Hidden Attractors in Discrete Dynamical Systems.
Marek Berezowski1, Marcin Lawnik2
1Faculty of Chemical Engineering and Technology, Cracow University of Technology, ul. Warszawska 24, 30-155 Kraków, Poland.
Entropy (Basel, Switzerland)
|June 2, 2021
Summary
This study reveals hidden attractors in discrete dynamical systems, such as Newton's recursion and numerical integration. Understanding chaos in these systems is crucial for preventing negative impacts in engineering and chemical processes.
Area of Science:
- Dynamical Systems and Chaos Theory
- Numerical Analysis
- Applied Mathematics
Background:
- Chaos theory aids understanding of dynamical systems, but chaos can cause detrimental effects.
- Hidden attractors in dynamic processes pose significant challenges, particularly in continuous systems.
- Existing research predominantly focuses on multidimensional continuous systems.
Purpose of the Study:
- To demonstrate the existence of hidden attractors in discrete dynamical systems.
- To extend the understanding of hidden attractors beyond continuous systems.
- To highlight the relevance of chaos in discrete numerical methods.
Main Methods:
- Analysis of discrete dynamical systems.
- Investigation of Newton's recursion for chaotic behavior.
- Examination of numerical integration schemes for hidden attractors.
Main Results:
- Hidden attractors were identified in discrete systems, specifically within Newton's recursion.
- The presence of hidden attractors was also confirmed in numerical integration methods.
- This work extends the known occurrences of hidden attractors to discrete domains.
Conclusions:
- Hidden attractors are present in discrete systems, contrary to the common focus on continuous systems.
- The findings have implications for the stability and predictability of discrete numerical methods.
- Further research into discrete chaotic systems is warranted to mitigate potential negative effects.
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