A Few Examples of MATLAB m-files:

PI-M01-M02

Some words about:

Cusp and Generalized Hopf bifurcations

The next file contains some Matlab Reminders: Matrix powers and exponentials, Eigenvalues, Singular Value Decomposition, Vector and Matrix Norms (m-file):

MatlabMathematicsReminders.m (these reminders can be skipped)

Elementary example of nonlinear syst. resolution (m-files):

nonlineareq.m solution.m

Example of Nonlinear Equations with Jacobian (m-files):

nleqwj.m solveroutine0.m

Example of Nonlinear Equations with Analytic Jacobian (m-files):

rosenbrockobj.m solveroutine1.m

Julia Sets (m-files):

JuliaSet.m JuliaSequence.m

Some nonlinear systems: Lorenz, Duffing, Chua circuit, Colpitts oscillator, Rayleigh oscillator, Van der Pol oscillator (for μ=0.2, μ=1, μ=10), and Henon attractor (m-files):

lorenzeqs.m duffing.m Chua.m Colpitts.m Rayleigh.m vdpeq1.m vdpeq2.m vdpeq3.m henonmap.m attractLIST.m (The last file starts the simulations)

Plot Van der Pol oscillator for both values μ=0.2, μ=1 (m-files):

vdpeq1.m vdpeq2.m vdpeq3.m vdpSUB.m

Evaluate Van der Pol oscillator for μ varying from 0 to 2 (m-files):

vdpeqMU.m vdpAnim.m [t,(y₁,y₂)] ; [y₁, y₂]

Evaluate Van der Pol oscillator for μ varying from 0 to 2.5. Show [t,y₁,y₂] (m-files):

vdpeqMU.m vdpAnim3D.m

Movie of the previous simulation (swf.file, wmv-file, 16'):

vdp3D.swf (vdp3D.wmv, [t,y₁,y₂]: vdp3D1.wmv , [y₁,y₂]: vdp3D2.wmv )

Evaluate Van der Pol oscillator for μ varying from -1.8 to 2.5. [t,y₁,y₂] (m-files):

vdpeqMU.m vdp3DLong.m

Movie of the previous simulation but for μ=-0.8 to 2.5 [t,y₁,y₂] (swf.file, wmv-file, 22'):

vdp3DL.swf (vdp3DL.wmv)

Linear system examples: center, node-sink, spiral-source, 2 saddles (m-files):

center.m centerDyn.m

nodesink.m nodesinkDyn.m

spiralsource.m spiralsourceDyn.m

saddle1.m saddle1Dyn.m

saddle2.m saddle2Dyn.m

Nonlinear system example with a (supercritical) Hopf bifurcation (m-files):

hopf.m hopfDyn.m

Movie of the simulation above (swf.file, wmv-file, 3'):

BasicHopf.swf BasicHopf.wmv

Example of restricted three-body problem (m-files):

events.m fct_tbp.m threebodyproblem.m

Torus (genus-1) whose surface is traversed by a point with a regular trajectory (m-files):

TorusPQ.m (In addition, see animation of a genus-1 whose radii change genus1m.swf)
Orbit of a nonlinear system (including Hopf Bifurcation), γ= -1,..,1, μ= -0.5,..,3 (m-file):

Attention: Consider as an exercise the question: Is the next simulation called "Dynamic2.m" valid or not?

Dynamic2.m

Movie of "Dynamic2" simulation above, same question as above (swf.file, wmv-file, 13'):

OrbitDyn2.swf OrbitDyn2.wmv

Cusp (singularity) Diagram example (m-file):

CuspDiag.m

Logistic dynamic for alpha=3.5,...,4 (m-files):

logis.m logeqsimu.m

Movie of the basic dynamic (observe the revolution of points) (swf.file, wmv-file, 1"06'):

logmapDyn.swf logmapDyn.wmv

Logistic orbit (m-files):

logisticorbit.m

Lyapunov Exponent of Logistic orbit (Lce) (m-files):

LyapLog1.m

Iterative maps of logistic equation (m-files):

bosn.m humpdemo.m

10 iterative maps of logistic equation (m-files):

fct_eqs.m bosses.m

Iterative maps (sliders m-files):

fct_eqs.m inter.m Huslider.m

Initial values and logistic map (sliders m-files):

logis.m inter1l.m Lslider.m

Logistic map and histogram varying with alpha (m-files):

hyyt.m hylog.m

Spectra, Poincaré sections, histograms of the logistic map for alpha=3.5,..,4 (m-files):

hyyt.m ehsp.m

Power spectrum for alpha=2.6,..,3.6 (m-files):

pppp.m densipp.m pppslid.m

Movie of the power spectrum for alpha=3.35,..,3.575, lobes (swf.file, wmv-file, 1"56'):

psd1.swf psd1.wmv

Movie of the power spectrum for alpha=3.5,..,4 (swf.file, wmv-file, 1"05'):

psd2.swf psd2.wmv

Fractional brownian motions; fBm, 1st and 2nd-difference histograms (sliders m-files):

