## How to Create Bifurcation Theory Catastrophe Curves in XL

Nine Parts:Helpful GuidanceThe TutorialCreate the Elliptic Umbilic Catastrophe Curve ChartCreate the Hyperbolic umbilic catastrophe worksheetCreate the Hyperbolic Umbilic Catastrophe ChartCreate the Parabolic umbilic catastrophe worksheetCreate the Parabolic Umbilic Catastrophe ChartRelated CurvesFinal CurveCommunity Q&A

You will learn to create catastrophe curves, which come from a branch of Bifurcation Theory in math. These are great curves to have in your portfolio in the realm of graphics.

Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes, analyzing how equation solutions depends on the parameters that appear in the equation. In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamics. Catastrophe theory originated with the work of the French mathematician Rene Thom in the ’60s, and became very popular via the efforts of Christopher Zeeman in the ’70s. It considers the special case where the long-run stable equilibrium can be identified with the minimum of a smooth, well-defined potential function. Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined geometrical structures.

Umbilic catastrophes are examples of corank 2 catastrophes. They can be observed in optics in the focal surfaces created by light reflecting off a surface in three dimensions and are intimately connected with the geometry of nearly spherical surfaces. Thom proposed that the elliptical umbilic modeled the creation of hair like structures. Which is why they became so interesting in the study of pupils and iris muscle fibers – they’re related to hair-like structures and also to focal surfaces which are nearly spherical; see the articles Model Your Own Iris in Excel and Improve Your Iris Model.

Part 1 Helpful Guidance 1 Make use of helper articles when proceeding through this tutorial: See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation. For more art charts and graphs, you might also want to click on Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category:Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page. Part 2 The Tutorial 1 Start by opening a new workbook in Excel from the desktop, from the dock, or from within your Applications folder inside the Microsoft folder. Double click on Excel (either the green X on the dock or the app title in the folder) and select File New Workbook. 2 In Preferences, set R1C1 to unchecked or Off, set Ribbon to checked or On and set Show Formula Bar to checked or On. 3 Click in the far upper left top corner above the 1 of row 1 and to the left of column A. Doing so will select the entire worksheet. Format Cells Number Number to decimal places 2, show comma. Format Cells Alignment Center. Title the first worksheet, “Elliptic umbilic catastrophe”. 4 Enter the Row 1 Column Headers: Enter to cell C1, 1; Enter to cell D1, 2; Enter to cell E1, 3; Enter to cell K1, Chart: V’X = ShrinkExpand_2*V+X_Offset_D; Enter to cell L1, Chart: V’Y = ShrinkExpand_1*V+Y_Offset_C; Enter to cell O1, .5; Enter to cell P1, -.5; Enter to cell Q1, .1250; Format Cells O1:Q1 Number Number Decimal Places 4; 5 Enter the Row 2 Column Headers: Enter to cell A2, x; Enter to cell B2, y; Enter to cell C2, a; Enter to cell D2, b; Enter to cell E2, c; Enter to cell F2, (x^3)/3; Enter to cell G2, x*y^2; Enter to cell H2, a*(x^2+y^2); Enter to cell I2, b*x; Enter to cell J2, c*y; Enter to cell K2, Formula: V=(x^3)/3-xy^2+a*(x^2+y^2)+bx+cy; Enter to cell L2, Formula: V=(x^3)/3-xy^2+a*(x^2+y^2)+bx+cy; Copy cell range A2:J2 and Paste it to cell range M2:V2; Select cell range A2:K2 and Format Cells Font Bold, Red; Select cell range L2:V2 and Format Cells Font Bold, Dark Blue; Enter to cell W2, ShrinkExpand_1; Enter to cell X2, .1; Enter to cell W3, X_Offset_C; Enter to cell X3, 2; Enter to cell W4, Y_Offset_C; Enter to cell X4, 2; Enter to cell W5, ShrinkExpand_2; Enter to cell X5, 2; Enter to cell W6, X_Offset_D; Enter to cell X6, 6; Enter to cell W7, Y_Offset_D; Enter to cell X7, 2 Select cell range W2:X7 and Insert Name Create Names in Left Column, OK; Format Cells Border Red Boldest Outline; Select Cell Range W2:X4 and Format Cells Border Red Boldest Outline; Select cell range X2:X7 and Format Cells Fill Yellow; (for input changes) Enter to cell Y2, Y-COMPONENT and Format Cells Font Bold, Color Dark Blue; Copy cell Y2 to cell range Y3:Y4; Enter to cell Y5, X-COMPONENT and Format Cells Font Bold, Color Red; Copy cell Y5 to cell range Y6:Y7; Copy cell X7 to cell X8, enter 50; Format cell Border Red Boldest Outline, Font Bold; Enter to cell Y8, CommonDivisor and select cell range X8:Y8 and Insert Name Create in Right Column, OK; 6 Enter Columnar Formulas: Enter to cell A3 the formula, =COS((ROW()-2)*PI()/180); Enter to cell B3 the formula, =SIN((ROW()-2)*PI()/180); Enter to cell C3 the value, 1; Enter to cell D3 the value, 2; Enter to cell E3 the value, 3; Enter to F3 the formula, =(A3^3)/3; Enter to G3 the formula, =A3*B3^2; Enter to H3 the formula, =C3*(A3^2+B3^2); Enter to I3 the formula, =D3*A3; Enter to J3 the formula, =E3*B3; Enter to K3 the formula, =ShrinkExpand_2*(F3-G3+H3+I3+J3)+X_Offset_D; Enter to L3 the formula, =ShrinkExpand_1*(R3-S3+T3+U3+V3)+Y_Offset_C; Copy cell range A3:J3 and Paste it to M3:V3; Enter to O3 the value, .5; Enter to P3 the value, -.5; Enter to Q3 the value, .125; Select cell range O3:Q3 and Format Cells Fill Yellow, Font Red, Bold; (for input) Select columns A:Y and Do Format Column AutoFit Selection; Copy A3:V3 and Paste it to A4; Enter to cell C4 the formula, =C3; Enter to cell D4 the formula, =D3; Enter to cell E4 the formula, =E3; Enter to cell O4 the formula, =O3; Enter to cell P4 the formula, =P3; Enter to cell Q4 the formula, =Q3; Format Cells O4:Q4 Fill Light Blue; Do menu item Window Split at cell A30; With the cursor in cell A31, Edit Go To cell A363; Select cell range A363:V4 and Edit Fill Down. Part 3 Create the Elliptic Umbilic Catastrophe Curve Chart

