This section addresses an introduction to the fasinated topic that originated from the work of the American mathematician and meteorologist Edward Norton Lorenz (1917--2008). He discovered what is known now as the deterministic chaos in mathematics and physics. The deterministic nature of dynamic systems does not make them predictable (determined) in practice. Solutions of differential systems involving more than two nonlinear differential equations may be qualitatively different from the planar solutions since slight pertubation may leads to exponential deviation. The chaos was summarised by Edward Lorenz as: Chaos – when the present determines the future, but the approximate present does not approximately determine the future.

The history of Lorenz discovery can be found in James Gleick (born 1954) book Gleick, formerly a science writer for the New York Times, depicts the beginnings of Chaos theory, which draws on the seemingly random patterns that characterize many natural phenomena. It explains the thought processes and investigative techniques of Chaos scientists, illustrating concepts like Julia sets, Lorenz attractors, and the Mandelbrot Set with sketches, photographs, and wonderful descriptive prose. This highly readable international best-seller is must-reading for any Clausewitzian. (See Alan D. Beyerchen's essay on the connection with Clausewitz.) ISBN: 0143113453.

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Introduction to Linear Algebra with Mathematica

Lorenz equations

We will wrap up this series of examples with a look at the fascinating Lorenz attractor. The Lorenz system (the Lorenz equations, note it is not Lorentz) is a three-dimensional system of ordinary differential equations that depends on three real positive parameters. They were first studied by the professor of MIT Edward Norton Lorenz (1917--2008) in 1963. Edward N. Lorenz, a meteorologist who tried to predict the weather with computers by solving a system of ordinary differential equations (now bearing his name) for certain parameter values and initial conditions, but instead gave rise to the modern field of chaos theory. For some parameter values, numerically computed solutions of the equations oscillate, apparently forever, in the pseudo-random way we now call "chaotic". In addition, there are some parameter values for which we see "preturbulence", a phenomenon in which trajectories oscillate chaotically for long periods of time before finally settling down to stable stationary or stable periodic behaviour, others in which we see "intermittent chaos", where trajectories alternate between chaotic and apparently stable periodic behaviours, and yet others in which we see "noisy periodicity", where trajectories appear chaotic though they stay very close to a non-stable periodic orbit.

An important problem in meteorology and in other applications of fluid dynamics is modeling a layer of fluid such as the earth's atmosphere. (Lorenz's derivation (1963) was based on considering a two-dimensional fluid cell (or layer) that is warmed from below and cooled from above. If the vertical temperature difference ΔT is small, then there is a linear variation of temperature with altitude, but no significant motion of the fluid layer. However, if ΔT is large enough, then the warmer air rises, displacing the cooler air above it, and steady convective motion results. If the temperature difference increases further, then eventually the steady convective flow breaks up and a more complex and turbulent motion ensues. Edward Lorenz was led to the nonlinear autonomous dynamic system:

\begin{equation} \label{EqLorenz.1} \begin{split} \frac{{\text d}x}{{\text d}t} &= \sigma\left( y-x \right) , \\ \frac{{\text d}y}{{\text d}t} &= x \left( \rho -z \right) -y, \\ \frac{{\text d}z}{{\text d}t} &= x\, y - \beta \, z . \end{split} \end{equation}
The three parameters σ, ρ, and β are respectively proportional to the Prandtl number, the Rayleigh number, and some physical proportions of the region under consideration; consequently, all three are taken to be positive.

Edward Lorenz was born in West Hartford, Connecticut. He studied mathematics at both Dartmouth College in New Hampshire and Harvard University in Cambridge, Massachusetts. From 1942 until 1946, he served as a meteorologist for the United States Army Air Corps. After his return from World War II, he decided to study meteorology. Lorenz earned two degrees in the area from the Massachusetts Institute of Technology. Dr. Lorenz was a staff member of M.I.T.’s meteorology department from 1948 to 1955, when he became an assistant professor. He was promoted to professor in 1962 and served as head of the department from 1977 to 1981. During the 1950s, Lorenz became skeptical of the appropriateness of the linear statistical models in meteorology, as most atmospheric phenomena involved in weather forecasting are non-linear. His work on the topic culminated in the publication of his 1963 paper "Deterministic Nonperiodic Flow" in Journal of the Atmospheric Sciences, and with it, the foundation of chaos theory. He was awarded the Kyoto Prize for basic sciences, in the field of earth and planetary sciences, in 1991.

