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Chaotic systems are dynamical systems that are highly sensitive to initial conditions. This sensitivity is known as the butterfly effect. A dynamical system may be mathematically expressed either by continuous set of equations, or by discrete system, called map. The term strange attractor was coined by David Ruelle and Floris Takens.
Lorenz System
The Lorenz system is defined by three non-linear differential equations (Lorenz equations), which were defined by Edward N. Lorenz in 1963. The Lorenz system is strange attractor with nonlinear and deterministic characteristics, thus the behavior is determined by the initial conditions, without any randomness involved.
The Lorenz system refers to chaotic behavior for certain values and initial conditions (Rayleigh ρ=28; Prandlt σ=10; and β=8/3). The Lorenz equations are commonly expressed as coupled non-linear differential equations.
dx / dt = σ (y – x)
dy / dt = x (ρ – z) – y
dz / dt = xy – βz
Projection of the Lorenz system on the x-z plane
Projection of the Lorenz system on the y-z plane
References
- E. N. Lorenz; Deterministic nonperiodic flow; J. Atmos. Sci., Vol. 20; 1963
- K. Mischaikow, M. Mrozek; Chaos in Lorenz equations: a computer assisted proof; Bull. Amer. Math. Soc.; 1995
- K. Mischaikow, M. Mrozek; Chaos in the Lorenz equations: a computer assisted proof. Part II: details; Mathematics of Computation; 67; 1998
- K. Mischaikow, M. Mrozek, A. Szymczak; Chaos in the Lorenz equations: a computer assisted proof. Part III: the classical parameter values; J. Diff. Equ. 169; 2001
- Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013
- http://www.3d-meier.de
- http://www.mathworld.wolfram.com
4D Sinusoidally Driven Lorenz System
dx / dt = -a (x – y – c sin(u))
dy / dt = bx -xz – y
dz / dt = xy – dz
du / dt = 𝜔
Two-wing hyperchaotic attrctor a = 8, b = 35, d = 8/3, c = 6, and 𝜔 = 4.5
where c sin 𝜔𝑡 is the sinusoidal function controller and c and 𝜔 are the amplitude and frequency of the sine function controller
Four-wing chaotic attractor a = 10, b = 30, d = 8/3, c = 17, and 𝜔 = 0.5
References
- Han Chunyan, Yu Simin, Wang Guangyi; A Sinusoidally Driven Lorenz System and Circuit Implementation; Mathematical Problems in Engineering; 2015
Multi-Scroll Modified Lorenz System
Based on the sawtooth wave function theory, the generalized first type of modified Lorenz system is described by
dx/dt = a (y − x)
dy/dt = (c − z) sgn(x)
dz/dt = g(x) − bz
where a = 0.5, b = 0.1, c = 2, sgn(x) is a sign function, and g(x) is a sawtooth wave function with A = 0.43, k = 2.5, An = 2.06
g0(x) = k|x|
g(x)= g0(x) +∑N(n=1) gn(x)
gn(x) = −Gn[2 + sgn(x − En) − sgn(x + En)]
Gn = A/An
En = nA/k
10-scroll modified Lorenz system
References
- A. S. Elwakil, S. Ozoguz, and M. P. Kennedy, Creation of a complex
butterfly attractor using a novel Lorenz-type system, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume. 49, 2002 - S. Ozoguz, A. S. Elwakil, and M. P. Kennedy, Experimental verification
of the butterfly attractor in a modified Lorenz system, International Journal Bifurcation and Chaos, Volume 12, No. 7, 2002 - J. Lu, G. Chen, Multi-scroll chaos generation: Theories, methods
and applications, International Journal Bifurcation and Chaos, Volume. 16, No. 4, 2006
Rotation Multiwing Lorenz System
θ is the rotation angle, and (x0, z0) is the central point of the rotation attractor; p is a constant, which can facilitate the design and implementation of the corresponding circuits; F0 is also an adjustable parameter.
