DEGENERATE COCYCLE WITH INDEX-1 AND LYAPUNOV EXPONENTS

2007 ◽  
Vol 07 (02) ◽  
pp. 229-245 ◽  
Author(s):  
NGUYEN HUU DU ◽  
TRINH KHANH DUY ◽  
VU TIEN VIET
Keyword(s):  
Lyapunov Exponents ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Initial Value ◽  
Leading Term ◽  
Random Difference ◽  
The Difference ◽  
Random Difference Equations

This paper deals with the solvability of initial-value problem and with Lyapunov exponents for linear implicit random difference equations, i.e. the difference equations where the leading term cannot be solved. An index-1 concept for linear implicit random difference equations is introduced and a formula of solutions is given. Paper is also concerned with a version of the multiplicative theorem of Oseledets type.

scholarly journals On a Theorem of Breiman and a Class of Random Difference Equations

2007 ◽  
Vol 44 (04) ◽  
pp. 1031-1046 ◽  
Author(s):  
Denis Denisov ◽  
Bert Zwart
Keyword(s):  
Difference Equations ◽  
Random Variables ◽  
Tail Behavior ◽  
Random Difference ◽  
Regularly Varying ◽  
Random Difference Equations

We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index α and that E{Yα+ε} < ∞ for some ε > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.

scholarly journals Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev ◽  
K. Stefanova
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Linear Difference Equation ◽  
Successive Approximations ◽  
Time Interval ◽  
Initial Value ◽  
Past Time ◽  
The Given ◽  
Unknown Solution

The object of investigation of the paper is a special type of difference equations containing the maximum value of the unknown function over a past time interval. These equations are adequate models of real processes which present state depends significantly on their maximal value over a past time interval. An algorithm based on the quasilinearization method is suggested to solve approximately the initial value problem for the given difference equation. Every successive approximation of the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given mixed problem. It is proved the quadratic convergence of the successive approximations. The suggested algorithm is realized as a computer program, and it is applied to an example, illustrating the advantages of the suggested scheme.

scholarly journals Existence, uniqueness and continuability of solutions of impulsive differential-difference equations

1999 ◽  
Vol 12 (3) ◽  
pp. 293-300 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Difference Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Initial Value

We consider an initial value problem for impulsive differential-difference equations, and obtain sufficient conditions for the existence, uniqueness, and continuability of solutions of such problem.

Second order properties of distribution tails and estimation of tail exponents in random difference equations

Extremes ◽  
2009 ◽  
Vol 12 (4) ◽  
pp. 361-400 ◽  
Author(s):  
Changryong Baek ◽  
Vladas Pipiras ◽  
Herwig Wendt ◽  
Patrice Abry
Keyword(s):  
Difference Equations ◽  
Second Order ◽  
Random Difference ◽  
Random Difference Equations

scholarly journals Solving the Topographic Potential Bias as an Initial Value Problem

2009 ◽  
Vol 44 (3) ◽  
pp. 75-84 ◽  
Author(s):  
L. Sjöberg
Keyword(s):  
Initial Value Problem ◽  
Potential Bias ◽  
Initial Value ◽  
Disturbing Potential ◽  
Computation Point ◽  
The Earth ◽  
Terrain Effect ◽  
Topographic Potential ◽  
The Difference ◽  
The Masses

Solving the Topographic Potential Bias as an Initial Value ProblemIf the gravitational potential or the disturbing potential of the Earth be downward continued by harmonic continuation inside the Earth's topography, it will be biased, the bias being the difference between the downward continued fictitious, harmonic potential and the real potential inside the masses. We use initial value problem techniques to solve for the bias. First, the solution is derived for a constant topographic density, in which case the bias can be expressed by a very simple formula related with the topographic height above the computation point. Second, for an arbitrary density distribution the bias becomes an integral along the vertical from the computation point to the Earth's surface. No topographic masses, except those along the vertical through the computation point, affect the bias. (To be exact, only the direct and indirect effects of an arbitrarily small but finite volume of mass around the surface point along the radius must be considered.) This implies that the frequently computed terrain effect is not needed (except, possibly, for an arbitrarily small inner-zone around the computation point) for computing the geoid by the method of analytical continuation.

scholarly journals Null recurrence and transience of random difference equations in the contractive case

2017 ◽  
Vol 54 (4) ◽  
pp. 1089-1110 ◽  
Author(s):  
Gerold Alsmeyer ◽  
Dariusz Buraczewski ◽  
Alexander Iksanov
Keyword(s):  
Markov Chain ◽  
Difference Equation ◽  
Difference Equations ◽  
Random Vectors ◽  
Positive Recurrence ◽  
Null Recurrence ◽  
Random Difference Equation ◽  
Random Difference ◽  
Random Difference Equations ◽  
Independent And Identically Distributed

Abstract Given a sequence (Mk, Qk)k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (Xn)n ≥ 0, defined by the random difference equation Xn = MnXn - 1 + Qn for n ≥ 1, where X0 is independent of (Mk, Qk)k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (Xn)n ≥ 0 is contractive in the sense that M1 ⋯ Mn → 0 almost surely, yet occasional large values of the Qn overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (Xn)n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.

scholarly journals Approximate Solution of the Fuzzy Triangular Initial Value Problem with Different Fractional Operator

2020 ◽  
Vol 9 (1) ◽  
pp. 1242-1249
Keyword(s):  
Initial Value Problem ◽  
Reproducing Kernel ◽  
Fractional Operator ◽  
Initial Value ◽  
Derivative Operator ◽  
Method Of Solution ◽  
The Difference ◽  
Definition Of ◽  
Reproducing Kernel Theory ◽  
Solution Behaviour

This study aims to conduct a comparison regarding the process of solving the fuzzy triangular initial value problem (FTIVP). The series solution of this problem is acquired through the reproducing kernel theory (RKT), although there have been past studies on FTIVP, there is no specialist study to compare solutions for the definition of different fractional operator. The comparisons where located through the difference in the use of an operator in the process of solution by using Riemann-Liouville integral operator (RLIO) and then by using Caputo fractional derivative operator (CFDO). Algorithm was presented to validate the method of solution and to view the effect of changing the operators on the solution behaviour in the two cases. During this comparison, the effectiveness of RKT was cleared and the notion of difference between using RLIO and CFDO were fixedly identified. Applications: The results identified in this research pronounced active difference in the behavior of errors, CDFO variations, and the behavior of error in favour of RLIO.

Sign in / Sign up

Export Citation Format

Share Document