(Nonlinear) Dynamics
Adam Getchell
You can Google for many, many more examples, and there are tons of software packages dedicated just to creating beautiful images with fractals.
More importantly, many, many natural phenomena of interest occur that are essentially non-linear. In the linear world, you can break the system down into composite parts, solve each part separately, and then recombine the solutions to get the final answer.
(Un)Fortunately, the universe is quite a bit more complicated. Although we have made great progress in science using simplified models with linear properties, to understand more we need a new kind of ... we need extend our current toolset a bit, and incorporate all the advances of the past into a cohesive approach.
Dynamics, then, is the study of systems that change and evolve in time.
There are two types of dynamical systems: differential equations in continuous time and iterated maps (aka difference equations) in discrete time. When we solve them, often we know solutions for a particular set of initial conditions. We can then construct an abstract phase space describing the possible solution trajectories of the system. The goal of dynamics is to do the reverse: from the system, draw the trajectories from the phase space. This kind of geometric reasoning will often give good information about the solutions without actually solving the system!
A trick: most physical systems of interest have a time component. But by parametrizing time (set $x_{n+1} = t$ such that $\frac{dx_{n+1}}{dt} = 1$) we can turn an n-dimensional time-dependent system into an (n+1)-dimensional time independent system. (But don't tell the GR-string theory-quantum gravity crowd ;-)
| Dynamics - A Capsule History (Table 1.1.1 from Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology) | ||
| 1666 | Newton | Invention of calculus, explanation of planetary motion |
| 1700s | Flowering of calculus and classical mechanics | |
| 1800s | Analytical studies of planetary motion | |
| 1890s | Poincare | Geometric approach, nightmares of chaos |
| 1920-1950 | Nonlinear oscillators in physics and engineering, invention of radio, radar, laser | |
| 1920-1960 | Birkhoff Kolmogorov Arnol'd Moser |
Complex behavior in Hamiltonian mechanics |
| 1963 | Lorenz | Strange attractor in simple model of convection |
| 1970s | Ruelle & Takens May Fiegenbaum Winfree |
Turbulence and chaos Chaos in logistic map Universality and renormalization, connection between chaos and phase transitions Experimental studies of chaos Nonlinear oscillators in biology Fractals |
| 1980s | Widespread interest in chaos, fractals, oscillators, and their applications | |
Consider the following equation: $$ \large \frac{dx}{dt} = \sin(x) $$ First, note that this equation is exactly solvable. Many systems of interest do not have closed form solutions. Anyway, by separating variables we get: $$ \large dt = \frac{dx}{\sin(x)} $$ Sage can solve this very quickly: ︡8a75474e-c3c9-4268-b6bf-63472a1fd60b︡{"html": "
![](\"Lorenz-attractor.png\")
\r\n\r\n\r\n
![](\"mandelbrot_set_1.jpg\")
\r\n\r\n![](\"water-lilies.jpg\")
\r\n
\r\n![](\"choascope.jpg\")
\r\n
\r\nYou can Google for many, many more examples, and there are tons of software packages dedicated just to creating beautiful images with fractals.\r\n
\r\nMore importantly, many, many natural phenomena of interest occur that are essentially non-linear. In the linear world, you can break the system down into composite parts, solve each part separately, and then recombine the solutions to get the final answer.\r\n
\r\n(Un)Fortunately, the universe is quite a bit more complicated. Although we have made great progress in science using simplified models with linear properties, to understand more we need a new kind of ... we need extend our current toolset a bit, and incorporate all the advances of the past into a cohesive approach.\r\n
\r\nDynamics, then, is the study of systems that change and evolve in time.\r\n
\r\nThere are two types of dynamical systems: differential equations in continuous time and iterated maps (aka difference equations) in discrete time. When we solve them, often we know solutions for a particular set of initial conditions. We can then construct an abstract phase space describing the possible solution trajectories of the system. The goal of dynamics is to do the reverse: from the system, draw the trajectories from the phase space. This kind of geometric reasoning will often give good information about the solutions without actually solving the system!\r\n
\r\nA trick: most physical systems of interest have a time component. But by parametrizing time (set $x_{n+1} = t$ such that $\\frac{dx_{n+1}}{dt} = 1$) we can turn an n-dimensional time-dependent system into an (n+1)-dimensional time independent system. (But don't tell the GR-string theory-quantum gravity crowd ;-)\r\n