1 Select cell range K3:L363 and either Insert Chart or do Chart Wizard or do the Ribbon Charts and Select All, Scatter, Smooth Lined Scatter. A chart will appear atop your data which you should move to the right, beside the data. Click inside the chart and Do Chart Layout and do Chart Title. Title it “Elliptic Umbilic Catastophe Curve”. Double click in the Plot Area and format fill sand yellow; double click in the outer chart area and Format Fill orange; click on the data series and then line and do Line Weight 3 pt, Gradient Rust Brown 0%, Dark Purple 74%, Medium Purple 100%. Your chart should resemble this one: 2 Create a new worksheet by tapping on the “+” tab at the bottom of the worksheets, to the right. Title the new worksheet, “Saves”. 3 Select the “Elliptic umbilic catastrophe” worksheet and select cell range A1:Y8, and select the new Saves worksheet and Paste the formulas into cell A2, then select the Elliptic umbilic catastrophe worksheet and select cell range A1:Y8, and select the new Saves worksheet and Paste the formulas but then do Paste Special Values into cell A12, which will be a fast way of doing the cell formats too (although there may be a faster way). Type Formulas: into cell A1 and type Values: into cell A11. Select the “Elliptic umbilic catastrophe” worksheet and select the new chart and with the Shift Key depressed, do Edit Copy Picture, then select the Saves worksheet and to the right or bottom of your data, depress the Shift Key again and do Edit Paste Picture. Part 4 Create the Hyperbolic umbilic catastrophe worksheet 1 Select the “Elliptic umbilic catastrophe” worksheet and select cell range A1:Y8 and copy it. Tap on the new worksheet “+” tab button and create a new worksheet; Paste to cell A1 the contents of the clipboard. Name the new worksheet, “Hyperbolic umbilic catastrophe”, w/o the quotes. Do Format Column AutoFit Selection; 2 Change Column Headings: Enter to cell F1, CHGD, and Format Cells Alignment Center; Copy cell F1 and paste it to cells G1:H1 and cell range R1:T1; Enter to cell F2, x^3; Enter to cell G2, y^3; Enter to cell H2, a*x*y; Enter to cell K2: Formula: V=x^3+y^3+a*x*y+b*x+c*y and copy it and select cell L2 and Paste Special Values; Select cell range F2:H2 and copy it and select cell R2 and Paste Special Values; 3 Change the Formulas: Enter to cell F3 the formula, =(A3^3); Enter to cell G3 the formula, =(B3^3); Enter to cell H3 the formula, =C3*A3*B3; Copy cell range F3:G3 and paste to cell range F4:H363; And paste to cell range R3:T363. Part 5 Create the Hyperbolic Umbilic Catastrophe Chart

