# Quartic polynomial approximation for fluctuations of separation of trajectories in chaos and correlation dimension

@article{Fouxon2019QuarticPA, title={Quartic polynomial approximation for fluctuations of separation of trajectories in chaos and correlation dimension}, author={Itzhak Fouxon and Siim Ainsaar and Jaan Kalda}, journal={Journal of Statistical Mechanics: Theory and Experiment}, year={2019}, volume={2019} }

We consider the cumulant generating function of the logarithm of the distance between two infinitesimally close trajectories of a chaotic system. Its long-time behavior is given by the generalized Lyapunov exponent providing the logarithmic growth rate of the kth moment of the distance. The Legendre transform of is a large deviations function that gives the probability of rare fluctuations where the logarithmic rate of change of the distance is much larger or much smaller than the mean rate…

## 4 Citations

### Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles.

- PhysicsPhysical review. E
- 2021

It is demonstrated that it is highly plausible that a pointwise limit for the growth of small perturbations of the Navier-Stokes equations exists and resolved the existing contradiction between the theory and observations, that predicts slow decrease of dimensionless Lyapunov exponent of turbulence with Re.

### Linear and nonlinear hydromagnetic stability in laminar and turbulent flows.

- Physics, MathematicsPhysical review. E
- 2021

Whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process is studied and a generalized Reynolds-Orr equation is derived for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy.

### Intermittency and collisions of fast sedimenting droplets in turbulence

- Physics
- 2022

We study theoretically and numerically spatial distribution and collision rate of droplets that sediment in homogeneous isotropic Navier-Stokes turbulence. It is assumed that, as it often happens in…

### On linear and non-linear hydromagnetic stability in laminar and turbulent flows

- Physics
- 2020

We derive several equations that determine the stability of purely hydrodynamic flow of an electrically conducting fluid with respect to creation of magnetic field. One equation determines the…

## References

SHOWING 1-10 OF 104 REFERENCES

### Kraichnan Flow in a Square: An Example of Integrable Chaos

- Mathematics
- 2006

The Kraichnan flow provides an example of a random dynamical system accessible to an exact analysis. We study the evolution of the infinitesimal separation between two Lagrangian trajectories of the…

### Large-deviation joint statistics of the finite-time Lyapunov spectrum in isotropic turbulence

- Physics
- 2015

One of the hallmarks of turbulent flows is the chaotic behavior of fluid particle paths with exponentially growing separation among them while their distance does not exceed the viscous range. The…

### Evolution to a singular measure and two sums of Lyapunov exponents

- Mathematics
- 2011

We consider dissipative dynamical systems represented by a smooth compressible flow in a finite domain. The density evolves according to the continuity (Liouville) equation. For a general,…

### Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

- Physics
- 1999

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged…

### Dynamics of threads and polymers in turbulence: power-law distributions and synchronization

- Physics
- 2012

We study the behavior of threads and polymers in a turbulent flow. These objects have finite spatial extension, so the flow along them differs slightly. The corresponding drag forces produce a finite…

### Generalized Lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices

- Mathematics
- 1988

We study the behavior of the generalized Lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsD. For products of random matrices…

### Characterisation of intermittency in chaotic systems

- Physics
- 1985

The authors discuss the characterisation of intermittency in chaotic dynamical systems by means of the time fluctuations of the response to a slight perturbation on the trajectory. A set of exponents…