SlideShare ist ein Scribd-Unternehmen logo
1 von 61
Discoveries and
Challenges in Chaos
      Theory




         Xiong Wang 王雄
Supervised by: Prof. Guanrong Chen
Centre for Chaos and Complex Networks
     City University of Hong Kong
Some basic questions?
 What’s  the fundamental mechanism in
  generating chaos?
 What kind of systems could generate
  chaos?
 Could a system with only one stable
  equilibrium also generate chaotic
  dynamics?
 Generally, what’s the relation between a
  chaotic system and the stability of its
  equilibria?                                2
Equilibria
 An equilibrium (or fixed point) of an
  autonomous system of ordinary differential
  equations (ODEs) is a solution that does not
  change with time.
 The ODE x = f ( x ) has an equilibrium
               &
  solution xe , if f ( xe ) = 0
 Finding such equilibria, by solving the f ( x) = 0
  equation analytically, is easy only in a few
  special cases.
                                                       3
Jacobian Matrix
 The  stability of typical equilibria of smooth
  ODEs is determined by the sign of real parts
  of the system Jacobian eigenvalues.
 Jacobian matrix:




                                                   4
Hyperbolic Equilibria
 The  eigenvalues of J determine linear
  stability of the equilibria.
 An equilibrium is stable if all eigenvalues
  have negative real parts; it is unstable if at
  least one eigenvalue has positive real part.
 The equilibrium is said to be hyperbolic if all
  eigenvalues have non-zero real parts.


                                                    5
Hartman-Grobman Theorem
 The local phase portrait of a hyperbolic
 equilibrium of a nonlinear system is
 equivalent to that of its linearized system.




                                                6
Equilibrium in 3D:
3 real eigenvalues




                     7
Equilibrium in 3D:
1 real + 2 complex-conjugates




                                8
9
10
Illustration of typical homoclinic
and heteroclinic orbits




                                     11
Review of the two theorems
 Hartman-Grobman     theorem says nonlinear
  system is the ‘same’ as its linearized model
 Shilnikov theorem says if saddle-focus +

  Shilnikov inequalities + homoclinic or
  heteroclinic orbit, then chaos exists
 Most   classical 3D chaotic systems belong
  to this type
 Most chaotic systems have unstable
  equilibria
                                                 12
Equilibria and eigenvalues of
several typical systems




                                13
 Lorenz System
     x = a( y − x)
      &
    
     y = cx − xz − y
      &
     z = xy − bz ,
    &
     a = 10, b = 8 / 3, c = 28

E. N. Lorenz, “Deterministic non-periodic flow,” J. Atmos. Sci., 20,
   130-141, 1963.

                                                                       14
Untable saddle-focus is
important for generating chaos




                                 15
 Chen System

  x = a ( y − x)
   &
 
  y = (c − a ) x − xz + cy
   &
  z = xy − bz ,
 &
   a = 35; b = 3; c = 28

G. Chen and T. Ueta, “Yet another chaotic attractor,” Int J. of Bifurcation and Chaos, 9(7),
   1465-1466, 1999.
T. Ueta and G. Chen, “Bifurcation analysis of Chen’s equation,” Int J. of Bifurcation and
   Chaos, 10(8), 1917-1931, 2000.
T. S. Zhou, G. Chen and Y. Tang, “Chen's attractor exists,” Int. J. of Bifurcation and Chaos,
   14, 3167-3178, 2004.                                                                         16
17
Rossler System




                 18
Do these two theorems
prevent “stable” chaos?
 Hartman-Grobman      theorem says nonlinear
  system is the same as its linearized model.
 But it holds only locally …not necessarily the
  same globally.
 Shilnikov theorem says if saddle-focus +

  Shilnikov inequalities + homoclinic or
  heteroclinic orbit, then chaos exists.
 But it is only a sufficient condition, not a
  necessary one.
                                                   19
Don’t be scarred by theorems
 So,  actually the
  theorems do not rule
  out the possibility of
  finding chaos in a
  system with a stable
  equilibrum.
 Just to grasp the
  loophole of the
  theorems …
                               20
Try to find a chaotic system
with a stable Equilibrium
 Some     criterions for the new system:
1.   Simple algebraic equations
2.   One stable equilibrium

To start with, let us first review some of the
  simple Sprott chaotic systems with only one
  equilibrium …