1 Select cell range K3:L363 and either Insert Chart or do Chart Wizard or do the Ribbon Charts and Select All, Scatter, Smooth Lined Scatter. A chart will appear atop your data which you should move to the right, beside the data. Click inside the chart and Do Chart Layout and do Chart Title. Title it “Hyperbolic Umbilic Catastophe Curve”. Double click in the Plot Area and format fill sky blue; double click in the outer chart area and Format Fill sky blue. Your chart should resemble this one: 2 Select the “Hyperbolic umbilic catastrophe” worksheet and select cell range A1:Y8, and select the new Saves worksheet and Paste the formulas into cell A25, then select the Elliptic umbilic catastrophe worksheet and select cell range A1:Y8, and select the new Saves worksheet and Paste the formulas but then do Paste Special Values into cell A35, which will be a fast way of doing the cell formats too (although there may be a faster way). Type Formulas: into cell A24 and type Values: into cell A34. Select the “Hyperbolic umbilic catastrophe” worksheet and select the new chart and with the Shift Key depressed, do Edit Copy Picture, then select the Saves worksheet and to the right or bottom of your data, depress the Shift Key again and do Edit Paste Picture. Part 6 Create the Parabolic umbilic catastrophe worksheet 1 Select the “Elliptic umbilic catastrophe” worksheet and select cell range A1:Y8 and copy it. Tap on the new worksheet “+” tab button and create a new worksheet; Paste to cell A1 the contents of the clipboard. Name the new worksheet, “Parabolic umbilic catastrophe”, w/o the quotes. At column F, do Insert Column. Also Insert Column to the right of c*y, and on the right side, to the right of c and c*y again; to the right of each c, enter d, and to the right of each c*y, enter d*y; Do Format Column AutoFit Selection; 2 Change Column Headings: Enter to cell F1, CHGD, and Format Cells Alignment Center; Copy cell F1 and paste it to cells G1:L1 and paste special to cell range T1:Z1; Enter to cell F2, d; Enter to cell G2, x^2*y; Enter to cell H2, y^4; Enter to cell I2, a*x^2; Enter to cell J2, b*y^2; Enter to cell K2, c*x; Enter to cell L2, d*y; Select cell range F2:L2 and copy it and select cell T2 and Paste Special Values; 3 Change the Formulas: Enter to cell F3 the formula, 4; Enter to cell G3 the formula, =(A3^2)*B3; Enter to cell H3 the formula, =B3^4; Enter to cell I3 the formula, =C3*A3^2; Enter to cell J3 the formula, =D3*B3^2; Enter to cell K3 the formula, =E3*A3; Enter to cell L3 the formula, =F3*B3; Copy cell range A3:L3 and paste to cell range A4:L363; Copy cell range G3:L3 and paste it to U3; Select cell range U3:Z363 and Edit Fill Down; Copy cell range A3:B363 and paste it to O3:P363; Enter to cell Q3, .5; Enter to cell R3, -.5; Enter to cell S3, .125; Enter to cell T3, -.125; Enter to cell Q4 the formula, =Q3; Enter to cell R4 the formula, =R3; Enter to cell S4 the formula, =S3; Enter to cell T4 the formula, =T3; Select cell range Q4:T4 and Format Cells Fill sky blue and Select cell range Q4:T363 and Edit Fill Down; Enter to cell M3 the formula, =ShrinkExpand_2*(G3+H3+I3+J3+K3+L3)+X_Offset_D; Enter to cell N3 the formula, =ShrinkExpand_1*(U3+V3+W3+X3+Y3+Z3)+Y_Offset_C; Select M3:N363 and Edit Fill Down. Part 7 Create the Parabolic Umbilic Catastrophe Chart

