An algebraically computable piecewise linear planar oscillator
نویسندگان
چکیده
A piecewise linear oscillator whose oscillation can be completely characterized by algebraic methods is studied. It represents up to the best of authors’s knowledge, the first example where all the oscillation properties can be determined for all the values of the bifurcation parameter, being not a perturbation of the harmonic oscillator. In fact, algebraic expressions for coordinates of representative points, period and characteristic multiplier of the corresponding periodic orbit are provided. Thus, the studied oscillator deserves to be considered an excellent benchmark for testing approximate methods of analysis in nonlinear oscillation theory. The analytical results obtained follow the point transformation method initiated in [1], see also [6]. We use a canonical form proposed in [2] which corresponds with a piecewise linear Liénard differential system. The basic idea facilitating the algebraic computations lies in the consideration of proportional spectra for the involved matrices and can be extended to higher dimension, see [7] and [8]. The oscillator studied has practical interest, as shown for an electronic circuit modeling a Van der Pol oscillator, see [3], which is a simplification of another circuit proposed by [5] modeling the conduction and nerve excitation [4]. Sección en el CEDYA 2011: EDO
منابع مشابه
Planelet Transform: A New Geometrical Wavelet for Compression of Kinect-like Depth Images
With the advent of cheap indoor RGB-D sensors, proper representation of piecewise planar depth images is crucial toward an effective compression method. Although there exist geometrical wavelets for optimal representation of piecewise constant and piecewise linear images (i.e. wedgelets and platelets), an adaptation to piecewise linear fractional functions which correspond to depth variation ov...
متن کاملCanard Explosion and Relaxation Oscillation in Planar, Piecewise-Smooth, Continuous Systems
Classical canard explosion results in smooth systems require the vector field to be at least C, since canard cycles are created as the result of a Hopf bifurcation. The work on canards in nonsmooth, planar systems is recent and has thus far been restricted to piecewise-linear or piecewise-smooth Van der Pol systems, where an extremum of the critical manifold arises from the nonsmoothness. In bo...
متن کاملXCS with Computable Prediction in Multistep Environments
XCSF extends the typical concept of learning classifier systems through the introduction of computable classifier prediction. Initial results show that XCSF’s computable prediction can be used to evolve accurate piecewise linear approximations of simple functions. In this paper, we take XCSF one step further and apply it to typical reinforcement learning problems involving delayed rewards. In e...
متن کاملThe topological entropy of iterated piecewise affine maps is uncomputable
We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real numbe...
متن کاملA subthreshold CMOS circuit for a piecewise linear neuromorphic oscillator with current-mode low-pass filters
We propose an analog current-mode subthreshold CMOS circuit implementing a piecewise linear neuromorphic oscillator. Our circuit was derived from a piecewise linear oscillator model proposed by Matsuoka, well known as a building block for constructing a robot locomotion controller. We modified Matsuoka’s oscillator to be suitable for analog current-mode integrated circuit implementation, and de...
متن کامل