# How long do numerical chaotic solutions remain valid

@article{Sauer1997HowLD, title={How long do numerical chaotic solutions remain valid}, author={Tim Sauer and Celso Grebogi and James A. Yorke}, journal={Physical Review Letters}, year={1997}, volume={79}, pages={59-62} }

Dynamical conditions for the loss of validity of numerical chaotic solutions of physical systems are already understood. However, the fundamental questions of {open_quotes}how good{close_quotes} and {open_quotes}for how long{close_quotes} the solutions are valid remained unanswered. This work answers these questions by establishing scaling laws for the shadowing distance and for the shadowing time in terms of physically meaningful quantities that are easily computable in practice. The scaling… Expand

#### 110 Citations

Shadowability of Chaotic Dynamical Systems

- Mathematics
- 2002

Abstract In studying their systems, physical scientists write differential equations derived from fundamental laws. These equations are then used to understand, analyze, predict, and control the… Expand

Unstable dimension variability in coupled chaotic systems.

- Physics, Medicine
- Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1999

This paper argues that unstable dimension variability can arise for small values of the coupling parameter and shows that unstable periodic orbits embedded in the dynamical invariant set of a coupled chaotic system can typically have different numbers of unstable directions. Expand

Pseudo-Deterministic Chaotic Systems

- Mathematics, Computer Science
- Int. J. Bifurc. Chaos
- 2003

It is argued that the effect of unstable dimension variability is more intense when the invariant chaotic set of the system loses transversal stability through a blowout bifurcation. Expand

Simulating a chaotic process

- Physics
- 2005

Computer simulations of partial differential equations of mathematical physics typically lead to some kind of high-dimensional dynamical system. When there is chaotic behavior we are faced with… Expand

Physical limit of prediction for chaotic motion of three-body problem

- Mathematics, Computer Science
- Commun. Nonlinear Sci. Numer. Simul.
- 2014

Mathematically reliable simulations of the chaotic trajectories of the three bodies suggest that, due to the butterfly effect of chaotic dynamic systems, the micro-level physical uncertainty of initial conditions might transfer into macroscopic uncertainty. Expand

Truncated chaotic trajectories in periodically driven systems with largely converging dynamics

- Physics
- 2000

Abstract Dynamical properties of numerically truncated trajectories are discussed for periodically driven chaotic systems with largely converging dynamics. Various trajectories having different… Expand

Predictability of orbits in coupled systems through finite-time Lyapunov exponents

- Physics
- 2013

The predictability of an orbit is a key issue when a physical model has strong sensitivity to the initial conditions and it is solved numerically. How close the computed chaotic orbits are to the… Expand

Obstruction to Deterministic Modeling of Chaotic Systems with an Invariant Subspace

- Mathematics, Computer Science
- Int. J. Bifurc. Chaos
- 2000

It is shown that unstable-dimension variability can occur in wide parameter regimes of chaotic systems with an invariant subspace such as systems of coupled chaotic oscillators, and this paper investigates this phenomenon by investigating a class of deterministic models. Expand

Exact Simulation of One-Dimensional Chaotic Dynamical Systems Using Algebraic Numbers

- Mathematics, Computer Science
- UCNC
- 2014

A method of true orbit generation that allowed us to perform exact simulations of discrete-time dynamical systems defined by one-dimensional piecewise linear and linear fractional maps with integer coefficients by generalizing the method proposed by Saito and Ito is introduced. Expand

Computation of true chaotic orbits using cubic irrationals

- Mathematics
- 2014

Abstract We introduce a method that enables us to generate long true orbits of discrete-time dynamical systems defined by one-dimensional piecewise linear fractional maps with integer coefficients.… Expand

#### References

SHOWING 1-6 OF 6 REFERENCES

"J."

- 1890

however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)… Expand

Phys Rev

- Lett.73, 1927
- 1994

Rev

- Lett.65, 1527 (1990); G. D. Quinlan and S. Tremaine, Mon. Not. R. Astron. Soc. 259, 505
- 1992

Proc

- Steklov Inst. Math. 90, 1 (1967); R. Bowen, J. Differential Equations 18, 333
- 1975

Proc

- Symp. Pure Math. (AMS 14, 5
- 1970

An Introduction to Probability Theory and Its Applications(Wiley

- New York,
- 1957