Projection x-z plane
dx/dt = Hx (H1 cos(θ) + H3 sin(θ))
dy/dt = H2
dz/dt = Hz (-H1 sin(θ) + H3 cos(θ))
H1 = 100(y-0.1x´)
H2 = 0.1(24-4c) x´- 0.1px´z´ + cy
H3 = F(x´) – 8/3z´
F(x´) = 0.01F0x´2 – ∑N(i=1) Fi (1+0.5sgn(0.1x´- Ei) – 0.5sgn(0.1x´ + Ei))
Hx = 1/0.0008(|x-x0 + 0.0004| – |x-x0 – 0.0004|)
Hz = 1/0.0008(|z-z0 + 0.0004| – |z-z0 – 0.0004|)
z´ = (|x – x0| + x0) sin(θ) + |z – z0| + z0) cos(θ)
x´= (|x – x0| + x0) cos(θ) – |z – z0| + z0) sin(θ)
θ = π/4, x0 = -1.2, z0 = -1.2, p = 20, c = 1, F0 = 400, N = 1, F1 = 20.5, E1 = 0.2 where initial values are (0,1,0,1,2.1)
References
- Shengqiu Dai, Kehui Sun, Shaobo He, Wei Ai; Complex Chaotic Attractor via Fractal Transformation; Entropy. 21.(11):1115 ; 2019
Chaos synchronization problem was first described by Fujisaka and Yemada in 1983, which did not receive any attention until Pecora and Carroll published their results on chaos synchronization. Synchronization of chaotic systems is a phenomenon that may occur when more chaotic oscillators are coupled. Because of the butterfly effect, which
causes the exponential divergence of the trajectories of the identical chaotic systems defined with the almost same initial conditions, synchronizing more chaotic systems may certainly be an issue.
Coupled Lorenz System
Coupled Lorenz system refers to the synchronization of chaotic systems which may not be supposted because of the sensitivity to the permutations on the initial conditions. Chaotic behaviour occurs in the response system regardless of synchronization. However certain subsystems of chaotic systems can be synchronized by linking them. Subsequently was discovered that synchronization of non-identical chaotic systems may also be possible.
The initial parameters are (Rayleigh ρ=28; Prandlt σ=10; β=8/3 and r1=35 and r2=1,15). The equations of the Coupled Lorenz System are commonly expressed
dx1 / dt = σ (y1-x1)
dy1 / dt = r1x1 – y1 – x1z1
dz1 / dt = -bz1 + x1y1
dx2 / dt = σ (y2 – x2) + ε (x1 – x2)
dy2 / dt = r2x2 – y2 – x2z2
dz2 / dt = -bz2 + x2y2
3-scroll coupled Lorenz system for ε=2.85
6-scroll coupled Lorenz system for ε=0.95
References
- J. Lü, G. Chen; Generating multiscroll chaotic attractors: Theories, Methods and Applications; Int. J. Bifur Chaos, 16 (4); 2006
- S. Yu, J. Lü, W. K. S. Tang, G. Chen; A general multiscroll Lorenz system family and its realization via digital signal processors; Chaos, 16; 2006
- Pecora L. M. and Carroll T. L.; Synchronization of Chaotic Systems. Physical Review Letters 64; 1990
- Khan A. and Singh P.; Non-Linear Dynamical System and Chaos Synchronization; International Journal of Bifurcation and Chaos 18; 2008
- Joos Vandewalle, Müştak E. Yalçin, Cellular Neural Networks, Multi-scroll Chaos and Synchronization,World Scientific Series on Nonlinear Science Series A: Volume 50, 2005
- http://www.3d-meier.de
- http://www.mathworld.wolfram.com
- https://chaoticatmospheres.com
Sprott C System
In 1994, Julien Clinton Sprott reported 19 cases of three-dimensional autonomous differential systems with either five terms and two quadratic nonlinearities or six
terms and one quadratic nonlinearity, whose solutions are chaotic. These systems were named as Sprott A-S systems, where the Sprott C system equations were defined as
follows
dx/dt = a(y − x)
dy/dt = xz
dz/dt = b – y2
The Sprott C system refers a close resemblance to the Lorenz system in that they exist as a butterfly strange attractor and two symmetrical equilibria.
