Mike Henderson
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mhender.bsky.social
Mike Henderson
@mhender.bsky.social
150 followers 200 following 520 posts
Retired Applied Mathematician. Computational Dynamical Systems.
Still trying to understand how things work.
https://multifario.sourceforge.io/henderson/

I might be wrong.
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mhender.bsky.social
These are out-takes from
www.worldscientific.com/doi/10.1142/...
A very weird surface. Points on it satisfy 3d ODE with bcs and four parameters. It can be embedded in 4d, but projected into 3d it crosses itself.

The colors have to do with a classification of the solutions - # loops and twists.
A complicated surface with portions colored differently. Symmetry planes are shown. A complicated surface with portions colored differently. Symmetry planes are shown. Mostly the same as the previous picture, but not all symmetries are shown and there are black lines showing the parameter value contours.
October 30, 2025 at 5:22 PM
mhender.bsky.social
This is the same algorithm (and code) as for the torus. The continuation is limited to the inside of a box, so you only see the yellow spheres until they cross that. The polyhedra around the spheres are more visible. They're used to find the boundary of the union of spherical balls.
October 27, 2025 at 1:42 PM
mhender.bsky.social
This is an animation of an algorithm for covering a torus with the projection of disks tangent to the surface. Red is the boundary, and the yellow contribute to the boundary. Blue ones are interior.

Each step picks a point on the boundary for a new disk. The boundary is a simple list.
October 26, 2025 at 7:17 PM
mhender.bsky.social
I 'm reading Hermann's "Geometry, Physics, and Systems", 1973. It's got a chapter on Thermo. I love this guy.

He doesn't like the (dU/dP)_V notation or mysticism (his word) about entropy.

But what's with the bibliography? All references are labelled 1? Later there are some 2's.
Geometry, Physics, and Systems Robert Hermann Department of Mathematics Rutgers University New Brunswick, New Jersey Marcel Dekker, Inc. New York 1973 Returning to the sumanifold phi given in the form (2.6),i.e., three of the thermodynamic variables expressed as functions of the other two, we see that conditions (21) and (2.3) lead to equations of the following dorrt: dU/dV - T dS/dV = P dU/dP = T dS/dP Similar relations are obtained bking the other pairs of thermodynamic variables independently Unfortunately, physicists and chemists have invented an awkward notation to hide this possibility of change of variable. For example, if (P,V) are independent variables, the partial derivatives of U(P,V) are denoted by some such symbol as (dU/dP)_V, which is read "the change in U with respect to P at constant V". This geometric framework is adequate to set up all of the "formal" material of thermodynamics, to be found in the textbooks, concerning the algebraic and differential relations between thermodynamic variables in equilibrium situations. This material is, in fact, rather limited in extent. Apparently, one of the emotional excesses of physicists in the 19th and early 20th centuries, was to overemphasize and make unduly mystical this material, particularly that part connected with "entropy." A reasonable approach towards entropy is to consider it simply the function that occurs in (3.1), in the role "conjugate" to T, relative to the 1-form theta.
October 1, 2025 at 6:19 PM
mhender.bsky.social
Aha. I probably shouldn't be doing this in public.

The phase constraint just is. We're talking about paths on the phase manifold, and not all paths are allowed? And for a path there's work done and a heat flow needed?

BTW This ⬇️isn't helping thank you.
A tortuous math sounding definition of thermodynamics Definition 1: A single-component substance is a quintet (D,T,P,mu,R), where (i) D, called the constitutive domain, is a nonvoid suset of (J)^+ times (m^3)^+; the first and second variables in D (usually denoted by e and v, respectively), are the specific internal energy and the specific volume, respectively; the elements of D are called states of the substance; (ii) T:D into (K)^+, the temperature, P:D into (Pa), the pressure, mu:D into (J), the chemical potential, the constitutive functions, are continuous. (iii) R, the regular constitutive domain, the subset of D on which the onstitutive functions are continuously differentiable and dT/de>0, dP/dv dT/de - dP/de dT/dv<0, holds is an open set dense in D.
September 16, 2025 at 4:44 PM
mhender.bsky.social
I'm used to conservation laws. Navier Stokes and so on. So div f=0 in spatial vars. Thermo eqs seem to instead be "Irrotational", or path independent, so curl f=0. And not in space, but in pairs of the phase variables? Related to the second partials being the same (d^2U/dSdV=d^2U/dV/dS).