WhiteNoise.m GenBrownian.m interMB.m MBhslider.m (see also detrending, nth-difference)

Chaotic Dynamics resulting from Delay model & R.May equation (ref. to Medio.1992);

1. Delay model for alpha=5 (m-files):

medioeq.m medio05.m

2. Delay model for alpha=4.95 (m-files):

medioeq.m medio0495.m

3. Delay model for alpha=4.85 (m-files):

medioeq.m medio0485.m

4. Delay model for alpha=4.75 (m-files):

medioeq.m medio0475.m

5. Delay model for alpha=4.5 period-doubling (m-files):

medioeq.m medio045.m

6. Delay model for alpha=4.35 (m-files):

medioeq.m medio0435.m

7. Delay model for alpha=4 (m-files):

medioeq.m medio04.m

8. Delay model for alpha=3 (m-files):

medioeq.m medio03.m

Dynamic view for alpha varying from 3 to 5 (m-files):

medioeq.m medioanim.m

Movie of the previous dynamic (swf.file, wmv-file, 1"39'):

medio.swf medio.wmv

Attractor reconstructed by using Singular Spectrum Analysis [Delay model & May equation for alpha=5] (m-files):

medioeq.m BT5med.m

Singular Spectrum Analysis applied to the cac40 stock-index (2806 daily values, Jan 1988 - April 1998) (m-files):

cac40.m BTcac40.m

../...

PII-S03-S04

Generate data of a one-dimensional Gaussian distribution, plot its normalized histogram estimate and true density (m-files):

EgaussD.m histpd.m EstimDensity.m

Epanechnikov, Biquadratic, Gaussian and Cubic kernels (m-file):

kernels.m

Gamma (or Euler) function (m-file):

Eugam.m

Poincaré sections, histograms for a stock-index, white noise, logist. map (m-file):

cac40.m WhiteNoise.m hyyt.m scatplot.m

Probability Density Function (m-file):

pdf.m

Natural measure of probability, u-shape convergence (swf.file, wmv-file, 6'):

Ushape.swf Ushape.wmv

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PIII-P05-P06

Fourier Transform example; coefficients, periodicity, periodogram, power (dat-file, m-file):

sunspot.dat FFTperiodicity.m (this file is based on a demo of Matlab.7)

Power Spectrum via Periodogram of signal (200Hz) embedded in additive noise (m-file):

periodogramEXP.m (this file is an example of the use of Matlab periodogram.m)

Magnitude and Phase of Transformed Data (m-file):

magphase.m

Sum of Wave functions (m-file):

Waves.m

Integral of Morlet Wave (m-file):

wavesurf.m

Non-normalized Window (m-file):

arbitrarywindow.m

Normalized Gauss window (m-file):

gausswindow.m

Gauss window with its derivative x10² (m-files):

gaussDgauss.m

Graphs of four window types (m-file):

windows.m

Simultaneous spread variations of a window and a wave function (sliders m-file):

inter32.m gabfon2.m g2slider.m

Simultaneous translation of a window and a wave function (sliders m-file):

inter3.m gabfonc.m gslider.m

2 Gabor functions (or Gabor atoms) (m-file):

gaboret.m

Schematic representation of the convolution of an arbitrary curve with a wave function (sliders m-files):

interwav1.m wav1.m wav1slider.m (demo).

Movie of previous files (swf.file, wmv-file, 12'):

Convo.swf Convo.wmv

. Matlab7 Toolbox, Wavelab701, 802 or 850 include versions of the "cwt.m" file.

. The cwt.m file used here is not a Wavelab or Matlab7 Toolbox version.

. Versions of Wigner-Ville, Cohen, alias-free generalized Time-Frequency distributions are proposed by Wavelab.

5 Gauss-pseudo-wavelet transforms of a stock-index (m-files):

cac40.m cwt.m CACechel.m wtfcacgauss.m

5 Morlet wavelet transforms of a stock-index (m-files):

cac40.m cwt.m CACechel.m wtfcacmorlet.m

Example of a spectrogram for the signal "transients" (m-files):

transients.asc SpectrogTrans.m

Wigner-Ville Time-Frequency Distribution of the signal "transients" (m-files):

wvdist.m transients.asc wgtrans.m (Time-Frequency plane: [0,1] vs. [0,1])

Wigner-Ville Time-Frequency Distribution of 2 Gabor atoms (m-files):

wvdist.m wg2ats.m (T-F. plane: [0,1] vs. [0,1/2]).