\r\n
| Dynamics - A Capsule History (Table 1.1.1 from Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology) | \r\n||
| 1666 | \r\nNewton | \r\nInvention of calculus, explanation of planetary motion | \r\n
| 1700s | \r\n\r\n | Flowering of calculus and classical mechanics | \r\n
| 1800s | \r\n\r\n | Analytical studies of planetary motion | \r\n
| 1890s | \r\nPoincare | \r\nGeometric approach, nightmares of chaos | \r\n
| 1920-1950 | \r\n\r\n | Nonlinear oscillators in physics and engineering, invention of radio, radar, laser | \r\n
| 1920-1960 | \r\nBirkhoff Kolmogorov Arnol'd Moser | \r\nComplex behavior in Hamiltonian mechanics | \r\n
| 1963 | \r\nLorenz | \r\nStrange attractor in simple model of convection | \r\n
| 1970s | \r\nRuelle & Takens May Fiegenbaum Winfree | \r\nTurbulence and chaos Chaos in logistic map Universality and renormalization, connection between chaos and phase transitions Experimental studies of chaos Nonlinear oscillators in biology Fractals | \r\n
| 1980s | \r\n\r\n | Widespread interest in chaos, fractals, oscillators, and their applications | \r\n
\r\nConsider the following equation:\r\n$$\r\n\\large\r\n\\frac{dx}{dt} = \\sin(x)\r\n$$\r\nFirst, note that this equation is exactly solvable. Many systems of interest do not have closed form solutions. Anyway, by separating variables we get:\r\n$$\r\n\\large\r\ndt = \\frac{dx}{\\sin(x)}\r\n$$\r\nSage can solve this very quickly:"}︡ ︠a0c0a0c8-ef2d-4afa-aaa5-59031605cafc︠ integral(csc(x),x) ︡bf9eac1b-b71e-4849-a8a5-163f021e0392︡{"stdout": "-log(csc(x) + cot(x))"}︡ ︠6b6ca7a5-63df-4a49-8cb5-890657ffaac2i︠ %html
Now, this is an indefinite integral, which means the actual solution is: $$ \large t = -\log|\csc(x) +\cot(x)| + C $$ We have to find the constant C by the usual means, let $x = x_0 @ t = 0$. Then $C = log|csc(x_0) + cot(x_0)|$, and the full solution is: $$ \large t = \log|\frac{\csc(x_0) + \cot(x_0)}{(\csc(x) + \cot(x)}| $$ But what does this mean exactly?
Quick, if $x_0=\pi/4$, what happens as $t\rightarrow\infty$? What happens as $t\rightarrow\infty$ for arbitrary $x_0$?
Let's think about this another way.
Instead, let's plot $\frac{dx}{dt}$ versus x, and think of the function as a vector field with fluid flowing steadily along the x-axis with velocity that varies according to the rule $\frac{dx}{dt} = \sin(x)$. ︡305d217a-54c5-4e13-b450-1c5d1df78951︡{"html": "
\r\nNow, this is an indefinite integral, which means the actual solution is:\r\n$$\r\n\\large\r\nt = -\\log|\\csc(x) +\\cot(x)| + C\r\n$$\r\nWe have to find the constant C by the usual means, let $x = x_0 @ t = 0$. Then $C = log|csc(x_0) + cot(x_0)|$, and the full solution is:\r\n$$\r\n\\large\r\nt = \\log|\\frac{\\csc(x_0) + \\cot(x_0)}{(\\csc(x) + \\cot(x)}|\r\n$$\r\nBut what does this mean exactly?\r\n
\r\nQuick, if $x_0=\\pi/4$, what happens as $t\\rightarrow\\infty$? What happens as $t\\rightarrow\\infty$ for arbitrary $x_0$?\r\n
\r\nLet's think about this another way.\r\n
\r\nInstead, let's plot $\\frac{dx}{dt}$ versus x, and think of the function as a vector field with fluid flowing steadily along the x-axis with velocity that varies according to the rule $\\frac{dx}{dt} = \\sin(x)$."}︡ ︠296113c6-cafb-4ade-a799-a93b70724e74︠ ''' Plot vector field picture of f Later, this should calculate arrows from zeroes of f automatically ''' plotrange = 3 x = var('x') f = sin(x) # Add plot g = plot(f, -plotrange * pi, plotrange * pi) # Add arrows startx = [] endx = [] arrow_len = 1 def create_arrow_endpoints(start, end, arrow_length, n, parity): if parity: start.append(n*pi - arrow_length) start.append(n*pi + arrow_length) end.append(n*pi) end.append(n*pi) else: start.append(n * pi) start.append(n * pi) end.append(n*pi - arrow_length) end.append(n*pi + arrow_length) for i in range(-plotrange, plotrange+1): odd = (True if i % 2 != 0 else False) create_arrow_endpoints(startx, endx, arrow_len, i, odd) y = [0 for y in range(len(startx))] arrowstart = zip(startx, y) arrowend = zip(endx, y) def arrow_gen(start, stop): ''' Creates an arrow starting at start and ending at stop with color and line width ''' return arrow(start, stop, rgbcolor=(1,0,0), width=0.05) for i in range(len(arrowstart)): g += arrow_gen(arrowstart.pop(), arrowend.pop()) g.show() ︡a69bf617-9b27-4b7b-8913-1936fc26aab5︡︡ ︠ab602e90-20ce-4d25-a8a9-31041bcbcd84i︠ %html
When $\frac{dx}{dt}$ is positive, flow is to the right (+x direction). When $\frac{dx}{dt}$ is negative, flow is to the left (-x direction). Where $\frac{dx}{dt}$ is zero, there is no flow, and these are called fixed points. Notice that the fixed points can be classified in terms of their stability:
This classification carries over to higher-dimensional systems, with additions. For now, note that stability is a local property only; if you perturb the system far enough, you will get different behavior.
