for the one-dimensional Schr\"odinger equation. For the numerical solution of the spectral problem, piecewise-linear finite elements are used. The computational algorithm was tested on the analytical solution of the problem of the spectrum of the hydrogen atom. For solution of matrix spectral problems, a free library for solving spectral problems of SLEPc is used. The results of calculations of the electron spectrum of a charged fullerene C60 are presented.

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On a conjecture related to the number of solutions of a nonlinear Dirichlet problem

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500012993 ◽

1983 ◽

Vol 95 (3-4) ◽

pp. 275-283 ◽

Cited By ~ 47

Author(s):

A. C. Lazer ◽

P. J. McKenna

Keyword(s):

Dirichlet Problem ◽

Elliptic Boundary ◽

Elliptic Boundary Value Problem ◽

Number Of Solutions ◽

Nonlinear Dirichlet Problem ◽

One Dimensional ◽

Elliptic Boundary Value ◽

The Past ◽

Semilinear Elliptic ◽

The One

SynopsisIn an earlier paper (1981), the present authors made a conjecture about the number of solutions of a semilinear elliptic boundary value problem which has been investigated extensively in the past decade. The conjecture is proved in the one-dimensional case.

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DYNAMICS OF PIECEWISE LINEAR DISCONTINUOUS MAPS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127404010837 ◽

2004 ◽

Vol 14 (07) ◽

pp. 2341-2351 ◽

Cited By ~ 18

Author(s):

LÁSZLÓ E. KOLLÁR ◽

GÁBOR STÉPÁN ◽

JÁNOS TURI

Keyword(s):

Lyapunov Exponent ◽

Control Parameter ◽

Piecewise Linear ◽

Two Dimensional ◽

One Dimensional ◽

Sensitive Dependence ◽

Processing Delay ◽

Sampled Systems ◽

The One ◽

Dense Orbits

In this paper, the dynamics of maps representing classes of controlled sampled systems with backlash are examined. First, a bilinear one-dimensional map is considered, and the analysis shows that, depending on the value of the control parameter, all orbits originating in an attractive set are either periodic or dense on the attractor. Moreover, the dense orbits have sensitive dependence on initial data, but behave rather regularly, i.e. they have quasiperiodic subsequences and the Lyapunov exponent of every orbit is zero. The inclusion of a second parameter, the processing delay, in the model leads to a piecewise linear two-dimensional map. The dynamics of this map are studied using numerical simulations which indicate similar behavior as in the one-dimensional case.

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A graphical compensation approach for smoothing the one-dimensional canonical piecewise-linear model

Sadhana ◽

10.1007/s12046-021-01700-6 ◽

2021 ◽

Vol 46 (3) ◽

Author(s):

V M Jimenez-Fernandez

Keyword(s):

Linear Model ◽

Piecewise Linear ◽

One Dimensional ◽

Piecewise Linear Model ◽

The One ◽

Compensation Approach

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SOME PROPERTIES OF A TWO-DIMENSIONAL PIECEWISE-LINEAR NONINVERTIBLE MAP

International Journal of Bifurcation and Chaos ◽

10.1142/s021812749600148x ◽

1996 ◽

Vol 06 (12a) ◽

pp. 2299-2319 ◽

Cited By ~ 26

Author(s):

CHRISTIAN MIRA ◽

CHRISTINE RAUZY ◽

YURI MAISTRENKO ◽

IRINA SUSHKO

Keyword(s):

Fixed Points ◽

Piecewise Linear ◽

Two Dimensional ◽

One Dimensional ◽

Critical Curves ◽

Unstable Set ◽

Rank One ◽

Characteristic Features ◽

The One ◽

Parameter Values

Properties of a piecewise-linear noninvertible map of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical point in the one-dimensional case). This map is of (Z0–Z2) type, i.e. the plane consists of a region without preimage, and a region giving rise to two rank one preimages. For the considered parameter values, the map has two saddle fixed points. The characteristic features of the “mixed chaotic area” generated by this map, and its bifurcations (some of them being of homoclinic and heteroclinic type) are examined. Such an area is bounded by the union of critical curves segments and segments of the unstable set of saddle cycles.

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