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Milds # Equilibrium point

Stability diagram classifying Poincaré maps of linear autonomous system $x'=Ax,$ as stable or unstable according to their features. Stability generally increases to the left of the diagram. Some sink, source or node are equilibrium points.

In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.

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• Equilibrium Point
• Equilibrium Solutions and Stability of Differential Equations (Differential Equations 36)
• Equilibrium Points for Nonlinear Differential Equations

#### Transcription

- SUPPLY AND DEMAND FUNCTIONS. OUR GOAL, GIVEN THE SUPPLY AND DEMAND FUNCTIONS FIND THE POINT OF EQUILIBRIUM. SUPPLY AND DEMAND CURVES PLAY A FUNDAMENTAL ROLE IN ECONOMICS THE SUPPLY CURVE INDICATES HOW MANY PRODUCERS WILL SUPPLY THE PRODUCT AT A PARTICULAR PRICE. THE DEMAND CURVE INDICATES HOW MANY CONSUMERS WILL BUY THE PRODUCT AT A GIVEN PRICE. THE EQUILIBRIUM POINT IS THE POINT IN QUANTITY FOR WHICH SUPPLIERS ARE WILLING TO SUPPLY AND CONSUMERS ARE WILLING TO BUY. IT IS THE INTERSECTION OF THE TWO FUNCTIONS SUPPLY EQUAL TO DEMAND. IF YOU TAKE A LOOK AT THIS GRAPH BELOW, YOU CAN SEE. IF WE LOOK AT THE SUPPLY CURVE. IF THE PRICE IS TOO LOW, SUPPLIERS WILL NOT SUPPLY THE ITEM. AS PRICE INCREASES, MORE AND MORE SUPPLIERS WILL PRODUCE THE ITEM. BUT LOOK AT THE DEMAND CURVE, THE PRICE IS TOO HIGH VERY FEW CONSUMERS WILL PURCHASE THE ITEM. BUT AS PRICE BEGINS TO DECREASE, THE NUMBER OF CONSUMERS WILLING TO PURCHASE INCREASES. THE POINT WE'RE CONCERNED ABOUT IS THE POINT WHERE EVERYBODY'S HAPPY. THE SUPPLIERS WILL SUPPLY AT A GIVEN PRICE AND CONSUMERS WILL PURCHASE AT A GIVEN PRICE FIND THE EQUILIBRIUM POINT FOR THE DEMAND AND SUPPLY FUNCTIONS FOR THE ULTRA FINE COFFEE MAKER WHERE Q REPRESENTS THE NUMBER OF COFFEE MAKERS PRODUCED IN HUNDREDS AND X IS THE PRICE IN DOLLARS. SO WE WANT TO FIND WHERE THE DEMAND CURVE IS EQUAL TO THE SUPPLY CURVE. NOW THERE ARE TWO WAYS OF DOING THIS. WE CAN DO IT ALGEBRAICALLY OR GRAPHICALLY. WE WILL DO IT BOTH. WELL IF WE WANT TO KNOW WHEN THESE TWO FUNCTIONS ARE EQUAL TO EACH OTHER, WE CAN SIMPLY SET 50 - 1/4 X = TO X - 25. THIS IS OUR DEMAND FUNCTION. THIS IS OUR SUPPLY FUNCTION. LET'S SOLVE FOR X. NOW THE FIRST THING I NOTICED HERE IS WE HAVE A FRACTION INVOLVED. I'M GOING TO CLEAR THE FRACTION BY MULTIPLYING BOTH SIDES OF THE EQUATION BY 4. THAT WOULD GIVE US 200 - X = 4X - 100. SOLVING FOR X, LET'S ADD X TO BOTH SIDES. THAT WOULD GIVE US 200 = 5X - 100. ADD 100 TO BOTH SIDES, 5X = 300, DIVIDE BY 5 X = 60. 60 IS THE PRICE OF THE EQUILIBRIUM POINT. THAT'S HOW MUCH THE ITEMS SHOULD SELL FOR. NOW REMEMBER THE EQUILIBRIUM POINT CONSISTS OF THE PRICE AND THE QUANTITY. IN ORDER TO FIND THE QUANTITY, WE HAVE TO SET X EQUAL TO 60 IN ONE OF THESE FUNCTIONS. LET'S GO AHEAD AND USE THE SUPPLY FUNCTION.   SO THE NUMBER OF ITEMS OR THE QUANTITY IS EQUAL TO 35. BUT REMEMBER FROM THE ORIGINAL PROBLEM, THIS WAS IN HUNDREDS. SO TO GET THE TRUE QUANTITY, WE HAVE TO MULTIPLY THIS BY A HUNDRED TO GET 3,500. SO THE EQUILIBRIUM POINT IS WHEN THE PRICE IS SET AT $60 AND THE QUANTITY PRODUCED IS 3,500. OUR TEXTBOOK LIKES TO USE THIS NOTATION X OF THE Q SUB E AS THE POINT OF EQUILIBRIUM. SO IN THIS CASE WE WOULD HAVE$60 AS THE PRICE AND 3,500 FOR THE QUANTITY. I DO WANT TO POINT OUT THIS IS THE OPPOSITE ORDER OF THE INITIAL EXAMPLE THAT WE SHOWED ON THE FIRST SCREEN. THE QUANTITY WAS THE X COORDINATE AND THE PRICE WAS THE Y COORDINATE. IT DOESN'T MATTER WHICH ORDER THESE ARE IN AS LONG AS WE'RE CONSISTENT. LET'S TAKE A LOOK AT THE SAME PROBLEM GRAPHICALLY. I'VE ALREADY GRAPHED THE DEMAND FUNCTION IN BLUE AND THE SUPPLY FUNCTION IN RED. THIS WOULD BE GRAPHICALLY, THIS WOULD BE THE EQUILIBRIUM POINT AND IT DOES VERIFY OUR ANSWER, LOOKS LIKE THE PRICE WOULD BE $60 AND THE QUANTITY ON THE Y AXIS WOULD BE 35 BUT OF COURSE THIS IS QUANTITY IN THE HUNDREDS SO IT DOES VERIFY THE PRICE OF$60 AND THE QUANTITY OF 3,500 COFFEE MAKERS. I HOPE THAT HELPS EXPLAIN AND REVIEW HOW TO FIND THE EQUILIBRIUM POINT GIVEN THE SUPPLY AND DEMAND FUNCTIONS. THANK YOU.

