1. Mathematical Analysis of a Demonstrative Chaotic Circuit Karen Kelleher and Dr. Thomas Kling, Department of Physics, Bridgewater State College, Bridgewater, MA 02325 The word bifurcation means “splitting into two branches”. Bifurcations are the change in qualitative structure of the flow as parameters are varied. Bifurcation points are fixed points that can be either created or destroyed or their stability can change. There are different kinds of bifurcations. The chaotic circuit designed by Mishina, Kohmoto, and Hashi has a recursion relation that is best described by a transcritical bifurcation. The transcritical bifurcation is the type of bifurcation where the stability of the fixed points themselves changes as the parameter is varied. The normal form for this bifurcation is. A normal form is an equation that is representative of all transcritical bifurcations. Introduction Transcritical Bifurcations Below is the bifurcation diagram for the transcritical bifurcation. The fixed point at r = 0 moves from stable when r 0. Conversely, the fixed point at x = r is unstable for r 0. The stable branches are solid lines and marked in red, while the unstable branches are marked in blue and dashed. Recursion Relation Recursion relations are also known as iterated maps or logistic maps. For these maps, X and Y are plotted on the vertical and n is plotted on the horizontal. The Microsoft Excel program on the laptop demonstrates the mapping of the circuit. Use the scrollbars to observe how the map changes with different values of each parameter. This is an example of the plots of the coupled iterative equations. The parameters can be varied to achieve examples of chaos. An analysis of the bifurcation structure of an electronic circuit designed by T. Mishina, T. Kohmoto, and T. Hashi from the American Journal of Physics, April 1985, pp. 332-334 is presented. The circuit models these two coupled iterative equations: where the parameters A, B, C and D represent dimensionless variable resistors. This equation maps current values (X n, Y n ) into new values (X n+1, Y n+1 ) and has fixed points – values of X n and Y n that do not change from one iteration to the next. Bifurcations, or changes in the number of fixed points, are created by the parameters that govern an equation’s behavior. As these parameters are varied, the circuit undergoes several different bifurcations leading to chaotic behavior. Abstract Period-doubling Cascade As the parameters are changed, the map begins to look periodic. Once X n repeats itself every two iterations it is called a period- 2 cycle. However, increasing the parameters more and more will achieve a period-4 and a period-8 and now the bifurcations are coming closer and closer together and this is called the period- doubling cascade and is the onset of the chaotic behavior. Bifurcation Code #________________________________________ # # Program reproduces the attractor for the # chaotic circuit # # Written April 22, 2005 # # With A not equal to C, B= D varying # #________________________________________ open(BIF, ">>BIF_d3.txt"); $nmax = 500; for($n = 1; $n<$nmax; $n++){ $x = rand; $y = rand; $A = 1; $B = 0 + 0.4*$n/$nmax; $C = 1.5; $D = 0 + 0.4*$n/$nmax; for ($i=1; $i<1000; $i++){ $x = 1-$A*$x**2-$B*($x-$y); $y = 1-$C*$x**2-$D*($y- $x); } for ($j=1; $j<20; $j++){ $x = 1-$A*$x**2-$B*($x- $y); $y = 1-$C*$x**2-$D*($y- $x); printf BIF "$D $x $y \n"; } close(BIF); References Mishina, T., T. Kohmoto, T. Hashi. “Simple electronic circuit for the demonstration of chaotic phenomenon” American Journal of Physics 53 (4) (1985): 332-334. Strogatz, Stephen H. Nonlinear Dynamics and Chaos. Cambridge: Westview, 1994. Conclusions The coupled iterative equations undergo several period- doublings before cascading into chaotic behavior. The fixed points change as the parameters are varied. In each of the bifurcation diagrams, the functions bifurcate at the same values regardless of the rate the values change.