                                                 21
Some Sprott systems




                      22
Idea
1.   Sprott systems I, J, L, N and R all have only
     one saddle-focus equilibrium, while systems
     D and E are both degenerate.
2.   A tiny perturbation to the system may be
     able to change such a degenerate
     equilibrium to a stable one.
3.   Hope it will work …


                                                 23
Finally Result




 When a = 0, it is the Sprott E system
 When a > 0, however, the stability of the

  single equilibrium is fundamentally different
 The single equilibrium becomes stable
                                                  24
Equilibria and eigenvalues of
the new system




                                25
The largest Lyapunov
exponent




                       26
The new system:
chaotic attractor with a = 0.006




                                   27
Bifurcation diagram
a period-doubling route to chaos




                                   28
Phase portraits and frequency
spectra




      a = 0.006      a=
                    0.02        29
Phase portraits and frequency
spectra




        a=          a=
       0.03        0.05         30
Attracting basins of the
equilibra




                           31
Conclusions
 We   have reported the finding of a simple
  3D autonomous chaotic system which, very
  surprisingly, has only one stable node-
  focus equilibrium.
 It has been verified to be chaotic in the
  sense of having a positive largest
  Lyapunov exponent, a fractional
  dimension, a continuous frequency
  spectrum, and a period-doubling route to
  chaos.                                       32
Theoretical challenges
To be further considered:
 Shilnikov homoclinic criterion?
 not applicable for this case

 Rigorous proof of the existence?
  Horseshoe?
 Coexistence of point attractor and strange
attractor?
 Inflation of attracting basin of the equilibrium?


                                                      33
Coexisting of point, cycle and
strange attractor




                                 34
Coexisting of point, cycle and
strange attractor




                                 35
Coexisting of point, cycle and
strange attractor




                                 36
one question answered,
more questions come …
Chaotic system with one stable
            equilibrium
Chaotic system with:
 No equilibrium?

 Two stable equilibria?

 Three stable equilibria?

 Any number of equilibria?

 Tunable stability of equilibria?
            Xiong Wang: Chaotic system with only one   37
                      stable equilibrium
Chaotic system with no
equilibrium




           Xiong Wang: Chaotic system with only one   38
                     stable equilibrium
Chaotic system with one
stable equilibrium




           Xiong Wang: Chaotic system with only one   39
                     stable equilibrium
Idea
 Really hard to find a chaotic system with a
  given number of equilibria in the sea of all
  possibility ODE systems …
 Try another way…

 To add symmetry to this one stable system.

 We can adjust the stability of the equilibria
  very easily by adjusting one parameter


                 Xiong Wang: Chaotic system with only one   40
                           stable equilibrium
The idea of symmetry



                     W =Z                  n




  W plane                                        Z plane
 W = (u,v) =                                 Z = (x,y) = x+yi
    u+vi                                      Symmetrical
  Original                                       system
  system
               Xiong Wang: Chaotic system with only one         41
                         stable equilibrium
                                                  (x,y,z)
symmetry




  Xiong Wang: Chaotic system with only one   42
            stable equilibrium
Stability of the two equilibria
 There are two symmetrical equilibria which
 are independent of the parameter a

 The    eigenvalue of Jacobian




 So,   a > 0 stable; a < 0 unstable

                  Xiong Wang: Chaotic system with only one   43
                            stable equilibrium
symmetry




a = 0.005 > 0, stable
      equilibria
       Xiong Wang: Chaotic system with only one   44
                 stable equilibrium
symmetry




a = - 0.01 < 0, unstable
        equilibria
        Xiong Wang: Chaotic system with only one
                  stable equilibrium
                                                   45
symmetry




  Xiong Wang: Chaotic system with only one   46
            stable equilibrium
Three symmetrical equilibria
with tunable stability




           Xiong Wang: Chaotic system with only one   47
                     stable equilibrium
symmetry




a = - 0.01 < 0, unstable
        equilibria
        Xiong Wang: Chaotic system with only one
                  stable equilibrium
                                                   48
symmetry




a = 0.005 > 0, stable
      equilibria
       Xiong Wang: Chaotic system with only one
                 stable equilibrium
                                                  49
Theoretically we can create
any number of equilibria …




           Xiong Wang: Chaotic system with only one   50
                     stable equilibrium
Conclusions
 Chaotic system with:
  No equilibrium - found

  Two stable equilibria - found

  Three stable equilibria - found

  Theoretically, we can create any
   number of equilibria …
  We can control the stability of equilibria
   by adjusting one parameter
                Xiong Wang: Chaotic system with only one   51
                          stable equilibrium
Chaos is a global phenomenon
A   system can be locally stable near the
  equilibrium, but globally chaotic far from
  the equilibrium.
 This interesting phenomenon is worth
  further studying, both theoretically and
  experimentally, to further reveal the
  intrinsic relation between the local stability
  of an equilibrium and the global complex
  dynamical behaviors of a chaotic system
                                                   52
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email: wangxiong8686@gmail.com


                                 53
ADDITIONAL BONUS:
ATTRACTOR GALLERY
                    54
55
56
57
58
59
60
61

Más contenido relacionado

Was ist angesagt?

