MATLAB Examples

The Lorenz Equations (Demo : lrz)

This demo computes two symmetric homoclinic orbits in the Lorenz equations

: $u_1' =  p_3 (u_2 - u_1)$,

: $u_2' =  p_1 u_1 - u_2 - u_1 u_3$,

: $u_3' =  u_1 u_2 - p_2 u_3$.

  • Here $p_1$ is the free parameter, and $p_2=8/3$, $p_3=10$.
  • The two homoclinic orbits correspond to the final, large period orbits on the two periodic solution families.

Create continuation object and set initial conditions.

a{1}=auto;

Print function file to screen.

type(a{1}.s.FuncFileName);
function [f,o,dfdu,dfdp]= func(par,u,ijac)
%
% equations file for lorenz demo lrz
%
f=[];
o=[];
dfdu=[];
dfdp=[];

f(1)= par(3) * (u(2)- u(1));
f(2)= par(1)*u(1) - u(2) - u(1)*u(3);
f(3)= u(1)*u(2) -  par(2)*u(3);


Set initial conditions.

[a{1}.s.Par0,a{1}.s.U0,a{1}.s.Out0]=stpnt;

Set constants.

a{1}.c=clrz1(a{1}.c);

Run equilibrium continuation.

a{1}=runauto(a{1});
 
    --------------- DYNAMICAL SYSTEMS TOOLBOX ---------------------     
 
USER NAME      : ECOETZEE
DATE           : 26/10/2010 10:10:28
 
 
<
  BR    PT  TY  LAB      PAR(01)      L2-NORM         U(01)         U(02)         U(03)
   1     1  EP    1   0.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00
   1     5  BP    2   1.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00
   1    13  EP    3   3.16000E+01   0.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00
  BR    PT  TY  LAB      PAR(01)      L2-NORM         U(01)         U(02)         U(03)
   2    42  HB    4   2.47368E+01   2.62685E+01   7.95602E+00   7.95602E+00   2.37368E+01
   2    45  EP    5   3.26008E+01   3.41635E+01   9.17980E+00   9.17980E+00   3.16008E+01
  BR    PT  TY  LAB      PAR(01)      L2-NORM         U(01)         U(02)         U(03)
   2    42  HB    6   2.47368E+01   2.62685E+01  -7.95602E+00  -7.95602E+00   2.37368E+01
   2    45  EP    7   3.26008E+01   3.41635E+01  -9.17980E+00  -9.17980E+00   3.16008E+01

 Total Time    0.938E-01
>

Create second object for restart

a{2}=auto;
a{2}.f8=a{1}.f8;
a{2}.c=clrz2(a{1}.c);

Compute periodic solutions; the final orbit is near-homoclinic from label 4

a{2}=runauto(a{2});
 
    --------------- DYNAMICAL SYSTEMS TOOLBOX ---------------------     
 
USER NAME      : ECOETZEE
DATE           : 26/10/2010 10:10:29
 
 
<
  BR    PT  TY  LAB      PAR(01)      L2-NORM     MAX U(01)     MAX U(02)     MAX U(03)      PERIOD
   4     2        8   2.47263E+01   2.62550E+01   8.24078E+00   8.35402E+00   2.42325E+01   6.53196E-01
   4     4        9   2.46221E+01   2.61217E+01   8.87099E+00   9.25396E+00   2.52683E+01   6.56815E-01
   4     6       10   2.43598E+01   2.57862E+01   9.55937E+00   1.02673E+01   2.62866E+01   6.66114E-01
   4     8       11   2.38815E+01   2.51733E+01   1.02649E+01   1.13402E+01   2.71688E+01   6.83815E-01
   4    10       12   2.31354E+01   2.42151E+01   1.09349E+01   1.23940E+01   2.77796E+01   7.13505E-01
   4    12       13   2.20780E+01   2.28511E+01   1.15132E+01   1.33396E+01   2.79816E+01   7.60644E-01
   4    14       14   2.06702E+01   2.10193E+01   1.19409E+01   1.40738E+01   2.76380E+01   8.35069E-01
   4    16       15   1.88821E+01   1.86471E+01   1.21484E+01   1.44796E+01   2.66089E+01   9.57569E-01
   4    18       16   1.67353E+01   1.56405E+01   1.20536E+01   1.44257E+01   2.47832E+01   1.18330E+00
   4    20       17   1.48138E+01   1.23916E+01   1.17081E+01   1.39788E+01   2.27276E+01   1.62929E+00
   4    22       18   1.39678E+01   9.14991E+00   1.14909E+01   1.36783E+01   2.17209E+01   2.78130E+00
   4    24       19   1.39266E+01   5.88316E+00   1.14637E+01   1.36364E+01   2.16632E+01   6.70329E+00
   4    26       20   1.39266E+01   4.27612E+00   1.14814E+01   1.36634E+01   2.16547E+01   1.26885E+01
   4    28       21   1.39266E+01   2.60523E+00   1.14798E+01   1.36596E+01   2.16716E+01   3.41837E+01
   4    30       22   1.39265E+01   1.77732E+00   1.14801E+01   1.36531E+01   2.15700E+01   7.34476E+01
   4    32       23   1.39266E+01   1.14209E+00   1.14546E+01   1.36068E+01   2.16277E+01   1.77872E+02
   4    34       24   1.39266E+01   7.31857E-01   1.14815E+01   1.36619E+01   2.16442E+01   4.33169E+02
   4    35  EP   25   1.39266E+01   6.29547E-01   1.14802E+01   1.36631E+01   2.16345E+01   5.85401E+02

