1. Localizing the Chaotic Strange Attractors of Multiparameter Nonlinear Dynamical Systems using Competitive Modes
A Literary Analysis

2. Table of Contents Introduction Strange Attractors and Chaos
Localization through Competitive Modes
Lorenz Attractor Example
Conclusions

3. Dynamical Systems 𝑥 =𝜎(𝑦−𝑥) 𝑦 =𝑥 𝜌−𝑧 −𝑦 𝑧 =𝑥𝑦−𝛽𝑧 Introduction
Strange Attractors
Competitive Modes
Conclusions
𝑥 =𝜎(𝑦−𝑥) 𝑦 =𝑥 𝜌−𝑧 −𝑦 𝑧 =𝑥𝑦−𝛽𝑧

4. Strange Attractors
Introduction
Strange Attractors
Competitive Modes
Conclusions
Definition: The strange attractor A of a n-dimensional system of differential equations is an attractor that is fractal in nature.

5. Chaos
Introduction
Strange Attractors
Competitive Modes
Conclusions
Definition: Chaos is the phenomenon where a dynamical system is extremely sensitive to initial conditions in some set 𝐶⊆ ℝ 𝑛 .

6. Chaos: Lorenz Introduction Strange Attractors Competitive Modes
Conclusions

7. Localization Through Competitive Modes

8. Oscillators 𝑥 +𝛼𝑥=0 with 𝛼≥0 Introduction Strange Attractors
Competitive Modes
Conclusions
𝑥 +𝛼𝑥=0 with 𝛼≥0

9. Oscillators Introduction Strange Attractors Competitive Modes
Conclusions

10. Oscillators to Competitive Modes
Introduction
Strange Attractors
Competitive Modes
Conclusions
𝑥 1 = 𝐹 1 𝑥 1 ,… 𝑥 𝑛 … 𝑥 𝑛 = 𝐹 𝑛 𝑥 1 ,… 𝑥 𝑛

11. Oscillators to Competitive Modes
Introduction
Strange Attractors
Competitive Modes
Conclusions
𝑥 1 + 𝑥 1 𝑔 1 𝑥 1 ,… 𝑥 𝑛 = ℎ 1 𝑥 2 ,… 𝑥 𝑛 … 𝑥 𝑛 + 𝑥 𝑛 𝑔 𝑛 𝑥 1 ,… 𝑥 𝑛 = ℎ 𝑛 𝑥 1 ,… 𝑥 𝑛−1

12. Competitive Mode Conjecture
Introduction
Strange Attractors
Competitive Modes
Conclusions
Conjecture: The conditions for a dynamical system to be chaotic are:
there exist at least two 𝑔 𝑖 ’s in the system
at least one 𝑔 𝑖 is a function of 𝑡
at least one ℎ 𝑖 is a function of the system variables
∃𝑡∈ℝ so that 𝑔 𝑖 𝑡 ≈ 𝑔 𝑗 𝑡 and 𝑔 𝑖 𝑡 , 𝑔 𝑗 𝑡 >0 for some 𝑔 𝑖 and 𝑔 𝑗

13. Competitive Modes: Lorenz
Introduction
Strange Attractors
Competitive Modes
Conclusions

14. Competitive Modes: Lorenz
Introduction
Strange Attractors
Competitive Modes
Conclusions
𝒈 𝟏 = 𝒈 𝟐 :
𝑥 2 =1− 𝜎 2
𝒈 𝟏 = 𝒈 𝟑 :
𝜎𝑧= 𝑥 2 + 𝜎 𝜎+𝜌 − 𝛽 2
𝒈 𝟐 = 𝒈 𝟑 :
𝜎𝑧=𝜎𝜌+1− 𝛽 2

15. Competitive Modes: Lorenz
Introduction
Strange Attractors
Competitive Modes
Conclusions

16. Conclusions and Research Questions
Introduction
Strange Attractors
Competitive Modes
Conclusions
Localization techniques do already exist, but are not necessarily simple.
Is the Competitive Mode Conjecture true?
Which systems fulfill the competitive mode conjecture?
Does it also apply to discrete systems?

17. Questions? Introduction Strange Attractors Competitive Modes
Conclusions

18. Localization through Trajectories
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions

19. Localization through Trajectories
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions
Self-Excited Attractors
Hidden Attractors

20. Localization of Chua Hidden Attractor
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions
𝑥 =𝛼(𝑦−𝑥−𝑓(𝑥)) 𝑦 =𝑥−𝑦+𝑧 𝑧 =−𝛽 𝑦−𝛾 𝑧 𝑓 𝑥 = 𝑚 1 𝑥+( 𝑚 0 + 𝑚 1 ) 𝑥+1 −|𝑥−1| 2

21. Localization of Chua Hidden Attractor
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions

22. Localization of Chua Hidden Attractor
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions

23. Nambu Systems
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions
𝑥 = 𝜕 𝐻 1 𝜕𝑦 𝜕 𝐻 2 𝜕𝑧 − 𝜕 𝐻 1 𝜕𝑧 𝜕 𝐻 2 𝜕𝑦 𝑦 = 𝜕 𝐻 1 𝜕𝑧 𝜕 𝐻 2 𝜕𝑥 − 𝜕 𝐻 1 𝜕𝑥 𝜕 𝐻 2 𝜕𝑧 𝑧 = 𝜕 𝐻 1 𝜕𝑥 𝜕 𝐻 2 𝜕𝑦 − 𝜕 𝐻 1 𝜕𝑦 𝜕 𝐻 2 𝜕𝑥

24. Nambu Systems
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions
Then 𝐻 1 =0 and 𝐻 2 =0 Meaning 𝐻 1 𝑥 𝑡 ,𝑦 𝑡 ,𝑧 𝑡 = 𝐻 1 𝑥 0 ,𝑦 0 ,𝑧 0 𝐻 2 𝑥 𝑡 ,𝑦 𝑡 ,𝑧 𝑡 = 𝐻 2 𝑥 0 ,𝑦 0 ,𝑧 0

25. Nambu Hamiltonian Localization
Introduction
Strange Attractors
Trajectories
Nambu Mechanics
Competitive Modes
Conclusions