Duffing oscillator and driven damped pendulum
Duffing oscillator and driven damped pendulumDuffing oscillator and driven damped pendulum
Duffing oscillator and driven damped pendulumYiteng Dang
 
Fault tolerant process control
Fault tolerant process controlFault tolerant process control
Fault tolerant process controlSpringer
 
Senior Seminar: Systems of Differential Equations
Senior Seminar:  Systems of Differential EquationsSenior Seminar:  Systems of Differential Equations
Senior Seminar: Systems of Differential EquationsJDagenais
 
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROLOUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROLijait
 
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)Matthew Leingang
 
Systems Of Differential Equations
Systems Of Differential EquationsSystems Of Differential Equations
Systems Of Differential EquationsJDagenais
 
Series solution to ordinary differential equations
Series solution to ordinary differential equations Series solution to ordinary differential equations
Series solution to ordinary differential equations University of Windsor
 
Controllability of Linear Dynamical System
Controllability of  Linear Dynamical SystemControllability of  Linear Dynamical System
Controllability of Linear Dynamical SystemPurnima Pandit
 

Was ist angesagt? (12)

Duffing oscillator and driven damped pendulum
Duffing oscillator and driven damped pendulumDuffing oscillator and driven damped pendulum
Duffing oscillator and driven damped pendulum
 
Ch07 5
Ch07 5Ch07 5
Ch07 5
 
Fault tolerant process control
Fault tolerant process controlFault tolerant process control
Fault tolerant process control
 
Ch05 2
Ch05 2Ch05 2
Ch05 2
 
Senior Seminar: Systems of Differential Equations
Senior Seminar:  Systems of Differential EquationsSenior Seminar:  Systems of Differential Equations
Senior Seminar: Systems of Differential Equations
 
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROLOUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL
 
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)
 
Systems Of Differential Equations
Systems Of Differential EquationsSystems Of Differential Equations
Systems Of Differential Equations
 
Series solution to ordinary differential equations
Series solution to ordinary differential equations Series solution to ordinary differential equations
Series solution to ordinary differential equations
 
NTU_paper
NTU_paperNTU_paper
NTU_paper
 
Controllability of Linear Dynamical System
Controllability of  Linear Dynamical SystemControllability of  Linear Dynamical System
Controllability of Linear Dynamical System
 
Ch07 9
Ch07 9Ch07 9
Ch07 9
 

Ähnlich wie CCCN Talk on Stable Chaos

2 general properties
2 general properties2 general properties
2 general propertieskatamthreveni
 
Equilibrium point analysis linearization technique
Equilibrium point analysis linearization techniqueEquilibrium point analysis linearization technique
Equilibrium point analysis linearization techniqueTarun Gehlot
 
lec_2 - Copy lyponve stability of system .pptx
lec_2 - Copy lyponve stability of system .pptxlec_2 - Copy lyponve stability of system .pptx
lec_2 - Copy lyponve stability of system .pptxabbas miry
 
lec_2 for stability of control system .pptx
lec_2 for stability of control system .pptxlec_2 for stability of control system .pptx
lec_2 for stability of control system .pptxabbas miry
 
Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential Equation
Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential  EquationHopf-Bifurcation Ina Two Dimensional Nonlinear Differential  Equation
Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential EquationIJMER
 
DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdf
DOMV No 8  MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD  - FREE VIBRATION.pdfDOMV No 8  MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD  - FREE VIBRATION.pdf
DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdfahmedelsharkawy98
 
M.pranavi[22R01A6735] maths^.pptx
M.pranavi[22R01A6735] maths^.pptxM.pranavi[22R01A6735] maths^.pptx
M.pranavi[22R01A6735] maths^.pptxRohanThota3
 
System of linear equations
System of linear equationsSystem of linear equations
System of linear equationsCesar Mendoza
 
System of linear equations
System of linear equationsSystem of linear equations
System of linear equationsCesar Mendoza
 
