# Universal criterion for the breakup of invariant tori in dissipative systems.

@article{Ketoja1992UniversalCF, title={Universal criterion for the breakup of invariant tori in dissipative systems.}, author={Ketoja}, journal={Physical review letters}, year={1992}, volume={69 15}, pages={ 2180-2183 } }

The transition from quasiperiodicity to chaos is studied in a two-dimensional dissipative map with the inverse-golden-mean rotation number. On the basis of a decimation scheme, it is argued that the (minimal) slope of the critical iterated circle map is proportional to the effective Jacobian determinant. Approaching the zero-Jacobian-determinant limit, the factor of proportion becomes a universal constant. Numerical investigation on the dissipative standard map suggests that this universal…

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## References

SHOWING 1-10 OF 21 REFERENCES

New directions in dynamical systems

- Mathematics
- 2008

Introduction 1. Universality and Renormalisation in Dynamical Systems David Rand 2. Smooth Dynamics on the Interval (with an emphasis on quadratic-like maps) Sebastian van Strien 3. Global…

New directions in dynamical systems

- Mathematics
- 1990

Bargaining with reading habit is no need. Reading is not kind of something sold that you can take or not. It is a thing that will change your life to life better. It is the thing that will give you…

Physica D

- Physica D
- 1992

The effective Jacobian controls the " universal crossover " in period-doubling systems as well; see C. Reick

- Phys. Rev. A
- 1992

Phys. Rev. A

- Phys. Rev. A
- 1989

J. Chem. Phys

- J. Chem. Phys
- 1986

Phys. Rev. Lett

- Phys. Rev. Lett
- 1986

Phys. Rev. Lett

- Phys. Rev. Lett
- 1986

Physica D

- Physica D
- 1986

Phys. Rep

- Phys. Rep
- 1985