If we plot the actual solution to the equation (now x is the x-axis, t is the y-axis), we get:
︡d140f342-f50a-4f2a-bfe8-df329d6d300a︡{"html": "
\r\nWhen $\\frac{dx}{dt}$ is positive, flow is to the right (+x direction). When $\\frac{dx}{dt}$ is negative, flow is to the left (-x direction). Where $\\frac{dx}{dt}$ is zero, there is no flow, and these are called fixed points. Notice that the fixed points can be classified in terms of their stability:\r\n
\r\n
\r\nThis classification carries over to higher-dimensional systems, with additions. For now, note that stability is a local property only; if you perturb the system far enough, you will get different behavior.\r\n
\r\nIf we plot the actual solution to the equation (now x is the x-axis, t is the y-axis), we get:\r\n
"}︡ ︠fb8f7f0b-2b54-49d4-adbc-cc44f7061014︠ @interact def f(c = slider(-10, 10, .25, label = 'x0')): print c show(plot(log(abs((((csc(c)+cot(c))/(csc(x)+cot(x)))))))) ︡52298455-df61-46f0-a55a-64cceef47267︡︡ ︠a6e19ce4-2df8-41cd-8440-4f3c5aacf3f7i︠ %html
Inspecting this, just like in the previous diagram we can see that for $\frac{\pi}{4}$ the system tends to accelerate at first towards the right, but then slow down as it approaches a limiting value. Of course, often times we don't have the luxury of an exact equation to plot!
In the previous case, we found the fixed points of the system, which helped us identify the behavior of the equation for various domains of interest. Next, we'll consider adding a parameter r to the system. We'll start off with a simple equation, $y = x^2 + r$:
︡98bd1dce-1e6d-4c1e-9ea1-79c93c064562︡{"html": "
\r\nInspecting this, just like in the previous diagram we can see that for $\\frac{\\pi}{4}$ the system tends to accelerate at first towards the right, but then slow down as it approaches a limiting value. Of course, often times we don't have the luxury of an exact equation to plot! \r\n
\r\nIn the previous case, we found the fixed points of the system, which helped us identify the behavior of the equation for various domains of interest. Next, we'll consider adding a parameter r to the system. We'll start off with a simple equation, $y = x^2 + r$:\r\n
"}︡ ︠1ff67466-98af-4e02-9510-60e35fbf40ac︠ @interact def f(r = slider(-3,3,.25, label = 'r')): show(plot(r + x**2, -3, 3)) ︡9c7c9aff-748f-4fda-9fc6-eaee195f221f︡︡ ︠53422eaf-7706-44f0-abc8-eb061a96729ei︠ %html
Notice that the location and classification of our fixed points depends upon the value of r.
Now let's look at a deceptively simple equation, the Logistic equation: $$ \large y = r*x*(1-x) $$︡ebbade34-9d66-4f0b-9f5c-60e11db1fc7b︡{"html": "
\r\nNotice that the location and classification of our fixed points depends upon the value of r.\r\n
\r\nNow let's look at a deceptively simple equation, the Logistic equation:\r\n$$\r\n\\large\r\ny = r*x*(1-x)\r\n$$\r\n"}︡ ︠9a1b7622-ebeb-49b7-9ec0-b7dfd62f1eb6︠ @interact def f(r = slider(0, 5,.1, label='r')): show(plot(r * x * (1.0 - x), 3, -3)) ︡260fc049-1df9-4015-9499-dc3658f1dc97︡︡ ︠57bf6047-ff83-4864-a486-1a445f804d98i︠ %html
Something strange is going on here; we are getting different values. One tool we can use to look at its stability is a CobWeb diagram.