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...ijcseit
 
Non linear Dynamical Control Systems
Non linear Dynamical Control SystemsNon linear Dynamical Control Systems
Non linear Dynamical Control SystemsArslan Ahmed Amin
 
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
 
Lagrangian Mechanics
Lagrangian MechanicsLagrangian Mechanics
Lagrangian MechanicsAmeenSoomro1
 
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
 
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
 

Ähnlich wie CCCN Talk on Stable Chaos (20)

Stable chaos
Stable chaosStable chaos
Stable chaos
 
2 general properties
2 general properties2 general properties
2 general properties
 
Equilibrium point analysis linearization technique
Equilibrium point analysis linearization techniqueEquilibrium point analysis linearization technique
Equilibrium point analysis linearization technique
 
Unit 5: All
Unit 5: AllUnit 5: All
Unit 5: All
 
lec_2 - Copy lyponve stability of system .pptx
lec_2 - Copy lyponve stability of system .pptxlec_2 - Copy lyponve stability of system .pptx
lec_2 - Copy lyponve stability of system .pptx
 
lyapunov.pdf
lyapunov.pdflyapunov.pdf
lyapunov.pdf
 
lec_2 for stability of control system .pptx
lec_2 for stability of control system .pptxlec_2 for stability of control system .pptx
lec_2 for stability of control system .pptx
 
Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential Equation
Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential  EquationHopf-Bifurcation Ina Two Dimensional Nonlinear Differential  Equation
Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential Equation
 
Oscillation ppt
Oscillation ppt Oscillation ppt
Oscillation ppt
 
Lect 2 bif_th
Lect 2 bif_thLect 2 bif_th
Lect 2 bif_th
 
DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdf
DOMV No 8  MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD  - FREE VIBRATION.pdfDOMV No 8  MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD  - FREE VIBRATION.pdf
DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdf
 
M.pranavi[22R01A6735] maths^.pptx
M.pranavi[22R01A6735] maths^.pptxM.pranavi[22R01A6735] maths^.pptx
M.pranavi[22R01A6735] maths^.pptx
 
System of linear equations
System of linear equationsSystem of linear equations
System of linear equations
 
System of linear equations
System of linear equationsSystem of linear equations
System of linear equations
 
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...
 
Non linear Dynamical Control Systems
Non linear Dynamical Control SystemsNon linear Dynamical Control Systems
Non linear Dynamical Control Systems
 
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...
 
Lagrangian Mechanics
Lagrangian MechanicsLagrangian Mechanics
Lagrangian Mechanics
 
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
 
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...
 

Mehr von Xiong Wang

Attractors distribution
Attractors distributionAttractors distribution
Attractors distributionXiong Wang
 
经典力学的三种表述以及对应的量子化
经典力学的三种表述以及对应的量子化经典力学的三种表述以及对应的量子化
经典力学的三种表述以及对应的量子化Xiong Wang
 
What is mass_lec1
What is mass_lec1What is mass_lec1
What is mass_lec1Xiong Wang
 
How the three
How the three How the three
How the three Xiong Wang
 
Toward a completed theory of relativity
Toward a completed theory of relativityToward a completed theory of relativity
Toward a completed theory of relativityXiong Wang
 
Above under and beyond brownian motion talk
Above under and beyond brownian motion talkAbove under and beyond brownian motion talk
Above under and beyond brownian motion talkXiong Wang
 
Chen’ family and rossler family
Chen’ family and rossler familyChen’ family and rossler family
Chen’ family and rossler familyXiong Wang
 
纵经沧海依然故我 (eadem mutata resurgo)
纵经沧海依然故我 (eadem mutata resurgo)纵经沧海依然故我 (eadem mutata resurgo)
纵经沧海依然故我 (eadem mutata resurgo)Xiong Wang
 

Mehr von Xiong Wang (9)

The Attractor
The AttractorThe Attractor
The Attractor
 
Attractors distribution
Attractors distributionAttractors distribution
Attractors distribution
 
经典力学的三种表述以及对应的量子化
经典力学的三种表述以及对应的量子化经典力学的三种表述以及对应的量子化
经典力学的三种表述以及对应的量子化
 
What is mass_lec1
What is mass_lec1What is mass_lec1
What is mass_lec1
 
How the three
How the three How the three
How the three
 
Toward a completed theory of relativity
Toward a completed theory of relativityToward a completed theory of relativity
Toward a completed theory of relativity
 
Above under and beyond brownian motion talk
Above under and beyond brownian motion talkAbove under and beyond brownian motion talk
Above under and beyond brownian motion talk
 
