MATLAB Examples

# Fixed Points of a Discrete Dynamical System (Demo : dd2)

This demo illustrates the computation of a solution family and its bifurcating families for a discrete dynamical system. Also illustrated is the continuation of Naimark-Sacker (or Hopf) bifurcations. The equations, a discrete predator-prey system, are

: ,

: ,

• In the first run is free.
• In the second run, both and are free.
• The remaining equation parameter, , is fixed in both runs.

Create continuation object and set initial conditions.

```a{1}=auto; ```

Print function file to screen. Note in this case that the user is supplying the derivative values, hence ijac > 0.

```type(a{1}.s.FuncFileName); ```
```function [f,o,dfdu,dfdp]= func(par,u,ijac) % % equations file for dd2 demo % f=[]; o=[]; dfdu=[]; dfdp=[]; f(1)=par(1)*u(1)*(1-u(1)) - par(2)*u(1)*u(2); f(2)=(1-par(3))*u(2) + par(2)*u(1)*u(2); if(ijac==0) return; end dfdu(1,1)=par(1)*(1-2*u(1))-par(2)*u(2); dfdu(1,2)=-par(2)*u(1); dfdu(2,1)=par(2)*u(2); dfdu(2,2)=1-par(3) + par(2)*u(1); if(ijac==1) return; end dfdp(1,1)=u(1)*(1-u(1)); dfdp(2,1)=0.0; dfdp(1,2)=-u(1)*u(2); dfdp(2,2)= u(1)*u(2); ```

Set initial conditions.

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

Set constants.

```a{1}.c=cdd21(a{1}.c); ```

Run equilibrium continuation.

```a{1}=runauto(a{1}); ```
``` --------------- DYNAMICAL SYSTEMS TOOLBOX --------------------- USER NAME : ECOETZEE DATE : 26/10/2010 10:09:53 < BR PT TY LAB PAR(01) L2-NORM U(01) U(02) 1 1 EP 1 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 1 13 BP 2 1.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 1 53 EP 3 5.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 BR PT TY LAB PAR(01) L2-NORM U(01) U(02) 2 38 BP 4 2.00000E+00 5.00000E-01 5.00000E-01 0.00000E+00 2 49 HB 5 3.00000E+00 6.66667E-01 6.66667E-01 0.00000E+00 2 70 EP 6 5.09510E+00 8.03733E-01 8.03733E-01 0.00000E+00 BR PT TY LAB PAR(01) L2-NORM U(01) U(02) 2 85 EP 7 1.65467E-01 5.04349E+00 -5.04349E+00 0.00000E+00 BR PT TY LAB PAR(01) L2-NORM U(01) U(02) 3 58 EP 8 3.99862E+00 5.02150E+00 5.00000E-01 4.99654E+00 BR PT TY LAB PAR(01) L2-NORM U(01) U(02) 3 58 EP 9 1.38253E-03 5.02150E+00 5.00000E-01 -4.99654E+00 Total Time 0.781E-01 > ```

Create second object for restart

```a{2}=a{1}; a{2}.c=cdd22(a{1}.c); ```

Run two parameter continuation

```a{2}=runauto(a{2}); ```
``` --------------- DYNAMICAL SYSTEMS TOOLBOX --------------------- USER NAME : ECOETZEE DATE : 26/10/2010 10:09:54 < BR PT TY LAB PAR(01) L2-NORM U(01) U(02) PAR(02) 5 100 EP 10 3.00000E+00 6.66667E-01 6.66667E-01 -1.52217E-33 9.96000E+00 Total Time 0.625E-01 > ```

Plot the solutions. The plotting routine needs to be modified, because it seems as if the end points are being connected (represented by the straight lines).

```p=plautobj; set(p,'xLab','Par','yLab','L2-norm'); ploteq(p,a); snapnow; ```