︡eda03bac-2f61-412d-b1e2-3009f1b4e824︡{"html": "
\r\nSomething strange is going on here; we are getting different values. One tool we can use to look at its stability is a CobWeb diagram.\r\n
"}︡ ︠f7621025-b00e-44c3-b522-67c1f6ec1ccfi︠ %hide # LogisticCobweb.py: # Plot the Logistic Map's quadratic function # And show iterating from an initial condition # Import modules from numpy import * # Plotting from pylab import * # Define the Logistic map's function def LogisticMap(r,x): return r * x * (1.0 - x) # Setup x array # Note how we make sure x = 1.0 is included x = arange(0.0,1.01,0.01) # We set r = 3.678, where two bands merge to one r = 3.678 # Set the initial condition state = 0.5 # Setup the plot # It's numbered 1 and is 6 x 6 inches, to make the plot square. # On my display this didn't turn out to be square, so I've fudged. figure(1,(6,5.8)) # Note how we turn the parameter value into a string # using the string formating commands. TitleString = 'Logistic map: f(x) = r*x*(1 - x) for r = %g and x0 = %g' % (r, state) title(TitleString) xlabel('X(n)') # set x-axis label ylabel('X(n+1)') # set y-axis label # We plot the identity y = x for reference plot(x, x, 'b--', antialiased=True) # Here's the Logistic Map itself plot(x, LogisticMap(r,x), 'g', antialiased=True) # ... and its second iterate # Establish the arrays to hold the line end points x0 = [ ] # The x_n value x1 = [ ] # The next value x_n+1 # Iterate for a few time steps nIterates = 20 # Plot lines, showing how the iteration is reflected off of the identity for n in xrange(nIterates): x0.append( state ) x1.append( state ) x0.append( state ) state = LogisticMap(r,state) x1.append( state ) # Plot the lines all at once plot(x0, x1, 'r', antialiased=True) savefig('LogisticBifn0', dpi=100) # Display plot in window show() ︡8a8ce9ea-c1f4-4bfa-ac2d-fae9b567dcb6︡︡ ︠719475d1-bf09-4212-b25a-dbacf6870e1bi︠ %html
Now lets go onto 2D systems. Let's consider the system with the parameter a: $$ \large \frac{dx}{dt} = -y + ax(x^2+y^2) $$ $$ \large \frac{dy}{dt} = x + ay(x^2+y^2) $$
︡4c864f7d-a65e-4765-be73-a738392755ff︡{"html": "
\r\nNow lets go onto 2D systems. Let's consider the system with the parameter a:\r\n$$\r\n\\large\r\n\\frac{dx}{dt} = -y + ax(x^2+y^2)\r\n$$\r\n$$\r\n\\large\r\n\\frac{dy}{dt} = x + ay(x^2+y^2)\r\n$$\r\n
"}︡ ︠ff47594c-8dc8-46d2-bc99-ed50010fbd80︠ a=-1 plot_vector_field((lambda x,y:-y+a*x*(x^2+y^2),lambda x,y:x+a*y*(x^2+y^2)), (x,-3,3),(y,-3,3)) ︡b0e3df12-8e5e-4d30-9727-47a252f23277︡︡ ︠d6174d99-b6ba-4830-a0c1-3a733625213ei︠ %html
Bifurcation and Period doubling
︡bffa4006-9e2c-42ac-b48c-2f49d697adee︡{"html": "
\r\nBifurcation and Period doubling\r\n
"}︡ ︠759b1aae-e918-46a7-8103-0c469f32bf13︠ %cython '''Plot a bifurcation diagram for the Logistic Map x_n+1 = r*x_n*(1 - x_n) ''' from numpy import * from pylab import * import math # Define LogisticMap def LogisticMap(double r, double x): #return r * x * (1.0 - x) return r * x - x*x*x # Setup parameter range rlow = 1 rhigh = 4.0 # Setup the plot # Stuff parameter range into a string via the string formating commands. TitleString = 'Logistic map, f(x) = r*x*(1 - x), ' TitleString += 'bifurcation diagram for r in [%g,%g]' % (rlow,rhigh) title(TitleString) # Label axes xlabel('Control parameter r') ylabel('{X(n)}') # Set the initial condition used across the different parameters ic = .01 # Establish the arrays to hold the set of iterates at each parameter value # The iterates we'll throw away nTransients = 200 # This sets how much the attractor is filled in nIterates = 250 # This sets how dense the bifurcation diagram will be nSteps = 500 # Sweep the control parameter over the desired range rInc = (rhigh-rlow)/float(nSteps) for r in arange(rlow,rhigh,rInc): # Set the initial condition to the reference value state = ic # Throw away the transient iterations for i in xrange(nTransients): state = LogisticMap(r,state) # Now store the next batch of iterates rsweep = [ ] # The parameter value x = [ ] # The iterates for i in xrange(nIterates): state = LogisticMap(r,state) rsweep.append(r) x.append( state ) plot(rsweep, x, 'k,') # Plot the list of (r,x) pairs as pixels # Use this to save figure as a bitmap png file savefig('LogisticBifn', dpi=100) # Turn on matplotlib's interactive mode. #ion() # Display plot in window show() ︡a227e078-7ca5-4df0-8d6b-14bfbbde413f︡︡ ︠c165f6b3-f2dc-4913-88e2-93796820a2aai︠ %html