Chen’ family and rossler family
Chen’ family and rossler familyChen’ family and rossler family
Chen’ family and rossler family
 
纵经沧海依然故我 (eadem mutata resurgo)
纵经沧海依然故我 (eadem mutata resurgo)纵经沧海依然故我 (eadem mutata resurgo)
纵经沧海依然故我 (eadem mutata resurgo)
 

Último

Dhavni Theory by Anandvardhana Indian Poetics
Dhavni Theory by Anandvardhana Indian PoeticsDhavni Theory by Anandvardhana Indian Poetics
Dhavni Theory by Anandvardhana Indian PoeticsDhatriParmar
 
The OERs: Transforming Education for Sustainable Future by Dr. Sarita Anand
The OERs: Transforming Education for Sustainable Future by Dr. Sarita AnandThe OERs: Transforming Education for Sustainable Future by Dr. Sarita Anand
The OERs: Transforming Education for Sustainable Future by Dr. Sarita AnandDr. Sarita Anand
 
Research Methodology and Tips on Better Research
Research Methodology and Tips on Better ResearchResearch Methodology and Tips on Better Research
Research Methodology and Tips on Better ResearchRushdi Shams
 
3.12.24 Freedom Summer in Mississippi.pptx
3.12.24 Freedom Summer in Mississippi.pptx3.12.24 Freedom Summer in Mississippi.pptx
3.12.24 Freedom Summer in Mississippi.pptxmary850239
 
Arti Languages Pre Seed Send Ahead Pitchdeck 2024.pdf
Arti Languages Pre Seed Send Ahead Pitchdeck 2024.pdfArti Languages Pre Seed Send Ahead Pitchdeck 2024.pdf
Arti Languages Pre Seed Send Ahead Pitchdeck 2024.pdfwill854175
 
How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17
How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17
How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17Celine George
 
BBA 205 BUSINESS ENVIRONMENT UNIT I.pptx
BBA 205 BUSINESS ENVIRONMENT UNIT I.pptxBBA 205 BUSINESS ENVIRONMENT UNIT I.pptx
BBA 205 BUSINESS ENVIRONMENT UNIT I.pptxProf. Kanchan Kumari
 
Metabolism , Metabolic Fate& disorders of cholesterol.pptx
Metabolism , Metabolic Fate& disorders of cholesterol.pptxMetabolism , Metabolic Fate& disorders of cholesterol.pptx
Metabolism , Metabolic Fate& disorders of cholesterol.pptxDr. Santhosh Kumar. N
 
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...Nguyen Thanh Tu Collection
 
AUDIENCE THEORY - PARTICIPATORY - JENKINS.pptx
AUDIENCE THEORY - PARTICIPATORY - JENKINS.pptxAUDIENCE THEORY - PARTICIPATORY - JENKINS.pptx
AUDIENCE THEORY - PARTICIPATORY - JENKINS.pptxiammrhaywood
 
2024.03.16 How to write better quality materials for your learners ELTABB San...
2024.03.16 How to write better quality materials for your learners ELTABB San...2024.03.16 How to write better quality materials for your learners ELTABB San...
2024.03.16 How to write better quality materials for your learners ELTABB San...Sandy Millin
 
The basics of sentences session 8pptx.pptx
The basics of sentences session 8pptx.pptxThe basics of sentences session 8pptx.pptx
The basics of sentences session 8pptx.pptxheathfieldcps1
 
LEAD6001 - Introduction to Advanced Stud
LEAD6001 - Introduction to Advanced StudLEAD6001 - Introduction to Advanced Stud
LEAD6001 - Introduction to Advanced StudDr. Bruce A. Johnson
 
25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...
25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...
25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...Nguyen Thanh Tu Collection
 
Material Remains as Source of Ancient Indian History & Culture.ppt
Material Remains as Source of Ancient Indian History & Culture.pptMaterial Remains as Source of Ancient Indian History & Culture.ppt
Material Remains as Source of Ancient Indian History & Culture.pptBanaras Hindu University
 
2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...
2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...
2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...Marlene Maheu
 
Alamkara theory by Bhamaha Indian Poetics (1).pptx
Alamkara theory by Bhamaha Indian Poetics (1).pptxAlamkara theory by Bhamaha Indian Poetics (1).pptx
Alamkara theory by Bhamaha Indian Poetics (1).pptxDhatriParmar
 
