Invariant Probability Density Function Research Articles

Stochastic generalized Kolmogorov systems with small diffusion: I. Explicit approximations for invariant probability density function

This paper presents Part I of a two-part series on studying the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. Part I establishes a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems, while Part II will focus on analyzing their non-autonomous periodic counterparts. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this paper introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation methods exhibit good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our methods in several application examples from biology and ecology.

The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

<abstract><p>The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.</p></abstract>

Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

The statistical behavior of families of maps is important in studying the stability properties of chaotic maps. For a piecewise expanding map τ whose slope >2 in magnitude, much is known about the stability of the associated invariant density. However, when the map has slope magnitude ≤2 many different behaviors can occur as shown in (Keller in Monatsh. Math. 94(4): 313–333, 1982) for W maps. The main results of this note use a harmonic average of slopes condition to obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Since these constants are determined explicitly the results can be extended to families of approximating maps.

On the Efetov–Wegner terms by diagonalizing a Hermitian supermatrix

The diagonalization of Hermitian supermatrices is studied. Such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory. However, it still poses serious problems since until now the calculation of all Rothstein contributions, known as Efetov–Wegner terms in physics, was quite cumbersome. We derive the supermatrix Bessel function with all Efetov–Wegner terms for an arbitrary rotation invariant probability density function. As applications we consider representations of generating functions for Hermitian random matrices with and without an external field as integrals over eigenvalues of Hermitian supermatrices. All results are obtained with all Efetov–Wegner terms which were previously unknown in such an explicit and compact representation.

A Microcontact Non-Gaussian Surface Roughness Model Accounting for Elastic Recovery

Most statistical contact analyses assume that asperity height distributions (g(z*)) follow a Gaussian distribution. However, engineered surfaces are frequently non-Gaussian with the type dependent on the material and surface state being evaluated. When two rough surfaces experience contact deformations, the original topography of the surfaces varies with different loads, and the deformed topography of the surfaces after unloading and elastic recovery is quite different from surface contacts under a constant load. A theoretical method is proposed in the present study to discuss the variations of the topography of the surfaces for two contact conditions. The first kind of topography is obtained during the contact of two surfaces under a normal load. The second kind of topography is obtained from a rough contact surface after elastic recovery. The profile of the probability density function is quite sharp and has a large peak value if it is obtained from the surface contacts under a normal load. The profile of the probability density function defined for the contact surface after elastic recovery is quite close to the profile before experiencing contact deformations if the plasticity index is a small value. However, the probability density function for the contact surface after elastic recovery is closer to that shown in the contacts under a normal load if a large initial plasticity index is assumed. How skewness (Sk) and kurtosis (Kt), which are the parameters in the probability density function, are affected by a change in the dimensionless contact load, the initial skewness (the initial kurtosis is fixed in this study) or the initial plasticity index of the rough surface is also discussed on the basis of the topography models mentioned above. The behavior of the contact parameters exhibited in the model of the invariant probability density function is different from the behavior exhibited in the present model.

Evaluation of channel coding and decoding algorithms using discrete chaotic maps

In this paper we address the design of channel encoding algorithms using one-dimensional nonlinear chaotic maps starting from the desired invariant probability density function (pdf) of the data sent to the channel. We show that, with some simple changes, it is straightforward to make use of a known encoding framework based upon the Bernoulli shift map and adapt it readily to carry the information bit sequence produced by a binary source in a practical way. On the decoder side, we introduce four already known decoding algorithms and compare the resulting performance of the corresponding transmitters. The performance in terms of the bit error rate shows that the most important design clue is related not only to the pdf of the data produced by the chosen discrete map: the own dynamics of the maps is also of the highest importance and has to be taken into account when designing the whole transmitting and receiving system. We also show that a good performance in such systems needs the extensive use of all the evidence stored in the whole chaotic sequence.

An original approximate method for estimating the invariant probability distribution of a large class of multi-dimensional nonlinear stochastic oscillators

This paper proposes an original method for obtaining analytical approximations of the invariant probability density function of multi-dimensional Hamiltonian dissipative dynamic systems under Gaussian white noise excitations, with linear non-conservative parts and nonlinear conservative parts. The method is based on an exact result and a heuristic argument. Its pertinence is attested by numerical tests.

Assessment of chaos-based FM signals for range–Doppler imaging

The authors analysed a set of random frequency modulated (FM) signals for wideband radar imaging and assessed their resolution capability and sidelobe distribution on the range–Doppler plane. To this effect deterministic, bounded, nonlinear iterated maps were first considered. The initial condition of each chaotic map was assigned to a random variable to obtain statistically independent samples with invariant probability density function. The resulting sequences, which have white time–frequency representations, are used to construct wideband stochastic FM signals. These FM signals are ergodic and stationary. The autocorrelation, spectrum and the ambiguity surface associated with each of the FM signals were characterised. It was also demonstrated that the ambiguity surface of an FM signal generated via a chaotic map with uniform sample distribution and tail-shifted chaotic attractor is comparable to the ambiguity function of a Gaussian FM signal.

OPTIMAL CONTROL OF CHAOTIC SYSTEMS

The problem of controlling a chaotic system is treated from a long term statistical basis. Unlike the OGY targeting method that exploits individual unstable orbits, this approach is concerned with targeting the density function of an invariant probability measure. Given a point transformation T, possessing a density function f, we choose [Formula: see text], a different probability density function, to be the target. Using optimization methods, we construct a point transformation [Formula: see text], close to T, whose invariant probability density function is [Formula: see text], or close to [Formula: see text].

A new approach to controlling chaotic systems

Given a piecewise monotonic transformation τ which is strongly transitive and preserves an invariant probability density function f, a construction is described which modifies τ very slightly but which changes f significantly in order to possess desired properties. This construction permits control of the global long term behavior of chaotic dynamical systems.

A dynamical system model for interference effects and the two-slit experiment of quantum physics

Given two probability density functions ƒ 1 and ƒ 2, a method is described for combining the densities in a physically meaningful way. The method involves the construction of underlying transformations τ 1 and τ 2, and then forming a dynamical system from these two transformations referred to as a random transformation. The invariant (stationary) probability density function for the random transformation is the “combined” density of ƒ 1 and ƒ 2. This method of combining probability density functions is used to model interference effects in physical systems. In particular, the dynamics of the two-slit experiment of quantum physics is modelled by an appropriate random transformation. Computer results are presented which qualitatively appear like experimental results. The notion of wave is not needed in the model.

Convergence of the sign algorithm for adaptive filtering with correlated data

Convergence of a decreasing gain sign algorithm (SA) for adaptive filtering is analyzed. The presence of the hard limiter in the algorithm makes a rigorous analysis difficult. Therefore, there are few results available. Such results normally include restrictive assumptions such as the assumptions that successive observation vectors are independent and the new error signal of the adaptive filter has a time invariant probability density function. The former assumption is not valid in the context of adaptive filtering since two successive observation vectors share most of their components, while the latter assumption is a restriction on the adaptive weights whose evolution is a priori unknown. In lieu of using these assumptions, an almost-sure convergence of the SA is proved under the assumption that the sequence of observation vectors is M-dependent. This assumption allows strong correlation between successive observations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A data-driven method for the steady state of randomly perturbed dynamics

We demonstrate a data-driven method to solve for the invariant probability density function of a randomly perturbed dynamical system. The key idea is to replace the boundary condition of numerical schemes by a least squares problem corresponding to a reference solution, which is generated by Monte Carlo simulation. With this method we can solve for the invariant probability density function in any local area with high accuracy, regardless of whether the attractor is covered by the numerical domain.