Dr. Lorenz is best known for the notion (in 1969) of the “butterfly effect,” the idea that a small disturbance like the flapping of a butterfly’s wings can induce enormous consequences. His chaos discovery was accidental. One day, Dr. Lorenz was running simulations of weather using a simple computer model. Another day, he wanted to repeat one of the simulations for a longer time, but instead of repeating the whole simulation, he started the second run in the middle, typing in numbers from the first run for the initial conditions. The computer program was the same, so the weather patterns of the second run should have exactly followed those of the first. Instead, the two weather trajectories quickly diverged on completely separate paths.

At first, he thought the computer was malfunctioning. Then he realized that he had not entered the initial conditions exactly. The computer stored numbers to an accuracy of six decimal places, like 0.506127, while, to save space, the printout of results shortened the numbers to three decimal places, 0.506. When typing in the new conditions, Dr. Lorenz had entered the rounded-off numbers, and even this small discrepancy, of less than 0.1 percent, completely changed the final result. Even though his model was vastly simplified, Dr. Lorenz realized that this meant perfect weather prediction was a fantasy. A perfect forecast would require not only a perfect model, but also perfect knowledge of wind, temperature, humidity and other conditions everywhere around the world at one moment of time. Even a small discrepancy could lead to completely different weather. In 1972, he gave a talk with a title that captured the essence of his ideas: “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?”

In his later years, Lorenz lived in Cambridge, Massachusetts. He was an avid outdoorsman, who enjoyed hiking, climbing, and cross-country skiing. He kept up with these pursuits until very late in his life, and managed to continue most of his regular activities until only a few weeks before his death. According to his daughter, Cheryl Lorenz, Lorenz had "finished a paper a week ago with a colleague." On April 16, 2008, Lorenz died at his home in Cambridge at the age of 90, having suffered from cancer.

The Lorenz equations are made up of three populations: x, y, and z, and three fixed coefficients: σ, ρ, and β. Remembering what we discussed previously, this system of equations has properties common to most other complex systems, such as lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, and chemical reactions. First, it is non-linear in two places: the second equation has a xz term and the third equation has a xy term. It is made up of a very few simple components. The system is three-dimensional and deterministic. Although difficult to see until we plot the solution, the equation displays broken symmetry on multiple scales. Because the equation is autonomous (no t term in the right side of the equations), there is no feedback in this case.


Because the three equations \eqref{EqLorenz.1} are so codependent, their trajectories orbit back and forth between two centers but never cross. Such properties (combined with the sensitivity to initial conditions) are what makes systems chaotic. This system’s behavior depends on the three constant values chosen for the coefficients. It has been shown that the Lorenz System exhibits complex behavior when the coefficients have the following specific values: \( \sigma = 10, \ \rho = 28 , \mbox{ and }\ \beta = 8/3. \) We now have everything we need to code up the ODE into Mathematica:


Example 1: We consider the Lorenz's equations under the following avlues of parameters:

\[ \sigma = 10, \quad \beta = 8/3, \rho = 28. \]
\[Sigma] = 10;
solution = NDSolve[{x'[t] == \[Sigma] (y[t] - x[t]), y'[t] == 28 x[t] - y[t] - x[t] z[t], z'[t] == x[t] y[t] - 8/3 z[t], x[0] == z[0] == 0, y[0] == 2}, {x, y, z}, {t, 0, 35}]
ParametricPlot3D[Evaluate[{x[t], y[t], z[t]} /. solution], {t, 0, 35}]
       Lorenz attractor.            Mathematica code