EDD8524 The Future of Educational Leader
EDD8524 The Future of Educational LeaderEDD8524 The Future of Educational Leader
EDD8524 The Future of Educational LeaderDr. Bruce A. Johnson
 

Último (20)

Dhavni Theory by Anandvardhana Indian Poetics
Dhavni Theory by Anandvardhana Indian PoeticsDhavni Theory by Anandvardhana Indian Poetics
Dhavni Theory by Anandvardhana Indian Poetics
 
The OERs: Transforming Education for Sustainable Future by Dr. Sarita Anand
The OERs: Transforming Education for Sustainable Future by Dr. Sarita AnandThe OERs: Transforming Education for Sustainable Future by Dr. Sarita Anand
The OERs: Transforming Education for Sustainable Future by Dr. Sarita Anand
 
Least Significance Difference:Biostatics and Research Methodology
Least Significance Difference:Biostatics and Research MethodologyLeast Significance Difference:Biostatics and Research Methodology
Least Significance Difference:Biostatics and Research Methodology
 
Research Methodology and Tips on Better Research
Research Methodology and Tips on Better ResearchResearch Methodology and Tips on Better Research
Research Methodology and Tips on Better Research
 
ANOVA Parametric test: Biostatics and Research Methodology
ANOVA Parametric test: Biostatics and Research MethodologyANOVA Parametric test: Biostatics and Research Methodology
ANOVA Parametric test: Biostatics and Research Methodology
 
3.12.24 Freedom Summer in Mississippi.pptx
3.12.24 Freedom Summer in Mississippi.pptx3.12.24 Freedom Summer in Mississippi.pptx
3.12.24 Freedom Summer in Mississippi.pptx
 
Arti Languages Pre Seed Send Ahead Pitchdeck 2024.pdf
Arti Languages Pre Seed Send Ahead Pitchdeck 2024.pdfArti Languages Pre Seed Send Ahead Pitchdeck 2024.pdf
Arti Languages Pre Seed Send Ahead Pitchdeck 2024.pdf
 
How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17
How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17
How to Customise Quotation's Appearance Using PDF Quote Builder in Odoo 17
 
BBA 205 BUSINESS ENVIRONMENT UNIT I.pptx
BBA 205 BUSINESS ENVIRONMENT UNIT I.pptxBBA 205 BUSINESS ENVIRONMENT UNIT I.pptx
BBA 205 BUSINESS ENVIRONMENT UNIT I.pptx
 
Metabolism , Metabolic Fate& disorders of cholesterol.pptx
Metabolism , Metabolic Fate& disorders of cholesterol.pptxMetabolism , Metabolic Fate& disorders of cholesterol.pptx
Metabolism , Metabolic Fate& disorders of cholesterol.pptx
 
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - HK2 (...
 
AUDIENCE THEORY - PARTICIPATORY - JENKINS.pptx
AUDIENCE THEORY - PARTICIPATORY - JENKINS.pptxAUDIENCE THEORY - PARTICIPATORY - JENKINS.pptx
AUDIENCE THEORY - PARTICIPATORY - JENKINS.pptx
 
2024.03.16 How to write better quality materials for your learners ELTABB San...
2024.03.16 How to write better quality materials for your learners ELTABB San...2024.03.16 How to write better quality materials for your learners ELTABB San...
2024.03.16 How to write better quality materials for your learners ELTABB San...
 
The basics of sentences session 8pptx.pptx
The basics of sentences session 8pptx.pptxThe basics of sentences session 8pptx.pptx
The basics of sentences session 8pptx.pptx
 
LEAD6001 - Introduction to Advanced Stud
LEAD6001 - Introduction to Advanced StudLEAD6001 - Introduction to Advanced Stud
LEAD6001 - Introduction to Advanced Stud
 
25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...
25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...
25 CHUYÊN ĐỀ ÔN THI TỐT NGHIỆP THPT 2023 – BÀI TẬP PHÁT TRIỂN TỪ ĐỀ MINH HỌA...
 
Material Remains as Source of Ancient Indian History & Culture.ppt
Material Remains as Source of Ancient Indian History & Culture.pptMaterial Remains as Source of Ancient Indian History & Culture.ppt
Material Remains as Source of Ancient Indian History & Culture.ppt
 
2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...
2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...
2024 March 11, Telehealth Billing- Current Telehealth CPT Codes & Telehealth ...
 