Projection of the Sprott C attractor on the x-z plane
Projection of the Sprott C system on the y-z plane
2-scroll Sprott C system
3-scroll Sprott C system for f(z) = 5 [ sgn (z-5) + sgn (z+5)]
The 4D multi scroll Sprott C system is distinguished by the equations
dx/dt = a (y − x)
dy/dt = xz – xw
dz/dt = b – y2
dw/dt= f(z) – w
where
f(z) = 5 sgn(z) for the 2-scroll Sprott C system
f(z) = 5 [ sgn (z-5) + sgn (z+5)] for the 3-scroll Sprott C system
f(z) = 5 [ sgn (z10) + sgn(z) + sgn (z+10)] for the 4-scroll Sprott C system
f(z) = 5 [ sgn (z15) + sgn (z-5) sgn (z+5)] + sgn (z+15)] for the 5-scroll Sprott C system
The 5-scroll Sprott C system f(z) = 5 [ sgn (z15) + sgn (z-5) sgn (z+5)] + sgn (z+15)]
References
- Julien Clinton Sprott; Strange Attractors: Creating Patterns in Chaos; M&T Press; 1993
- Joos Vandewalle, Müştak E. Yalçin, Cellular Neural Networks, Multi-scroll Chaos and Synchronization,World Scientific Series on Nonlinear Science Series A: Volume 50, 2005
- http://sprott.physics.wisc.edu
- http://www.3d-meier.de
- http://cpb.iphy.ac.cn
Nosé-Hoover System
Sprott developed a search method which appliedin order to get sets of coupled chaotic ordinary differential equations. Each of the sets contained three coupled quadratic equations. Of these, Sprott’s A system, is the set of equations that Posch, William Hoover, and Vesely used to describe a one-dimensional Nosé-Hoover oscillator. The Sprott A system is a particular case of the Nose-Hoover oscillator.
Projection of the Nosé-Hoover system on the x-z plane
Projection of the Nosé-Hoover system on the y-z plane
References
- S. Nosé; A molecular dynamics method for simulations in the canonical ensemble; Molecular Physics; 1984
- W. G. Hoover; Nonlinear conductivity and entropy in the two-body Boltzmann gas; Journal of Statistical Physics; 1986
- H. A. Posch, W.oover & F. J. Vesely; Canonical dynamics of the Nosé oscillator: Stability, order, and chaos; Physical Review A; 1986
- Sprott, J. C.; Some simple chaotic flows; Phys. Rev.; 1994
- Mahdi Adam; Valls Clàudia; The integrability of the Nose-Hoover equation; Journal of Geometry and Physics, Vol. 62; 2011
- http://www.gsd.uab.cat
Rössler System
Rössler system is a system of three non-linear differential equations defined by Otto Rössler German biochemist in 1976.
dx/dt = – y – z
dy/dt = x + a y
dz/dt = b + z (x – c)
Rössler studied the chaotic system with a=0.2, b=0.2, c=5.7
Hyperchaos was first described in 1979 by O. E. Rössler in a four-dimensional system with two unstable equilibrium points. The degree of hyperchaos can be quantified by the value of the Lyapunov exponent.
References
- O. E. Rössler; An Equation for COntinous Chaos; Physics Letters; 1976
- O. E. Rössler; An Equation for Hyperchaos; Physics Letters; 1979
- O. E. Rössler; Different types of chaos in two simple differential equations; Zeitschrift für Naturforsch A; 1976
- Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013
- https://www.library.cornell.edu
Three-Scroll Unified Chaotic System (TSUCS)
The Three-Scroll Unified Chaotic System (TSUCS) was introduced by Lin Pan, Wuneng Zhou, Jian’an Fang and Dequan Li in 2010. The Three-Scroll Unified Chaotic System (TSUCS), distinguished by three equilibrium points, represents continued transition from the Lorenz subsystem to the Chen subsystem.
The Three-Scroll Unified Chaotic System is chaotic independent on the input variables (a=40, b=55, c=11/6, d=0.16, e=0.65, f=20). The equations of the Three-Scroll Unified Chaotic System (TSUCS) are commonly expressed
dx / dt = a (y – x) + dxz
dy / dt = bx – xz +fy
dz / dt = -ex2 + xy + cz
Projection of the Three-Scroll Unified Chaotic System (TSUCS) on the x-z plane
Projection of the Three-Scroll Unified Chaotic System (TSUCS) on the y-z plane
References
- L. Pan, W. Zhou, J. Fang, and D. Li; A new three-scroll unified chaotic system coined; International Journal of Nonlinear Science; vol. 10; 2010
- D. Li, A three-scroll chaotic attractor; Physics Letters A; vol. 372; 2008
- S. Dadras and H.R. Momeni; Generating one-, two-, three- and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system; Chinese Physics B, vol. 19; 2009
- http://www.3d-meier.de
- http://www.mathworld.wolfram.com
- https://chaoticatmospheres.com
- https://www.library.cornell.edu/
- http://cpb.iphy.ac.cn
Chua’s Circuit
Chua´s circuit was proposed in 1984 by Leon O. Chua. The Chua´s circuit is determined by three equlibrium points.