Example 2: We seek solutions of the Lorenz equations as power series

\begin{align*} x(t) &= \sum_{i\ge 0} \frac{a_i}{i!}\left( t - t_0 \right)^i , \\ y(t) &= \sum_{k\ge 0} \frac{b_k}{k!} \left( t - t_0 \right)^k , \\ z(t) &= \sum_{m\ge 0} \frac{c_m}{k!} \left( t - t_0 \right)^m \end{align*}
into the system of equations. This yeilds the following recurrence:
\begin{align*} a_k &= - \sigma \,a_{k-1} + \sigma\,b_{k-1} , \qquad k \ge 1, \quad a_0 = x(t_0 ) , \\ b_{k+1} &= r\,a_k - b_k - k! \,\sum_{i=0}^k \frac{a_i \,c_{k-i}}{i!\,(k-i)!} , \qquad k \ge 0, \quad b_0 = y(t_0 ) , \\ c_{k+1} &= b\, c_k + k! \,\sum_{i=0}^k \frac{a_i \,b_{k-i}}{i!\,(k-i)!} , \qquad k \ge 0, \quad c_0 = z(t_0 ) \end{align*}


Example 3: The Glukhovsky--Dolzhanksy system is of the following form
\[ \begin{split} \dot{x} = -\sigma \left( x - y \right) -a yz , \\ \dot{y} = rx - y -xz , \\ \dot{z} = - bz + xy , \end{split} \]
where σ, a, r, b are physical parameters. Comparing to well-known Lorenz system, it has an additional non-linear term, which leads to essential differences in analytical structure and dynamics of the system. The Glukhovsky--Dolzhansky system describes following physical processes: convective fluid motion in an ellipsoidal rotating cavity, a rigid body r otation in a resisting medium, the forced motion of a gyrostat, a convective motion in harmonically oscillating horizontal fluid layer. Initially, this system was obtained by Glukhovsky and Dolzhansky as a three-mode model of convection for a fluid in an ellipsoidal rotating cavity, which can be interpreted as one of the models of ocean flows.
Example 4: Consider a dissipative chaotic system with no equilibrium given by
\[ \begin{split} \dot{x} = y, \\ \dot{y} = -x + yz , \\ \dot{z} = x^2 - 4\,y^2 +1 , \end{split} \qquad x(0) = 0, \quad y(0) =2, \quad z(0) =0 . \]
Example 5: A neat mechanical model of the Lorenz equations was invented by Willem Malkus and Lou Howard at MIT (Cambridge, MA, USA) in the 1970s. A Lorenzian (or "chaotic") waterwheel is a physical model that perfectly corresponds to the Lorenz equations. A chaotic waterwheel is just like a normal waterwheel except for the facts that the buckets leak. Water pours into the top bucket at a steady rate and gives the system energy while water leaks out of each bucket at a steady rate and removes energy from the system. If the parameters of the wheel are set correctly, the wheel will exhibit chaotic motion: rather than spinning in one direction at a constant speed, the wheel will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of behaviors in an unpredictable manner. Its derivation is given in Strogatz's book Nonlinear Dynamics and Chaos, section 9.1:
\[ \begin{split} \dot{a} = omega\,b - K\,a, \\ \dot{b} = -\omega\,a -K\,b + q , \\ \dot{\omega} = -\frac{\nu}{I}\,\omega + \frac{\pi G\,r}{I}\, a , \end{split} \]
I is the moment of inertia of the wheel,
θ is the angle of the wheel,
\( \omega = \dot{\theta} \) is the angular velocity (increases counterclockwise, as does θ),
K is is the liquid’s leakage rate,
ν is the rotational damping rate,
r is the radius of the wheel,
G is the effective gravity constant.

The dependent variables 𝑎 = 𝑎1 and b = b1 are the Fourier amplitudes of the first modes of the liquid’smass distribution function of water around the rim of the wheel, defined such that the mass between θ1 and θ2 is \( M(t) = \int_{\therta_1}^{\theta_2} m(\theta , t)\,{\text d}\theta , \)

\[ m(\theta , t ) = \sum_{n\ge 0} a_n (t)\,\cos n\theta + b_n \sin n\theta . \]
Here q = q1 is the Fourier amplitude of the first mode of the liquid inflow mass distribution function
\[ Q(\theta ) = \sum_{n\ge 0} q_n \cos nm\theta . \]
So Q(θ) is the inflow (rate at which water is pumped in by the nozzles above position θ.    ■