Alamkara theory by Bhamaha Indian Poetics (1).pptx
Alamkara theory by Bhamaha Indian Poetics (1).pptxAlamkara theory by Bhamaha Indian Poetics (1).pptx
Alamkara theory by Bhamaha Indian Poetics (1).pptx
 
EDD8524 The Future of Educational Leader
EDD8524 The Future of Educational LeaderEDD8524 The Future of Educational Leader
EDD8524 The Future of Educational Leader
 

CCCN Talk on Stable Chaos

  • 1. Discoveries and Challenges in Chaos Theory Xiong Wang 王雄 Supervised by: Prof. Guanrong Chen Centre for Chaos and Complex Networks City University of Hong Kong
  • 2. Some basic questions?  What’s the fundamental mechanism in generating chaos?  What kind of systems could generate chaos?  Could a system with only one stable equilibrium also generate chaotic dynamics?  Generally, what’s the relation between a chaotic system and the stability of its equilibria? 2
  • 3. Equilibria  An equilibrium (or fixed point) of an autonomous system of ordinary differential equations (ODEs) is a solution that does not change with time.  The ODE x = f ( x ) has an equilibrium & solution xe , if f ( xe ) = 0  Finding such equilibria, by solving the f ( x) = 0 equation analytically, is easy only in a few special cases. 3
  • 4. Jacobian Matrix  The stability of typical equilibria of smooth ODEs is determined by the sign of real parts of the system Jacobian eigenvalues.  Jacobian matrix: 4
  • 5. Hyperbolic Equilibria  The eigenvalues of J determine linear stability of the equilibria.  An equilibrium is stable if all eigenvalues have negative real parts; it is unstable if at least one eigenvalue has positive real part.  The equilibrium is said to be hyperbolic if all eigenvalues have non-zero real parts. 5
  • 6. Hartman-Grobman Theorem  The local phase portrait of a hyperbolic equilibrium of a nonlinear system is equivalent to that of its linearized system. 6
  • 7. Equilibrium in 3D: 3 real eigenvalues 7
  • 8. Equilibrium in 3D: 1 real + 2 complex-conjugates 8
  • 9. 9
  • 10. 10
  • 11. Illustration of typical homoclinic and heteroclinic orbits 11
  • 12. Review of the two theorems  Hartman-Grobman theorem says nonlinear system is the ‘same’ as its linearized model  Shilnikov theorem says if saddle-focus + Shilnikov inequalities + homoclinic or heteroclinic orbit, then chaos exists  Most classical 3D chaotic systems belong to this type  Most chaotic systems have unstable equilibria 12
  • 13. Equilibria and eigenvalues of several typical systems 13
  • 14.  Lorenz System  x = a( y − x) &   y = cx − xz − y &  z = xy − bz , & a = 10, b = 8 / 3, c = 28 E. N. Lorenz, “Deterministic non-periodic flow,” J. Atmos. Sci., 20, 130-141, 1963. 14
  • 15. Untable saddle-focus is important for generating chaos 15
  • 16.  Chen System  x = a ( y − x) &   y = (c − a ) x − xz + cy &  z = xy − bz , & a = 35; b = 3; c = 28 G. Chen and T. Ueta, “Yet another chaotic attractor,” Int J. of Bifurcation and Chaos, 9(7), 1465-1466, 1999. T. Ueta and G. Chen, “Bifurcation analysis of Chen’s equation,” Int J. of Bifurcation and Chaos, 10(8), 1917-1931, 2000. T. S. Zhou, G. Chen and Y. Tang, “Chen's attractor exists,” Int. J. of Bifurcation and Chaos, 14, 3167-3178, 2004. 16
  • 17. 17
  • 19. Do these two theorems prevent “stable” chaos?  Hartman-Grobman theorem says nonlinear system is the same as its linearized model.  But it holds only locally …not necessarily the same globally.  Shilnikov theorem says if saddle-focus + Shilnikov inequalities + homoclinic or heteroclinic orbit, then chaos exists.  But it is only a sufficient condition, not a necessary one. 19
  • 20. Don’t be scarred by theorems  So, actually the theorems do not rule out the possibility of finding chaos in a system with a stable equilibrum.  Just to grasp the loophole of the theorems … 20
  • 21. Try to find a chaotic system with a stable Equilibrium  Some criterions for the new system: 1. Simple algebraic equations 2. One stable equilibrium To start with, let us first review some of the simple Sprott chaotic systems with only one equilibrium … 21
  • 23. Idea 1. Sprott systems I, J, L, N and R all have only one saddle-focus equilibrium, while systems D and E are both degenerate. 2. A tiny perturbation to the system may be able to change such a degenerate equilibrium to a stable one. 3. Hope it will work … 23