3-double-scroll Chua´s circuit
7-scroll Chua´s circuit
Further research may refer to multi-dimensional multi-scroll systems.
References
- Zou Y., Luo X., Jiang P., Wang B., Chen G., Fang J. & Quan H.;Controlling the chaotic n-scroll Chua’s circuit; Int. J. Bifurcation and Chaos 13; 2003
- Suykens J. A. K., Huang A. & Chua L. O.; A family of n-scroll attractors from a generalized Chua’s circuit; Int. J. Electron. Commun. 51; 1997
- Kennedy M. P.; Three steps to chaos, Part II: A Chua’s circuit primer; IEEE Trans. Circuits Syst.-I 40; 1993
- Cafagna D. & Grassi G.; New 3D-scroll attractors in hyperchaotic Chua’s circuit forming a ring; Int. J. Bifurcation and Chaos 13; 2003
- Matsumuto T.; A chaotic attractor from Chua’s circuit; IEEE Trans. Circuits Syst. 31(12); 1984
- http://www.3d-meier.de
- http://cpb.iphy.ac.cn
- http://www.mathworld.wolfram.com
- https://www.library.cornell.edu/
- https://www.actaphys.uj.edu.pl
Rotation Multi-Scroll Chua System
dx/dt = (3/20)Sx (F1 cos(θ) + F2 sin(θ))
dy/dt = (3/20)F3
dz/dt = (3/20)Sz (-F1 sin(θ) + F2 cos(θ))
F1 = 10(x´- h(z´))
F2 = (20/3)y + z´ – x´
F3 = -16x´
h(x) = mx-mn(sgn(x) + sgn(x-2n) + sgn(x+2n))
Sx = 1/0.08(|(20/3)(x-1) – x0 + 0.04| – |(20/3)(x-1) – x0 – 0.04|)
Sz = 1/0.08(|(20/3)z-z0 + 0.04| – |(20/3)z-z0 – 0.04| )
z´ = (|(20/3)(x-1) – x0| + x0) cos(θ) – (|(20/3)z – z0| + z0) sin(θ)
x´= (|(20/3)(x-1) – x0| + x0) sin(θ) + (|(20/3)z – z0| + z0) cos(θ)
θ=π/4, x0=-14.5, z0=-13, m=0.3, n=5 where initial values are (0,1,0,1,0.1)
Projection x-z plane
Projection y-z plane
References
- Shengqiu Dai, Kehui Sun, Shaobo He, Wei Ai; Complex Chaotic Attractor via Fractal Transformation; Entropy. 21.(11):1115 ; 2019
Matsumoto–Chua–Kobayashi System
Matsumoto–Chua–Kobayashi (MCK) circuit refers to a hyperchaotic variation of Chua´s circuit with equations
dx/dt = α (g(y − x)−z)
dy/dt = β(−g(y −x)− w)
dz/dt = γ(x + z)
dw/dt = δy
where g = ε(y − x) + 0.5(θ − ε)(|y − x + μ| − |y − x − μ|)
By obtaining generalized g function results in n-scroll hyperchaotic systems.
Double-scroll Matsumoto–Chua–Kobayashi (MCK) circuit
References
- T. Matsumoto, Leon O. Chua, T. Kobayashi, IEEE Trans. Circuits Systems, 1986
- Gianluca Setti, Gianluca Mazzin, Riccardo Rovatti, Nonlinear Dynamics of Electronic Systems, Proceedings of the IEEE Workshop, 2000, https://doi.org/10.1142/4453
- Simin Yu, Jinhu Lü, Guanrong Chen, A family of n-scroll hyperchaotic attractors and their realization, Physics Letters, 2007, http://dx.doi.org/10.1016/j.physleta.2006.12.029
Thomas’ Cyclically Symmetric Attractor
Thomas’ Cyclically Symmetric attractor, proposed by René Thomas, is cyclically symmetric in the x,y, and z variables.
References
- Thomas René; Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’; Int. J. Bifurcation and Chaos 9 (10); 1999
- H. Zhang, D. Liu and Z. Wang; Controlling Chaos Suppression, Synchronization and Chaotification; Springer; 2009
- Kaneko T., Tsuda I.; Complex Systems: Chaos and Beyond. A Constructive Approach with Applications in Life Sciences; New York: Springer; 2001
- http://www.3d-meier.de
- http://www.mathworld.wolfram.com
- https://www.library.cornell.edu/
Dadras-Momeni System
The Dadras-Momeni system refers to an autonomous multi-scroll chaotic system distinguished by five equilibrium points. By varying only the system parameter c has am substantial impact on the quantity of scrolls.