  1. Chen, G., Kuznetsov, N.V., and Mokaev, T.N., Hidden Attractors on One Path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich Systems, International Journal of Bifurcation and Chaos, 2017, Vol. 27, No. 08, 1750115; doi: 10.1142/S0218127417501152
  2. Garashchuk, I.R., Kudryashov, N.A., and Sinelshchikov, D.I., On the analytical properties and some exact solutions of the Glukhovsky-Dolzhansky system, Journal of Physics: Conference Series, 2017, Vol. 788, 5 pages; doi:10.1088/1742-6596/788/1/012013
  3. Gleick, J., Chaos: Making a New Science, 1987/2008, Viking, New York.
  4. Glukhovskii, A. B. and Dolzhanskii, F. V. 1980. Three-component geostrophic model of convection in a rotating fluid. Academy of Sciences, USSR, Izvestiya, Atmo- spheric and Oceanic Physics (in Russian) 16, 311--318.
  5. Guellal, S., Grimalt, P., and Cherruault, Y., “Numerical study of Lorenz’s equation by the Adomian method”, Computers and Mathematics with Applications, 1997, Vol. 33, No. 3, pp. 25--29.
  6. Hashim, I., Noorani, M.S.M., Ahmad, R., Bakar, S.A., Ismail, E.S., and Zakaria, A.M., “Accuracy of the Adomian decomposition method applied to the Lorenz system”, Chaos, Solitons and Fractals, 2006, Vol. 28, No. 5, pp. 1149--1158.
  7. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion, European Physical Journal: Special Topics, 224(8), 1421–1458 (2015).
  8. Lorenz, E. N., Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 1963, 20 (2), 130--141.
  9. Mokaev, Timur, Localization and dimension estimation of attractors in the Glukhovsky-Dolzhansky system, Dissertation, 2016, by University of Jyväskylä, Finland.
  10. Munmuangsaen, B., Srisuchinwong, B., A hidden chaotic attractor in the classical Lorenz system, Chaos, Solitons, and Fractals, 2018, Vol. 107, pp. 61--66,
  11. Sparrow, C., The Lorenz Equations. Bifurcations, Chaos, and Strange Attractors, 1982, Springer-Verlag, New York.
  12. Sprott, J.C., Strange attractors with various equilibrium types, European Physical Journal: Special Topics, 2015, Vol. 224, pp. 1409--1419; doi: 10.1140/epjst/e2015-02469-8
  13. Vadasz, P. (2010). Analytical prediction of the transition to chaos in Lorenz equations, Applied Mathematics Letters, 2010, Vol. 23, Issue 5, pp. 503--507.