1 Select cell range M3:N363 and either Insert Chart or do Chart Wizard or do the Ribbon Charts and Select All, Scatter, Smooth Lined Scatter. A chart will appear atop your data which you should move to the right, beside the data. Click inside the chart and Do Chart Layout and do Chart Title. Title it “Parabolic Umbilic Catastophe Curve”. Double click in the Plot Area and format fill red; double click in the outer chart area and Format Fill canary yellow. Your chart should resemble this one: 2 Select the “Parabolic umbilic catastrophe” worksheet and select cell range A1:Y8, and select the new Saves worksheet and Paste the formulas into cell A50, then select the Parabolic umbilic catastrophe worksheet and select cell range A1:Y8, and select the new Saves worksheet and Paste the formulas but then do Paste Special Values into cell A55, which will be a fast way of doing the cell formats too (although there may be a faster way). Type Formulas: into cell A49 and type Values: into cell A54. Select the “Parabolic umbilic catastrophe” worksheet and select the new chart and with the Shift Key depressed, do Edit Copy Picture, then select the Saves worksheet and to the right or bottom of your data, depress the Shift Key again and do Edit Paste Picture. Part 8 Related Curves 1 Here are the settings and details and other charts you can create with similar / related formulations: Formula: V=(x^3)/3-xy^2+a*(x^2+y^2)+bx+cy; a,b,c on left = 1,2,3 and on the right a,b,c = .5,-.5,.125;

Formula: V=(x)/3-xy+a*(x^2+y^2)+bx+cy; a,b,c on left = 1,2,3 and on the right a,b,c = .24,.2398,30; ShrinkExpandA=2, X_Offset_A=0, Y_Offset_A=2, ShrinkExpandB=2, X_Offset_B=0, Y_Offset_B=6;

Formula: V=(x)/3-xy+a*(x^2+y^2)+bx+cy; a,b,c on left = 1,2,3 and on the right a,b,c = .24,.2398,30; ShrinkExpandA=2, X_Offset_A=0, Y_Offset_A=0, ShrinkExpandB=2, X_Offset_B=0, Y_Offset_B=0;

Butterfly catastrophe: Formula: V=x^6+ax^4+bx^3+cx^2+dx; a,b,c,d on left = 1.5, -2, .5, 2 and on the right a,b,c,d = .5, -.5, .125, -.01; ShrinkExpand_1=.1, X_Offset_C=2, Y_Offset_C=2, ShrinkExpand_2=2, X_Offset_D=6, Y_Offset_D=2; y on left = column L and y on right = column M;

Part 9 Final Curve

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The topic is actually more involved than what is shown here and it would be a good idea to do further research to gain a better understanding of Bifurcation Theory and Catastrophe Theory, as well as Chaos Theory and Lie Groups. For example, take the following regarding elementary catastrophes: Catastrophe theory analyses degenerate critical points of the potential function — points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the germs of the catastrophe geometries. When the degenerate points are not merely accidental, but are structurally stable, the degenerate points exist as organizing centers for particular geometric structures of lower degeneracy, with critical features in the parameter space around them. If the potential function depends on two or fewer active variables, and 5 or fewer active parameters, then there are only 11 generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed.