Vector Iterative Maps and Autonomous DEs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== Consider the 2 component iterative map with the functions below QyQ+SSJmRzYiLCgqJkkieEdGJSIiI0kieUdGJSIiJiIiIiomRihGK0YqRikhIiItSSRleHBHRiU2IywmKiZGKEYsRipGLEYsRi5GLEYsRiw= LCgqJkkieEc2IiIiI0kieUdGJSIiJiIiIiomRiRGKEYnRiYhIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmKiZGJEYpRidGKUYpRitGKUYp QyQ+SSJnRzYiLCoqJkkieEdGJSIiIkkieUdGJSIiJEYpKiZGKCIiI0YqRi1GKSomRihGK0YqRilGKSEiJEYpRik= LCoqJkkieEc2IiIiIkkieUdGJSIiJEYmKiZGJCIiI0YnRipGJiomRiRGKEYnRiZGJiEiJEYm Let us find a fixed point for the map. Remember this is not a foor of f and g but a root of f-x and g-y. QyQ+SSZmaXhlZEc2Ii1JJ2Zzb2x2ZUdGJTYkPCQvSSJmR0YlSSJ4R0YlL0kiZ0dGJUkieUdGJTwkL0YsIiIiL0YvRjJGMg== PCQvSSJ4RzYiJCIrSlt2KTMiISIqL0kieUdGJSQiKyopb3JwNUYo QyY+SSZ4c3Rhckc2Ii1JJHJoc0clKnByb3RlY3RlZEc2IyZJJmZpeGVkR0YlNiMiIiJGLT5JJnlzdGFyR0YlLUYnNiMmRis2IyIiI0Yt JCIrSlt2KTMiISIq JCIrKilvcnA1ISIq Let's now investigate the stability of this fixed point by doing a simulation of the map for a few iterations starting near the fixed point. QyY+SSR4aXRHNiIsJkkmeHN0YXJHRiUiIiIkRighIiRGKEYoPkkkeWl0R0YlLCZJJnlzdGFyR0YlRihGKUYoRig= JCIrSlt2KjMiISIq JCIrKilvcnE1ISIq QyQ/KEkiaUc2IiIiIkYmIiImSSV0cnVlRyUqcHJvdGVjdGVkR0MnPkkleG5ld0dGJS1JJmV2YWxmR0YpNiMtSSVldmFsR0YpNiRJImZHRiU8JC9JInhHRiVJJHhpdEdGJS9JInlHRiVJJHlpdEdGJT5JJXluZXdHRiUtRi42Iy1GMTYkSSJnR0YlRjQ+RjdGLD5GOkY8LUkmcHJpbnRHRik2JEY3RjohIiI= NiQkIitIMiEzNCIhIiokIitnPiNbMyJGJQ== NiQkIis/VVtyNiEiKiQiKz1OIjM/IkYl NiQkIit4VCpvdSIhIiokIitqQyJ6JEhGJQ== NiQkIitzeEUpKmUhIigkIitgJ0gqSCQpISIp NiQkIit4IlJNWCMiJkc4IyQiK25aeCUpPiIiIg== It seems pretty clear from the simulation that the fixed point is unstable. We can confirm this by finding the eigenvalues of the jacobian matrix at the fixed point. Unlike last time, when we computed the entries of the jacobian matrix one by one, we can use built-in Maple commands to compute it. QyQtSSV3aXRoRzYiNiNJL1ZlY3RvckNhbGN1bHVzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi QyQ+SSViaWdKRzYiLUkpSmFjb2JpYW5HRiU2JDckSSJmR0YlSSJnR0YlNyRJInhHRiVJInlHRiUiIiI= LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciK0NDSjJW QyQ+SSNKMUc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUklZXZhbEdGKDYkSSViaWdKR0YlPCQvSSJ4R0YlSSZ4c3RhckdGJS9JInlHRiVJJnlzdGFyR0YlIiIi LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciK2tFSjJW Now do the eigenanalysis QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLkxpbmVhckFsZ2VicmFHRiUhIiI= LUktRWlnZW52ZWN0b3JzRzYiNiNJI0oxR0Yk NiQtJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkdGKTYjL0kkJWlkR0YpIitXUEoyVi1JJ01hdHJpeEdGJjYjL0YuIisvYEoyVg== So we can see that there is an eigenvalue larger than one in absolute value (in fact, both are) so the equilibrium is indeed unstable, as our simulation predicted. Autonomous DE, vector example consider now the DE for x(t) and y(t) with the RHS functions f anf g above. The equilibrium points are roots of f=0, g=0. One is found below. QyQ+SSxlcXVpbGlicml1bUc2Ii1JJ2Zzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkPCQvSSJmR0YlIiIhL0kiZ0dGJUYvPCQvSSJ4R0YlIiIiL0kieUdGJUY1RjU= PCQvSSJ4RzYiJCIrMzVTcjYhIiovSSJ5R0YlJCIrTCQzLFEpISM1 QyY+SSR4ZXFHNiItSSRyaHNHJSpwcm90ZWN0ZWRHNiMmSSxlcXVpbGlicml1bUdGJTYjIiIiRi0+SSR5ZXFHRiUtRic2IyZGKzYjIiIjRi0= JCIrMzVTcjYhIio= JCIrTCQzLFEpISM1 Let's set up the numerical approximation of the DE with initial conditions (1,1). 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Now let's plot x(t) and y(t) as a function of t. Note that the solution moves away from the equilibrium. This is evidence that it is unstable. Note that there is a finite time singularity in the solution that Maple detects. LUkob2RlcGxvdEc2IjYlSSVzb2xuR0YkNyQ3JEkidEdGJC1JInhHRiQ2I0YpNyRGKS1JInlHRiRGLC9GKTsiIiEiIiI= LUkob2RlcGxvdEc2IjYlSSVzb2xuR0YkNyQ3JEkidEdGJC1JInhHRiQ2I0YpNyRGKS1JInlHRiRGLC9GKTsiIiEiIiI= It is a bit more interesting to polt the solution in the phase plane, done below. In time, the solution moves up and to the right on the curve shown. LUkob2RlcGxvdEc2IjYlSSVzb2xuR0YkNyM3JC1JInhHRiQ2I0kidEdGJC1JInlHRiRGKy9GLDsiIiEiIiI= LUkob2RlcGxvdEc2IjYlSSVzb2xuR0YkNyM3JC1JInhHRiQ2I0kidEdGJC1JInlHRiRGKy9GLDsiIiEiIiI= To get more global information about the solutions to the DE, we can use a direction field. The built in direction field command below only shows the direction of the solution motion in time, not the magnitude. I was disappointed in this. 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I explored other ways to get the plot I wanted. The following command uses "VectorCalculus" that was declared above QyQ+SSNGMUc2Ii1JLFZlY3RvckZpZWxkR0YlNiQ3JC1JImZHRiU2JEkieEdGJUkieUdGJS1JImdHRiVGLCZJKmNhcnRlc2lhbkdGJUYsIiIi LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIrN3QoKilSJQ== QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIi This approach gives a much more meaningful plot. The equilibrium point is clearly seen as the arrow with zero length here. It is clear from this picture that the equilibrium is a saddle node (unstable). 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We can confirm this by looking at the eigenanalysis of the Jacobian at the equilibrium point QyQ+SSNKMkc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUklZXZhbEdGKDYkSSViaWdKR0YlPCQvSSJ4R0YlSSR4ZXFHRiUvSSJ5R0YlSSR5ZXFHRiUiIiI= LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciK18oeSopUiU= One negative and one positive eigenvalue characterizes a saddle (unstable) equilibrium. 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