Multiscroll attractor

<h2 id="chen-attractor">Chen attractor</h2>
The Chen system is defined as follows<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

 
 
 
 
 
 
 d
 x
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 a
 (
 y
 (
 t
 )
 −
 x
 (
 t
 )
 )
 
 
 {\displaystyle {\frac {dx(t)}{dt}}=a(y(t)-x(t))}

d
 y
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 (
 c
 −
 a
 )
 x
 (
 t
 )
 −
 x
 (
 t
 )
 z
 (
 t
 )
 +
 c
 y
 (
 t
 )
 
 
 {\displaystyle {\frac {dy(t)}{dt}}=(c-a)x(t)-x(t)z(t)+cy(t)}

d
 z
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 x
 (
 t
 )
 y
 (
 t
 )
 −
 b
 z
 (
 t
 )
 
 
 {\displaystyle {\frac {dz(t)}{dt}}=x(t)y(t)-bz(t)}

Plots of Chen attractor can be obtained with the <a href="/facts/Runge-Kutta_method/YRjbyHpY">Runge-Kutta method</a>:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
parameters: a = 40, c = 28, b = 3
initial conditions: x(0) = -0.1, y(0) = 0.5, z(0) = -0.6

<h2 id="other-attractors">Other attractors</h2>
Multiscroll attractors also called n-scroll attractor include the Lu Chen attractor, the modified Chen chaotic attractor, PWL Duffing attractor, <a href="/facts/Rabinovich-Fabrikant_equation/rpRpBoky">Rabinovich Fabrikant attractor</a>, modified Chua chaotic attractor, that is, multiple scrolls in a single attractor.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h3>Lu Chen attractor</h3>

An extended Chen system with multiscroll was proposed by Jinhu Lu (吕金虎) and Guanrong Chen<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
Lu Chen system equation

 
 
 
 
 
 
 d
 x
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 a
 (
 y
 (
 t
 )
 −
 x
 (
 t
 )
 )
 
 
 {\displaystyle {\frac {dx(t)}{dt}}=a(y(t)-x(t))}

d
 y
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 x
 (
 t
 )
 −
 x
 (
 t
 )
 z
 (
 t
 )
 +
 c
 y
 (
 t
 )
 +
 u
 
 
 {\displaystyle {\frac {dy(t)}{dt}}=x(t)-x(t)z(t)+cy(t)+u}

parameters：a = 36, c = 20, b = 3, u = -15.15
initial conditions：x(0) = .1, y(0) = .3, z(0) = -.6

<h3>Modified Lu Chen attractor</h3>

System equations:<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

 
 
 
 
 
 
 d
 x
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 a
 (
 y
 (
 t
 )
 −
 x
 (
 t
 )
 )
 ,
 
 
 {\displaystyle {\frac {dx(t)}{dt}}=a(y(t)-x(t)),}

d
 y
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 (
 c
 −
 a
 )
 x
 (
 t
 )
 −
 x
 (
 t
 )
 f
 +
 c
 y
 (
 t
 )
 ,
 
 
 {\displaystyle {\frac {dy(t)}{dt}}=(c-a)x(t)-x(t)f+cy(t),}

In which

 
 
 
 f
 =
 d
 0
 z
 (
 t
 )
 +
 d
 1
 z
 (
 t
 −
 τ
 )
 −
 d
 2
 sin
 ⁡
 (
 z
 (
 t
 −
 τ
 )
 )
 
 
 {\displaystyle f=d0z(t)+d1z(t-\tau )-d2\sin(z(t-\tau ))}

params := a = 35, c = 28, b = 3, d0 = 1, d1 = 1, d2 = -20..20, tau = .2
initv := x(0) = 1, y(0) = 1, z(0) = 14

<h3>Modified Chua chaotic attractor</h3>

In 2001, Tang et al. proposed a modified Chua chaotic system<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

 
 
 
 
 
 
 d
 x
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 α
 (
 y
 (
 t
 )
 −
 h
 )
 
 
 {\displaystyle {\frac {dx(t)}{dt}}=\alpha (y(t)-h)}

d
 y
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 x
 (
 t
 )
 −
 y
 (
 t
 )
 +
 z
 (
 t
 )
 
 
 {\displaystyle {\frac {dy(t)}{dt}}=x(t)-y(t)+z(t)}

d
 z
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 −
 β
 y
 (
 t
 )
 
 
 {\displaystyle {\frac {dz(t)}{dt}}=-\beta y(t)}

In which

 
 
 
 h
 :=
 −
 b
 sin
 ⁡
 
 (
 
 
 
 
 π
 x
 (
 t
 )
 
 
 2
 a
 
 
 
 +
 d
 
 )
 
 
 