Bifurcation for Cosine map
It's remarkable how similiar these two bifurcation plots look, even for systems with quite different character. The resemblance is more than skin deep. Due to a 1973 theorem by Metropolis et. al., the periodic attractors always occur in the same sequence! This sequence is called the U-sequence:
$$ \large 1, 2, 2x2, 6, 5, 3, 2x3, 5, 6, 4, 6, 5, 6 ... $$
Thus, for systems in the low range of chaotic behavior, the form of $f(x)$ is irrelevant!
Furthermore, they all period-double at same rate! This rate is given by the Feigenbaum constant, $\delta = 4.699$.
Amazingly, regardless of the details of the physical system, they exhibit bifurcations (period-doubling) in the same order, at the same rate!
Chaos is defined as aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions (positive Liapunov exponent).
Chaos is not equivalent to instability. For example, $\frac{dx}{dt}=x$ is deterministic and shows separation of nearby trajectories. It's not chaotic because the trajectories are repelled to inifinity and never return. So $\infty$ acts like an attracting fixed point, whereas true chaotic behavior is aperiodic (which excludes fixed points as well as periodic behavior).
The canonical example is the Lorenz equations: $$\frac{dx}{dt} = \sigma(y-x) $$ $$\frac{dy}{dt} = rx - y -xz $$ $$\frac{dz}{dt} = xy - bz $$
In sum, there's a lot more we don't know about dynamical systems and chaos. Hopefully many of you will help find answers!
| Table 1.3.1 from Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology) | |||||
| Number of variables $\rightarrow$ | |||||
| n=1 | n=2 | n>=3 | n>>1 | Continuum | |
| Linear | Growth, decay, or equilibrium | Oscillations | Collective phenomena | Wave and patterns | |
| Exponential growth RC circuit Radioactive decay |
Linear oscillators Mass and spring RLC circuit 2-body problem (Kepler, Newton) |
Civil engineering, structures Electrical engineering |
Coupled harmonic oscillators Solid-state physics Molecular dynamics Equilibrium statistical mechanics |
Elasticity Wave equations Electromagnetism (Maxwell) Quantum mechanics (Schrodinger, Heisenberg, Direc) Heat and diffusion Acoustics Viscous fluids |
|
| Nonlinear | Fixed points Bifurcations Overdamped systems, relaxational dynamics Logistic equation for single species |
Pendulum Anharmonic oscillators Limit cycles Biological oscillators (neurons, heart cells) Predator-prey cycles Nonlinear electronics (van der Pol, Josephson) |
Chaos Strange attractors (Lorenz) 3-body problem (Poincare) Chemical kinetics Iterated maps (Feigenbaum) Fractals (Mandelbrot) Forced nonlinear oscillators (Levinson, Smale) Practical uses of chaos Quantum chaos? |
Coupled nonlinear oscillators Lasers, nonlinear optics Nonequilibrium statistical mechanics Nonlinear solid-state physics (semiconductors) Josephson arrays Heart cell synchronization Neural networks Immune system Ecosystems Economics |
Spatio-temporal complexity Nonlinear waves (shocks, solitons) Plasmas Earthquakes General Relativity (Einstein) Quantum field theory Reaction-diffusion, biological and chemical waves Fibrillation Epilepsy Turbulent fluids (Navier-Stokes) Life |
Thanks to Professor James Crutchfield and his PHY 250 Nonlinear Dynamics class
Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology. Studies in Nonlinearity. Westview Press, 2001.