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Simplest Chaotic System with a Hyperbolic Sine and Its Applications in DCSK Scheme IET Communications 12(7) · January 2018 DOI: 10.1049/iet-com.2017.0455 Rössler OE . An equation for continuous chaos. Phys Lett A 1976;57:397–8 . [3] Sprott JC . Some simple chaotic jerk functions. Am J Phys 1997;65:537–43 . [4] Sprott JC . Elegant chaos: algebraically simple chaotic flows. Singapore: World Scientific; 2010 . [5] Lorenz EN , Emanuel KA . Optimal sites for supplementary weather observa- tions: simulation with a small model. J Atmos Sci 1998;55:399–414 . [6] Thomas R . Deterministic chaos seen in terms of feedback circuits: analysis, synthesis, ‘labyrnth chaos. Int J Bifurcat Chaos Appl Sci Eng 1999;9:1889–905 . [7] Chlouverakis KE , Sprott JC . Chaotic hyperjerk systems. Chaos Solit Frac 2006;28:739–46 . [8] Mumuangsaen B , Srisuchinwong B . Elementary chaotic snap flows. Chaos Solit Frac 2011;44:995–1003 . [9] Rössler OE . An equation for hyperchaos. Phys Lett A 1979;71:155–7 . [10] Liu Z , Lai YC , Mat ´ιas MA . Universal scaling of Lyapu nov exponents in coupled chaotic oscillators. Phys Rev E 2003;67(045203(R)):1–4 . [11] Cuomo KM , Oppenheim AV . Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett 1993;71:65–8 . [12] Blakely JN , Eskridge MB , Corron NJ . A simple Lorenz circuit and its radio fre- quency implementation. Chaos 2007;17(023112):1–5 . [13] Matsumoto T , Chua LO , Komuro M . The double scroll. IEEE Trans Circuits Syst 1985;32:797–818 . [14] Bartissol P , Chua LO . The double hook. IEEE Trans Circuits Syst 1988;35:1512–22 . [15] Chua LO , Lin G-N . Canonical realization of Chua’s circuit family. IEEE Trans Cir- cuits Syst 1990;37:885–902 . [16] Elwakil AS , Kennedy MP . High frequency Wien-type chaotic oscillator. Electron Lett 1998;34:1161–2 . [17] Srisuchinwong B , Treetanakorn R . Current-tunable chaotic jerk circuit based on only one unity-gain amplifier. Electron Lett 2014;50:1815–17 . [18] Srisuchinwong B , Nopchinda D . Current-tunable chaotic jerk oscillator. Electron Lett 2013;49:587–9 . [19] Sprott JC . Simple chaotic systems and circuits. Am J Phys 20 0 0;68:758–63 . [20] Sprott JC . A new chaotic jerk circuit. IEEE Trans Circuits Syst II, Exp Briefs 2011;58:240–3 . [21] Kuznetsov NV , Leonov GA , Vagaitsev VI . Analytical-numerical method for at- tractor localization of generalize d Chua’s system. IFAC Proc 2010;4:29–33 . [22] Leonov GA , Kuznetsov NV . Hidden attractors in dynamical systems: from hidden oscillation in Hilbert–Kolmogorov, Aizerman and Kalman prob- lems to hidden chaotic attractor in Chua circuits. Int J Bifur Chaos 2013;23(1330 0 02):1–69 . [23] Leonov GA , Kuznetsov NV , Vagaitsev VI . Localization of hidden Chua’s attrac- tors. Phys Lett A 2011;375:2230–3 . [24] Leonov GA , Kuznetsov NV , Mokaev TN . Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur Phys J Spec Top 2015;224:1421–58 . [25] Kiseleva MA , Kuznetsov NV , Leonov GA , Neittaanmäki P . Hidden oscillations in drilling system actuated by induction motor. IFAC Proc 2013;46:86–89. . [26] Jafari S , Sprott JC , Nazarimehr F . Recent new examples of hidden attractors. Eur Phys J Spec Top 2015;224:1469–76 . [27] Kuznetsov NV . Hidden attractors in fundamental problems and engineering models: a short survey, AETA 2015: recent advances in electrical engineering and related sciences. Lect Notes Elect Eng 2016;371:13–25 . [28] Pisarchik AN . Controlling the multistability of nonlinear systems with coexist- ing attractors. Phys Rev E 2001;64(046203):1–5 . [29] Pisarchik AN , Feudel U . Control of multistability. Phys Rep 2014;540:167–218 . [30] Jafari S , Sprott JC , Molaie M . A simple chaotic flow with a plane of equilibria. Int J Bifur Chaos 2016;26(1650 098):1–6 . [31] Leonov GA , Kuznetsov NV , Vagaitsev VI . Hidden attractor in smooth Chua sys- tems. Phys D 2012;241:1482–6 . [32] Sprott JC . Strange attractors with various equilibrium