  • 24. Finally Result  When a = 0, it is the Sprott E system  When a > 0, however, the stability of the single equilibrium is fundamentally different  The single equilibrium becomes stable 24
  • 25. Equilibria and eigenvalues of the new system 25
  • 27. The new system: chaotic attractor with a = 0.006 27
  • 29. Phase portraits and frequency spectra a = 0.006 a= 0.02 29
  • 30. Phase portraits and frequency spectra a= a= 0.03 0.05 30
  • 31. Attracting basins of the equilibra 31
  • 32. Conclusions  We have reported the finding of a simple 3D autonomous chaotic system which, very surprisingly, has only one stable node- focus equilibrium.  It has been verified to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous frequency spectrum, and a period-doubling route to chaos. 32
  • 33. Theoretical challenges To be further considered:  Shilnikov homoclinic criterion? not applicable for this case  Rigorous proof of the existence? Horseshoe?  Coexistence of point attractor and strange attractor?  Inflation of attracting basin of the equilibrium? 33
  • 34. Coexisting of point, cycle and strange attractor 34
  • 35. Coexisting of point, cycle and strange attractor 35
  • 36. Coexisting of point, cycle and strange attractor 36
  • 37. one question answered, more questions come … Chaotic system with one stable equilibrium Chaotic system with:  No equilibrium?  Two stable equilibria?  Three stable equilibria?  Any number of equilibria?  Tunable stability of equilibria? Xiong Wang: Chaotic system with only one 37 stable equilibrium
  • 38. Chaotic system with no equilibrium Xiong Wang: Chaotic system with only one 38 stable equilibrium
  • 39. Chaotic system with one stable equilibrium Xiong Wang: Chaotic system with only one 39 stable equilibrium
  • 40. Idea  Really hard to find a chaotic system with a given number of equilibria in the sea of all possibility ODE systems …  Try another way…  To add symmetry to this one stable system.  We can adjust the stability of the equilibria very easily by adjusting one parameter Xiong Wang: Chaotic system with only one 40 stable equilibrium
  • 41. The idea of symmetry W =Z n W plane Z plane W = (u,v) = Z = (x,y) = x+yi u+vi Symmetrical Original system system Xiong Wang: Chaotic system with only one 41 stable equilibrium (x,y,z)
  • 42. symmetry Xiong Wang: Chaotic system with only one 42 stable equilibrium
  • 43. Stability of the two equilibria  There are two symmetrical equilibria which are independent of the parameter a  The eigenvalue of Jacobian  So, a > 0 stable; a < 0 unstable Xiong Wang: Chaotic system with only one 43 stable equilibrium
  • 44. symmetry a = 0.005 > 0, stable equilibria Xiong Wang: Chaotic system with only one 44 stable equilibrium
  • 45. symmetry a = - 0.01 < 0, unstable equilibria Xiong Wang: Chaotic system with only one stable equilibrium 45
  • 46. symmetry Xiong Wang: Chaotic system with only one 46 stable equilibrium
  • 47. Three symmetrical equilibria with tunable stability Xiong Wang: Chaotic system with only one 47 stable equilibrium
  • 48. symmetry a = - 0.01 < 0, unstable equilibria Xiong Wang: Chaotic system with only one stable equilibrium 48
  • 49. symmetry a = 0.005 > 0, stable equilibria Xiong Wang: Chaotic system with only one stable equilibrium 49
  • 50. Theoretically we can create any number of equilibria … Xiong Wang: Chaotic system with only one 50 stable equilibrium
  • 51. Conclusions Chaotic system with:  No equilibrium - found  Two stable equilibria - found  Three stable equilibria - found  Theoretically, we can create any number of equilibria …  We can control the stability of equilibria by adjusting one parameter Xiong Wang: Chaotic system with only one 51 stable equilibrium
  • 52. Chaos is a global phenomenon A system can be locally stable near the equilibrium, but globally chaotic far from the equilibrium.  This interesting phenomenon is worth further studying, both theoretically and experimentally, to further reveal the intrinsic relation between the local stability of an equilibrium and the global complex dynamical behaviors of a chaotic system 52
  • 53. Xiong Wang 王雄 Centre for Chaos and Complex Networks City University of Hong Kong Email: wangxiong8686@gmail.com 53
  • 55. 55
  • 56. 56
  • 57. 57
  • 58. 58
  • 59. 59
  • 60. 60
  • 61. 61