︡60af171c-467a-4df6-a899-eeaf73d76e4a︡{"html": "
\r\nBifurcation for Cosine map\r\n
\r\n
\r\nIt's remarkable how similiar these two bifurcation plots look, even for systems with quite different character. The resemblance is more than skin deep. Due to a 1973 theorem by Metropolis et. al., the periodic attractors always occur in the same sequence! This sequence is called the U-sequence:\r\n
\r\n$$\r\n\\large\r\n1, 2, 2x2, 6, 5, 3, 2x3, 5, 6, 4, 6, 5, 6 ...\r\n$$\r\n
\r\nThus, for systems in the low range of chaotic behavior, the form of $f(x)$ is irrelevant!\r\n
\r\nFurthermore, they all period-double at same rate! This rate is given by the Feigenbaum constant, $\\delta = 4.699$.\r\n
\r\nAmazingly, regardless of the details of the physical system, they exhibit bifurcations (period-doubling) in the same order, at the same rate!\r\n\r\n
\r\n
\r\nChaos is defined as aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions (positive Liapunov exponent).\r\n
\r\nChaos is not equivalent to instability. For example, $\\frac{dx}{dt}=x$ is deterministic and shows separation of nearby trajectories. It's not chaotic because the trajectories are repelled to inifinity and never return. So $\\infty$ acts like an attracting fixed point, whereas true chaotic behavior is aperiodic (which excludes fixed points as well as periodic behavior).\r\n
\r\nThe canonical example is the Lorenz equations:\r\n$$\\frac{dx}{dt} = \\sigma(y-x)\r\n$$\r\n$$\\frac{dy}{dt} = rx - y -xz\r\n$$\r\n$$\\frac{dz}{dt} = xy - bz\r\n$$\r\n
\r\nIn sum, there's a lot more we don't know about dynamical systems and chaos. Hopefully many of you will help find answers!\r\n
\r\n
| Table 1.3.1 from Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology) | \r\n|||||
| \r\n | Number of variables $\\rightarrow$ | \r\n||||
| \r\n | n=1 | \r\nn=2 | \r\nn>=3 | \r\nn>>1 | \r\nContinuum | \r\n
| Linear | \r\nGrowth, decay, or equilibrium | \r\nOscillations | \r\n\r\n | Collective phenomena | \r\nWave and patterns | \r\n
| Exponential growth RC circuit Radioactive decay | \r\nLinear oscillators Mass and spring RLC circuit 2-body problem (Kepler, Newton) | \r\nCivil engineering, structures Electrical engineering | \r\nCoupled harmonic oscillators Solid-state physics Molecular dynamics Equilibrium statistical mechanics | \r\nElasticity Wave equations Electromagnetism (Maxwell) Quantum mechanics (Schrodinger, Heisenberg, Direc) Heat and diffusion Acoustics Viscous fluids | \r\n|
| Nonlinear | \r\nFixed points Bifurcations Overdamped systems, relaxational dynamics Logistic equation for single species | \r\nPendulum Anharmonic oscillators Limit cycles Biological oscillators (neurons, heart cells) Predator-prey cycles Nonlinear electronics (van der Pol, Josephson) | \r\nChaos Strange attractors (Lorenz) 3-body problem (Poincare) Chemical kinetics Iterated maps (Feigenbaum) Fractals (Mandelbrot) Forced nonlinear oscillators (Levinson, Smale) Practical uses of chaos Quantum chaos? | \r\nCoupled nonlinear oscillators Lasers, nonlinear optics Nonequilibrium statistical mechanics Nonlinear solid-state physics (semiconductors) Josephson arrays Heart cell synchronization Neural networks Immune system Ecosystems Economics | \r\nSpatio-temporal complexity Nonlinear waves (shocks, solitons) Plasmas Earthquakes General Relativity (Einstein) Quantum field theory Reaction-diffusion, biological and chemical waves Fibrillation Epilepsy Turbulent fluids (Navier-Stokes) Life\r\n |
\r\nThanks to Professor James Crutchfield and his PHY 250 Nonlinear Dynamics class\r\n
\r\n
\r\nStrogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology. Studies in Nonlinearity. Westview Press, 2001.\r\n
"}︡
![examples](\"http://en.wikipedia.org/wiki/Image:Mandel_zoom_00_mandelbrot_set.jpg\"){kind=link}