 Total Time    0.509E+01
>

Create third object for restart

a{3}=auto;
a{3}.f8=a{1}.f8;
a{3}.c=clrz3(a{3}.c);

Compute the symmetric periodic solution family from label 6

a{3}=runauto(a{3});
 
    --------------- DYNAMICAL SYSTEMS TOOLBOX ---------------------     
 
USER NAME      : ECOETZEE
DATE           : 26/10/2010 10:10:34
 
 
<
  BR    PT  TY  LAB      PAR(01)      L2-NORM     MAX U(01)     MAX U(02)     MAX U(03)      PERIOD
   6     2        8   2.47263E+01   2.62550E+01  -7.66589E+00  -7.55614E+00   2.42325E+01   6.53196E-01
   6     4        9   2.46221E+01   2.61217E+01  -6.98291E+00  -6.63767E+00   2.52683E+01   6.56815E-01
   6     6       10   2.43598E+01   2.57862E+01  -6.16259E+00  -5.57720E+00   2.62866E+01   6.66114E-01
   6     8       11   2.38815E+01   2.51733E+01  -5.21877E+00  -4.41802E+00   2.71688E+01   6.83815E-01
   6    10       12   2.31354E+01   2.42151E+01  -4.18398E+00  -3.22592E+00   2.77796E+01   7.13505E-01
   6    12       13   2.20780E+01   2.28511E+01  -3.10528E+00  -2.07754E+00   2.79816E+01   7.60644E-01
   6    14       14   2.06702E+01   2.10193E+01  -2.04466E+00  -1.05682E+00   2.76380E+01   8.35069E-01
   6    16       15   1.88821E+01   1.86471E+01  -1.08612E+00  -2.53862E-01   2.66089E+01   9.57569E-01
   6    18       16   1.67353E+01   1.56405E+01  -3.58904E-01   2.34383E-01   2.47832E+01   1.18330E+00
   6    20       17   1.48138E+01   1.23916E+01  -4.02246E-02   3.50485E-01   2.27276E+01   1.62929E+00
   6    22       18   1.39678E+01   9.14991E+00  -1.09822E-04   3.28597E-01   2.17209E+01   2.78130E+00
   6    24       19   1.39266E+01   5.88316E+00   2.71087E-12   3.26551E-01   2.16632E+01   6.70329E+00
   6    26       20   1.39266E+01   4.27612E+00   3.10163E-11   3.26608E-01   2.16547E+01   1.26885E+01
   6    28       21   1.39266E+01   2.60523E+00   1.78906E-10   3.26143E-01   2.16716E+01   3.41837E+01
   6    30       22   1.39265E+01   1.77732E+00   2.29274E-15   3.16121E-01   2.15700E+01   7.34476E+01
   6    32       23   1.39266E+01   1.14209E+00   6.72816E-13   3.25737E-01   2.16277E+01   1.77872E+02
   6    34       24   1.39266E+01   7.31857E-01   1.18433E-09   3.26767E-01   2.16442E+01   4.33169E+02
   6    35  EP   25   1.39266E+01   6.29547E-01   8.84302E-12   3.26839E-01   2.16345E+01   5.85401E+02

 Total Time    0.512E+01
>

Create fourth object for restart

a{4}=auto;
a{4}.f8=a{3}.f8;
a{4}.c=clrz4(a{4}.c);

Compute the symmetric periodic solution family from label 9

a{4}=runauto(a{4});
 
    --------------- DYNAMICAL SYSTEMS TOOLBOX ---------------------     
 
USER NAME      : ECOETZEE
DATE           : 26/10/2010 10:10:40
 
 
<
  BR    PT  TY  LAB      PAR(01)      L2-NORM     MAX U(01)     MAX U(02)     MAX U(03)      PAR(02)
   6     5  BP   26   2.46124E+01   2.61228E+01  -7.89124E+00  -7.88743E+00   2.36299E+01   2.64389E+00
   6    10       27   2.47827E+01   2.61614E+01  -5.03850E+00  -3.96979E+00   2.91900E+01   2.90160E+00
   6    15       28   2.73327E+01   2.76484E+01  -2.13282E+00   5.87442E-01   3.94746E+01   4.16093E+00
   6    17  EP   29   3.03044E+01   2.94769E+01  -1.35817E+00   2.46713E+00   4.61441E+01   4.94313E+00

 Total Time    0.228E+01
>

Plot the solution Create plaut object and plot solution.

p=plautobj;
set(p,'xLab','Par','yLab','L2norm');
ploteq(p,a);