types. Eur Phys J Spec Top 2015;224:1409–19 . [33] Molaie M , Jafari S , Sprott JC , Golpayegani SMRH . Simple chaotic flows with one stable equilibrium. Int J Bifur Chaos 2013;23(1350188):1–7 . [34] Sprott JC , Jafari S , Pham VT , Hosseini ZS . A chaotic system with a single unsta- ble node. Phys Lett A 2015;379:2030–6 . [35] Jafari S , Sprott JC . Simple chaotic flows with a line equilibrium. Chaos Solit Frac 2013;57:79–84 . [36] Gotthans T , Sprott JC , Petrzela J . Simple chaotic flow with circle and square equilibrium. Int J Bifur Chaos 2016;26(1650137):1–8 . [37] Jafari S , Sprott JC , Pham VT , Volos C , Li C . Simple chaotic 3D flows with sur- faces of equilibria. Nonlinear Dyn 2016;86:1349–58 . [38] B. Munmuangsean, B. Srisuchinwong, to be published. [39] Sprott JC . Chaos and time-series analysis. New Yor k: Oxford University Press; 2003 . [40] Sprott JC , Xiong A . Classifying and quantifying basins of attraction. Chaos 2015;25(083101):1–7 . [41] Li C , Sprott JC . Finding coexisting attractors using amplitude control. Nonlinear Dyn. 2014;77:2059–64 . [42] Souza SLT , Batista AM , Caldas IL , Viana RL , Kapitaniak T . Noise-induced basin hopping in a vibro-impact system. Chaos Solit Frac 2007;32:758–67 . [43] Li C , Sprott JC . Multistability in the Lorenz system: a broken butterfly. Int J Bifur Chaos 2014;24(1450131):1–7 . [1] G. Adomian, Nonlinear stochastics systems theory and application to physics, Dor- drecht: Kluwer, 1989. [2] G. Adomian. A review of the decomposition method in applied mathematics, J Math Anal Appl 135 (1988), 501–544. [3] G. Adomian, Solving frontier problems of physics: the decomposition method, Boston: Kluwer, 1994. [4] J. Biazar, E. Babolian , R. Islam. Solution of the system of ordinary differential equations by Adomian decomposition method, Appl Math Comput 147 (2004), 713– 719. [5] J. Biazar, R. Montazeri. A computational method for solution of the prey and preda- tor problem, Appl Math Comput 163 (2005), 841–847. [6] D. J. Evans, K. R. Raslan. The Adomian decomposition method for solving delay differential equations, Intern J Computer Math 82 (2005), 49–54. [7] S. Guellal, P. Grimalt, Y. Cherruault. Numerical study of Lorenz’s equation by the Adomian method, Comput Math Appl 33 (1997), 25–29. [8] I. Hashim, M. S. M. Noorani, R. Ahmad, S. A. Bakar, E. S. Ismail and A. M. Za- karia. Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos, Solitons and Fractals 28 (2006), 1149–1158. [9] I. Hashim, M. S. M. Noorani and M. R. Said Al-Hadidi. Solving the generalized Burgers-Huxley equation using the Adomian decomposition method, Mathematical and Computer Modelling 43(11-12) (2006), 1404–1411. [10] D. Kaya, S. M. El-Sayed. Numerical soliton-like solutions of the potential Kadomtsev- Petviashvili equation by the decomposition method, Phys Letts A 320 (2003), 192– 199. [11] E. N. Lorenz. Deterministic nonperiodic flow, J Atmospheric Sci 20(2) (1963), 130– 141. [12] S. Olek. An accurate solution to the multispecies Lotka-Volterra equations, SIAM Review 36(3) (1994), 480–488. [13] A. Répaci. Nonlinear dynamical systems: on the accuracy of Adomian’s decomposi- tion method, Appl Math Lett 3(4) (1990), 35–39. [14] N. Shawagfeh, G. Adomian. Non-perturbative analytical solution of the general Lotka- Volterra three-species system, Appl Math Comput 76 (1996), 251–266. [15] N. Shawagfeh, D. Kaya. Comparing numerical methods for the solutions of systems of ordinary differential equations, Appl Math Lett 17 (2004), 323–328. [16] P. Vadasz, S. Olek. Convergence and accuracy of Adomian’s decomposition method for the solution of Lorenz equation, Int J Heat Mass Transfer 43 (2000), 1715–1734. [17] A. M. Wazwaz. The numerical solution of special fourth-order boundary value prob- lems by the modified decomposition method, Intern J Computer Math 118 (2002), 345–356. [18] A. M. Wazwaz. Exact solutions with solitons and periodic structures for the Zakharov- Kuznetsov (ZK) equation