The Dadras-Momeni system is described by the following nonlinear differential equations,
dx/dt = y − ax + byz
dy/dt = cy − xz + z
dz/dt = dxy − hz
References
- Dadras S., Momeni H. R.: A novel three-dimensional autonomos chaotic system generating two, three and four-scroll attractors; Phys. Lett. A 373; 2009
- Dadras S., Momeni, H.; Four-scroll hyperchaos and four-scroll chaos evolved from a novel 4D nonlinear smooth autonomous system; Phys. Lett. A 374; 2010
- Dadras S., Momeni H. R., Qi, G.; Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos.; Nonlinear Dyn. 62; 2010
- http://www.3d-meier.de
Newton-Leipnik System
The Newton–Leipnik system is a chaotic system with two strange attractors. The Newton–Leipnik system is described by differencial equations
dx / dt = – ax + y + 10 yz
dy / dt = −x – 4y + 5 xz
dz / dt = bz – 5 xy
In 1981, Newton and Leipnik constructed a set of differential equations from Euler’s rigid body equations which were modified with a linear feedback. In 2002, B. Marlin established the existence of closed orbits which were not asymptotically stable for this system. In 2002, Chen studied chaos control and synchronization of the Newton–Leipnik system by using a stable-manifoldbased method. More recent studies by Wang and Tian indicated that this chaotic system can be controlled to unstable period orbits and torus with a suited linear controller.
References
- G. Chen, X. Dong; On feedback control of chaotic continuous time systems; IEEE Trans. Circuit Systems; 1993;
- M. T. Yassen; Chaos control of Chen chaotic dynamical system. Chaos; Solitons & Fractals; 2003
- Xuedi Wang, Chao Ge; Adaptive control and synchronization of the Newton-Leipnik systems, Journal of Information and Computing Science; Vol. 3; No. 4; 2008
Chen-Lee System
Projection of the Chen-Lee system on the x-y plane
Projection of the Chen-Lee system on the x-z plane
References
- Tam L., Chen J., Chen H., & Tou W.; Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation; Chaos, Solitons and Fractals; Vol. 38; 2008
Li C., Chen G.; Chaos in the fractional order Chen system and its control; Chaos Solitons Fractals 22; 2004
- https://www.library.cornell.edu/
- http://cpb.iphy.ac.cn
- www.3d-meier.de
Modified Multi-Scroll Chen System
Chen system was introduced in 1999 while studying how to control the Lorenz system. The modified multi-scroll Chen system is defined by equations
dx/dt = a(y – x)
dy/dt = (c – a)x – xu + cy
dz/dt = xy – bz
where u = d1z – d2 sin(z)
The quantity of the equilibrium points depends on the values of the parameter d2
Projection of the modified 6-scrollChen system on the x-z plane
Projection of the modified 6-scroll Chen system on the y-z plane
The delay differential equation may generate hyperchaotic dynamic behavior. The modified multi-scroll Chen system equations
u(t) = d0z(t) + d1z(t −τ ) − d2 sin(z(t − τ ))
where d0, d1, d2 are constants and τ is the time delay.
References
- Liu, Shen, Zhang; Multi-scroll chaotic and hyperchaotic attractors generated from Chen system; International Journal of Bifurcation and Chaos; Vol. 22; No. 2; 2012
- Wang L. & Yang X.; Generation of multi-scroll delayed chaotic oscillator; 2006
- Rossler, O. E.; An equation for hyperchaos; 1979
- Lu J., Han F., Yu X. & Chen G.; Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method; 2004
Bouali System 2
The Bouali system, introduced by Safieddine Bouali in research paper in Nonlinear Dynamics in 2012, derives from the established Lotka-Volterre oscillator.