 {\displaystyle h:=-b\sin \left({\frac {\pi x(t)}{2a}}+d\right)}

params := alpha = 10.82, beta = 14.286, a = 1.3, b = .11, c = 7, d = 0
initv := x(0) = 1, y(0) = 1, z(0) = 0

<h3>PWL Duffing chaotic attractor</h3>

Aziz Alaoui investigated PWL Duffing equation in 2000:<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
PWL Duffing system:

 
 
 
 
 
 
 d
 x
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 y
 (
 t
 )
 
 
 {\displaystyle {\frac {dx(t)}{dt}}=y(t)}

d
 y
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 −
 
 m
 
 1
 
 
 x
 (
 t
 )
 −
 (
 1
 
 /
 
 2
 (
 
 m
 
 0
 
 
 −
 
 m
 
 1
 
 
 )
 )
 (
 
 |
 
 x
 (
 t
 )
 +
 1
 
 |
 
 −
 
 |
 
 x
 (
 t
 )
 −
 1
 
 |
 
 )
 −
 e
 y
 (
 t
 )
 +
 γ
 cos
 ⁡
 (
 ω
 t
 )
 
 
 {\displaystyle {\frac {dy(t)}{dt}}=-m_{1}x(t)-(1/2(m_{0}-m_{1}))(|x(t)+1|-|x(t)-1|)-ey(t)+\gamma \cos(\omega t)}

params := e = .25, gamma = .14+(1/20)i, m0 = -0.845e-1, m1 = .66, omega = 1; c := (.14+(1/20)i)，i=-25...25;
initv := x(0) = 0, y(0) = 0;

<h3>Modified Lorenz chaotic system</h3>

Miranda & Stone proposed a modified Lorenz system:<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

 
 
 
 
 
 
 d
 x
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 1
 
 /
 
 3
 ∗
 (
 −
 (
 a
 +
 1
 )
 x
 (
 t
 )
 +
 a
 −
 c
 +
 z
 (
 t
 )
 y
 (
 t
 )
 )
 +
 (
 (
 1
 −
 a
 )
 (
 x
 (
 t
 
 )
 
 2
 
 
 −
 y
 (
 t
 
 )
 
 2
 
 
 )
 +
 (
 2
 (
 a
 +
 c
 −
 z
 (
 t
 )
 )
 )
 x
 (
 t
 )
 y
 (
 t
 )
 )
 
 
 {\displaystyle {\frac {dx(t)}{dt}}=1/3*(-(a+1)x(t)+a-c+z(t)y(t))+((1-a)(x(t)^{2}-y(t)^{2})+(2(a+c-z(t)))x(t)y(t))}

1
            
              3
              
                
                  x
                  (
                  t
                  
                    )
                    
                      2
                    
                  
                  +
                  y
                  (
                  t
                  
                    )
                    
                      2
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\frac {1}{3{\sqrt {x(t)^{2}+y(t)^{2}}}}}}

d
 y
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 1
 
 /
 
 3
 (
 (
 c
 −
 a
 −
 z
 (
 t
 )
 )
 x
 (
 t
 )
 −
 (
 a
 +
 1
 )
 y
 (
 t
 )
 )
 +
 (
 (
 2
 (
 a
 −
 1
 )
 )
 x
 (
 t
 )
 y
 (
 t
 )
 +
 (
 a
 +
 c
 −
 z
 (
 t
 )
 )
 (
 x
 (
 t
 
 )
 
 2
 
 
 −
 y
 (
 t
 
 )
 
 2
 
 
 )
 )
 
 
 {\displaystyle {\frac {dy(t)}{dt}}=1/3((c-a-z(t))x(t)-(a+1)y(t))+((2(a-1))x(t)y(t)+(a+c-z(t))(x(t)^{2}-y(t)^{2}))}

d
 z
 (
 t
 )
 
 
 d
 t
 
 
 
 =
 1
 
 /
 
 2
 (
 3
 x
 (
 t
 
 )
 
 2
 
 
 y
 (
 t
 )
 −
 y
 (
 t
 
 )
 
 3
 
 
 )
 −
 b
 z
 (
 t
 )
 
 
 {\displaystyle {\frac {dz(t)}{dt}}=1/2(3x(t)^{2}y(t)-y(t)^{3})-bz(t)}

parameters： a = 10, b = 8/3, c = 137/5;
initial conditions： x(0) = -8, y(0) = 4, z(0) = 10