## Formal definition

The point ${\tilde {\mathbf {x} }}\in \mathbb {R} ^{n}$ is an equilibrium point for the differential equation

${\frac {d\mathbf {x} }{dt}}=\mathbf {f} (t,\mathbf {x} )$ if $\mathbf {f} (t,{\tilde {\mathbf {x} }})=\mathbf {0}$ for all $t$ .

Similarly, the point ${\tilde {\mathbf {x} }}\in \mathbb {R} ^{n}$ is an equilibrium point (or fixed point) for the difference equation

${\textstyle \mathbf {x} _{k+1}=\mathbf {f} (k,\mathbf {x} _{k})}$ if $\mathbf {f} (k,{\tilde {\mathbf {x} }})={\tilde {\mathbf {x} }}$ for $k=0,1,2,\ldots$ .

Equilibria can be classified by looking at the signs of the eigenvalues of the linearization of the equations about the equilibria. That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances), by finding the eigenvector(s) associated with each eigenvalue.

An equilibrium point is hyperbolic if none of the eigenvalues have zero real part. If all eigenvalues have negative real parts, the point is stable. If at least one has a positive real part, the point is unstable. If at least one eigenvalue has negative real part and at least one has positive real part, the equilibrium is a saddle point and it is unstable. If all the eigenvalues are real and have the same sign the point is called a node.