The initial variables of the Bouali equations are (a=3, α=2.2, b=1, β=1, c=0.002 and s=1). The equations of the Bouali systemis expressed by three-nonlinear differential equations.
dx / dt = x (a − y) + αz
dy / dt = −y (b − x2)
dz / dt = −x (c − sz) − βz
References
- Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI 10.1007/s11071-012-0625-6
Safieddine Bouali; Feedback Loop in Extended van der Pol’s Equation Applied to an Economic Model of Cycles; International Journal of Bifurcation and Chaos; Vol. 9; 1999
http://chaos-3d.e-monsite.com
Bouali System 3
The Bouali system refers to the sensitive dependency on initial conditions (SDIC). The equations of the Bouali system are expressed by three-nonlinear differential equations.
dx/dt = α x (1 – y) – β z
dy/dt = – γ y (1 – x2)
dz/dt = μ x
The Bouali system for α=3, β=2.2, γ=1 μ=0.001
The bifurcation of the dynamical system seems analog to the “Ruelle-Takens-Newhouse route to chaos”; the modification of μ value validates the relevance of the “Ruelle-Takens-Newhouse route to chaos”.
References
- Bouali, Safieddine; A 3D Strange Attractor with a Distinctive Silhouette. The Butterfly Effect Revisited; 2013;
- Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI: 10.1007/s11071-012-0625-6
- Safieddine Bouali; Feedback Loop in Extended van der Pol’s Equation Applied to an Economic Model of Cycles; International Journal of Bifurcation and Chaos; Vol. 9; 1999; DOI: 10.1142/S0218127499000535
- Newhouse, S., Ruelle, D., Takens, F., Occurrence of strange Axiom A attractors near quasiperiodic flows on Tm, m≥3, Communications in Mathematical Physics 64, No. 1, 1978;
- Kapitaniak T., Ponce E. and Wojewoda J., Route to chaos via strange non-chaotic attractors, Journal of Physics A: Mathematical and General, Volume 23, No. 8, 1990;
- http://chaos-3d.e-monsite.com
Hyperchaotic 4D Bouali System A
A hyperchaotic systems are defined as chaotic behavior with at least two positive Lyapunov exponents. The initial 4D strange system was described in 1979 by Otto Rössler. Some of hyperchaotic systems are thereafter defined mainly by the extension of another dimension; hyperchaotification techniques. The hyperchaotic 4D Bouali system A does not refer to the application of the hyperchaotification techniques. However, the present system retains a modified 2D Lotka-Volterra oscillator.
The hyperchaotic 4D Bouali system described by four nonlinear differential equations
dx/dt = x (1 – y) + α z
dy/dt = β (x² – 1) y
dz/dt = γ (1 – y) w
dw/dt = η z
where x, y, z, w are the variables, and α, β, γ, and η real parameters. The core of the hyperchaotic 4D Bouali system A refers to modification of the Lotka-Volterra oscillator.
dx/dt = x (1 – y)
dy/dt = ( x² – 1) y
Projection of the hyperchaotic 4D Bouali system on the w-y plane
Projection of the hyperchaotic 4D Bouali system on the x-y plane
Projection of the hyperchaotic 4D Bouali system on the y-z plane
Projection of the hyperchaotic 4D Bouali system on the w-z plane
Projection of the hyperchaotic 4D Bouali system on the w-y plane
References
- Safieddine Bouali; A New Hyperchaotic Attractor with Complex Patterns; 2015;
- Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI: 10.1007/s11071-012-0625-6
Safieddine Bouali; Feedback Loop in Extended van der Pol’s Equation Applied to an Economic Model of Cycles; International Journal of Bifurcation and Chaos; Vol. 9; 1999; DOI: 10.1142/S0218127499000535
Rössler O. E.; An Equation for Hyperchaos; Physics Letters; 1979
- http://www.mathworld.wolfram.com
- http://chaos-3d.e-monsite.com
- http://cpb.iphy.ac.cn
- http://www.3d-meier.de
- https://www.library.cornell.edu
Xing-Yun System
The Xing-Yun system is a four-wing autonomous chaotic system which each equation contains a cubic product term. The Xing-Yun four-wing autonomous chaotic system indicates a broad magnitude bandwidth compared to the Lorenz system and other four-wing chaotic systems.
dx / dt = a(y − x) + yzz
dy / dt = b(x + y) − xzz
dz / dt = -cz + dy + xyz
Projection of the Xing-Yun system on the x-z plane for a=50, b=13, c=13, d=6
Projection of the Xing-Yun system on the y-z plane for a=50, b=13, c=13, d=6
References
- Liu Xing-Yun; A New 3D Four-Wing Chaotic System with Cubic Nonlinearity and Its Circuit Implementation; Chinese physics letters Vol. 26; No. 9; 2009
- Vaněček and Čelikovský; Control Systems: From Linear Analysis to Synthesis of Chaos; 1996
- http://cpb.iphy.ac.cn
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