<h2 id="gallery">Gallery</h2>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.chuacircuits.com/howtobuild4.php">The double-scroll attractor and Chua's circuit</a></li>
<li>Lozi, R.; Pchelintsev, A.N. (2015). <a href="https://hal.archives-ouvertes.fr/hal-01323625/document">"A new reliable numerical method for computing chaotic solutions of dynamical systems: the Chen attractor case"</a>. International Journal of Bifurcation and Chaos. 25 (13): 1550187–1550412. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2015IJBC...2550187L">2015IJBC...2550187L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS0218127415501874">10.1142/S0218127415501874</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:12339358">12339358</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Matsumoto, Takashi (December 1984). "A Chaotic Attractor from Chua's Circuit" (PDF). IEEE Transactions on Circuits and Systems. CAS-31 (12). IEEE: 1055–1058. doi:10.1109/TCS.1984.1085459. <a href="http://www.eecs.berkeley.edu/~chua/papers/Matsumoto84.pdf" target="_blank">http://www.eecs.berkeley.edu/~chua/papers/Matsumoto84.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Chua, Leon; Motomasa Komoru; Takashi Matsumoto (November 1986). "The Double-Scroll Family" (PDF). IEEE Transactions on Circuits and Systems. CAS-33 (11). <a href="http://www.eecs.berkeley.edu/~chua/papers/Chua86.pdf" target="_blank">http://www.eecs.berkeley.edu/~chua/papers/Chua86.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Chua, Leon (2007). "Chua circuits". Scholarpedia. 2 (10): 1488. Bibcode:2007SchpJ...2.1488C. doi:10.4249/scholarpedia.1488. <a href="https://doi.org/10.4249%2Fscholarpedia.1488" target="_blank">https://doi.org/10.4249%2Fscholarpedia.1488</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Chua, Leon (2007). "Fractal Geometry of the Double-Scroll Attractor". Scholarpedia. 2 (10): 1488. Bibcode:2007SchpJ...2.1488C. doi:10.4249/scholarpedia.1488. <a href="https://doi.org/10.4249%2Fscholarpedia.1488" target="_blank">https://doi.org/10.4249%2Fscholarpedia.1488</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Leonov G.A.; Vagaitsev V.I.; Kuznetsov N.V. (2011). "Localization of hidden Chua's attractors" (PDF). Physics Letters A. 375 (23): 2230–2233. Bibcode:2011PhLA..375.2230L. doi:10.1016/j.physleta.2011.04.037. <a href="http://www.math.spbu.ru/user/nk/PDF/2011-PhysLetA-Hidden-Attractor-Chua.pdf" target="_blank">http://www.math.spbu.ru/user/nk/PDF/2011-PhysLetA-Hidden-Attractor-Chua.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Chen G., Ueta T. Yet another chaotic attractor. Journal of Bifurcation and Chaos, 1999 9:1465. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">CHEN, GUANRONG; UETA, TETSUSHI (July 1999). "Yet Another Chaotic Attractor". International Journal of Bifurcation and Chaos. 09 (7): 1465–1466. Bibcode:1999IJBC....9.1465C. doi:10.1142/s0218127499001024. ISSN 0218-1274. <a href="https://dx.doi.org/10.1142/s0218127499001024" target="_blank">https://dx.doi.org/10.1142/s0218127499001024</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">CHEN, GUANRONG; UETA, TETSUSHI (July 1999). "Yet Another Chaotic Attractor". International Journal of Bifurcation and Chaos. 09 (7): 1465–1466. Bibcode:1999IJBC....9.1465C. doi:10.1142/s0218127499001024. ISSN 0218-1274. <a href="https://dx.doi.org/10.1142/s0218127499001024" target="_blank">https://dx.doi.org/10.1142/s0218127499001024</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">阎振亚著 《复杂非线性波的构造性理论及其应用》第17页 SCIENCEP 2007年 <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Chen, Guanrong; Jinhu Lu (2006). "Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 775–858. Bibcode:2006IJBC...16..775L. doi:10.1142/s0218127406015179. Retrieved 2012-02-16. <a href="http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf" target="_blank">http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Chen, Guanrong; Jinhu Lu (2006). "Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 775–858. Bibcode:2006IJBC...16..775L. doi:10.1142/s0218127406015179. Retrieved 2012-02-16. <a href="http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf" target="_blank">http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Chen, Guanrong; Jinhu Lu (2006). "Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 775–858. Bibcode:2006IJBC...16..775L. doi:10.1142/s0218127406015179. Retrieved 2012-02-16. <a href="http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf" target="_blank">http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Chen, Guanrong; Jinhu Lu (2006). "Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 793–794. Bibcode:2006IJBC...16..775L. CiteSeerX 10.1.1.927.4478. doi:10.1142/s0218127406015179. Retrieved 2012-02-16. <a href="http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf" target="_blank">http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">J. Lu, G. Chen p. 837 <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">J.Liu and G Chen p834 <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
</ol>

Multiscroll attractor open-in-new

Multiscroll attractor