Wigner-Ville Time-Frequency Distribution of 2 slighly different Gabor atoms whose internal frequencies progressively increase (m-files):

wvdist.m wg2atsmv.m (T-F. plane: [0,1] vs. [0,1]).

Movie of files above, observe the (calculation) interference terms (swf-file, wmv-file, 4'):

WGatoms.swf WGatoms.wmv (T-F. plane: [0,1] vs. [0,1]).

Same Wigner-Ville Dist. but in the Time-Frequency plane: [0,1] vs. [0,1/2] (m-files):

wvdist.m wg2atsmvHALF.m (T-F. plane: [0,1] vs. [0,1/2]).

Movie of the files above (swf-file, wmv-file, 4'):

WGatomsHALF.swf WGatomsHALF.wmv

Alias-Free Generalized Discrete Time-Frequency Distribution of 2 Gabor atoms (m-files):

tfdist.m td2ats.m

Cohen class Time-Frequency Distribution of 2 Gabor atoms (m-files):

CohenDist.m Interpol2.m repCH.m

Cohen class Time-Frequency Distribution of 2 slighly different Gabor atoms whose internal frequencies progressively increase (m-files):

CohenDist.m Interpol2.m repCHmv.m

Movie of the previous files of the Cohen class Time-Frequency Dist. (swf-file, wmv-file, 4'):

ChGatoms.swf ChGatoms.wmv (compare with its spectrogram: SpectgAts.swf)

Cohen class Time-Frequency Distribution of a signal that consists of: (1) two slighly different Gabor atoms whose internal frequencies progressively increase, (2) a Dirac, (3) a sinusoid, (4) and a noise that increases at each repeated sequence. (23') (m-files):

CohenDist.m Interpol2.m CHMultSig.m

Movie of the previous files of the Cohen class Time-Frequency Dist. (swf-file, wmv-file, 4'):

CHsignals.swf CHsignals.wmv (compare with its Wigner-Ville dist.: WIGsignalsFULL.swf WIGsignalsFULL.wmv (T-F plane [0,1] vs. [0,1]) WIGsignalsHALF.swf WIGsignalsHALF.wmv ([0,1] vs. [0,1/2]). See calculation interference terms).

Cohen class Time-Frequency Distribution of the signal "transients" (m-files):

CohenDist.m Interpol2.m transients.asc ConhenDisttransients.m

Cohen class Time-Frequency Distribution of the topologist's sine curve (m-files):

CohenDist.m Interpol2.m topoCH.m

Cohen class Time-Frequency Distribution of the signal "linchirp" (m-files):

CohenDist.m Interpol2.m linchirp.asc ConhenDistlinchirp.m

Cohen class Time-Frequency Distribution of 2048 Data of cac40 1st-differences (m-files): CohenDist.m Interpol2.m cac40.m ConhenDistCAC.m

Cohen class Time-Frequency Distribution of 2847 Data of cac40 1st-differences (m-files): CohenDist.m Interpol2.m cac40.m ConhenDistCAC2847.m

../...

PIV-E07-E08

Reminders about Cobb-Douglas & CES functions :

Production fcts. or via pdf file

Reference Solow Model (1956) and Steady State (m-files):

ces.m cobbdouglas.m Vest.m netVest.m TnestVest2.m SolowR.m

Animation of Solow Model (1956) when the savings rate s varies (sliders, m-files):

cobbdouglas.m VestCB.m netVestCB.m TnestVestCB.m SolowSUB.m SwSLID.m

Movie of previous files (swf-file, wmv-file, 08'):

slw.swf slw.wmv
Some words about:

Goodwin Model (1967)

Goodwin Model (1967) (m-files):

goodwin.m good1sim.m

Goodwin Model (1990) (m-files):

goodwin2.m good2sim.m

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Additional files will be subsequently added.

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1.Each group of MATLAB m-files above provides an independent simulation.

2.The simulations start by the m-file placed on the right.

3.Some m-files use syntaxes developed via versions anterior to Matlab.7.0. R14.

4.This material is only intented for didactic uses.

Movie examples (menu): Views

Views of each animation swf http-page

- Reach swf-files to save as target ..

- Reach wmv-files to save as target ..

- Adobe Shock-wave player is required to display swf-files, if needed Download

Last revised on 5 January 2010.

isnld.com

Interdisciplinary Studies of Nonlinear Dynamics

A Few Examples of MATLAB m-files:

PI-M01-M02

PII-S03-S04

PIII-P05-P06

PIV-E07-E08