Head hurts
Four equations for dU, dH, dF and dG (intternal energy, enthalpy, Helmholtz energy, Gibbs free energy), each in terms of phase variables T, P, V, S (temperature, pressure, volume, entropy) and two of the differentials (dT, dP, dV, dS). These say that there is a scalar function U whose integral along a path between two points in the (V,S) is independent of the path. I think. Definitions of the energies F=U-TS (Helmhotz energy) G=H-TS (Gibbs free energy) and H=U+PV (enthalpy)
September 15, 2025 at 5:50 PM
mhender.bsky.social
Did it again. Gotta watch out for black text on a transparent background.
Maxwell's relations (https://en.wikipedia.org/wiki/Maxwell_relations). Four relations between the first partial derivatives of T (temperature),P (pressure),V (volume) and S (entropy) along coordinate directions and the second partials of U (internal energy),H (enthalpy),F (Helmholtz free energy), and G (Gibbs free energy) with respect to T,P,V,S.
September 14, 2025 at 3:59 PM
mhender.bsky.social
Always wondered why they wrote thermodynamics equations as differentials. Of course Navier Stokes writes a temperature equation as a PDE, but ..

Am reading www.fys.ku.dk/~andresen/BA... and trying to get my head around work and heat flow being differential forms on the manifold of state (???!)
Maxwell's relations (https://en.wikipedia.org/wiki/Maxwell_relations). Four relations between the first partial derivatives of T (temperature),P (pressure),V (volume) and S (entropy) along coordinate directions and the second partials of U (internal energy),H (enthalpy),F (Helmholtz free energy), and G (Gibbs free energy) with respect to T,P,V,S.
September 14, 2025 at 3:52 PM
mhender.bsky.social
These are from the periodically forced damped pendulum, showing how points near the unstable fixed point (up) evolve. W/o forcing they'd spiral to the stable fixed point (down).
The surface is rendered as white with black polygons. Hidden line removal but still black and white.
A surface that rolls up. Black polygons on a white surface. The surface has wavy corrugations. Seen from off diagonal with time coming toward the viewer and right to left. A surface that rolls up. Black polygons on a white surface. The surface has wavy corrugations. Seen from aove the side with time moving from right to left. A surface that rolls up. Black polygons on a white surface. The surface has wavy corrugations. Seen from off diagonal with time coming toward the viewer and left to right. The hoops are the heterclinic connections in the undamped unforced pendulum. They connect the unstable (up) fixed point with the stable (down) fixed point, and are shown at the end of each period of the forcing.
August 31, 2025 at 5:47 PM
mhender.bsky.social
These are three very old (1987) figures showing the cusp, swallowtail and butterfly catastrophes and the complex sheets associated with them. These were done in CATIA. Yep. Not many choices back then.
A blue surface and a red surface, showing the cusp catastrophe. The projection on the horizontal is shown as a yellow curve (the cusp). The surfaces touch on the yellow curve. A blue surface and a red surface, showing the swallowtail catastrophe. The projection on the horizontal is shown as a yellow curve (the swallowtail). The surfaces touch on the yellow curve. A blue surface and a red surface, showing the cuspbutterfly catastrophe. The projection on the horizontal is shown as a yellow curve (the butterfly). The surfaces touch on the yellow curve.
August 31, 2025 at 5:25 PM
mhender.bsky.social
‪
Cubes about points on a shiny sphere. Rendered with POV-Ray.
#BlueSkyArtShow #Shiny
A shiny sphere with orange fins sticking out. All on a grey/white checkerboard pattern.
August 30, 2025 at 4:43 PM
mhender.bsky.social
And of course there's always the "famous" multifario demo:
multifario.sourceforge.io/henderson/mu...
August 27, 2025 at 2:40 AM
mhender.bsky.social
I used my admittedly limited JavaScript skills to make a vector field editor:
multifario.sourceforge.io/henderson/ve...
Changing each vector independently isn't the best, but it is what I set out to do. Uses a tensor product Bezier approx.

Next to find and mark the fixed points.
August 27, 2025 at 2:35 AM
mhender.bsky.social
I never really understood this picture -- it's part of the configurations of a axially clamped twisted rod. So like a garden hose held in two hands. The horizontal is how hard you have to pull, and the vertical plane is the torque you have to apply. The curves are various special configurations.
A red surface bounded by red, blue and yellow curves.
August 23, 2025 at 3:12 AM
mhender.bsky.social
Computing a stable invariant circle in a map. I write a system for a point on a circle, and it's 9 iterates under the map. Long vector. The solution is a curve in the large space. Plotting pairs of coordinates gives the ellipses. The black zig zag line is one solution of the large system.
A green circle and five blue ellipses with red normals. Labelled from 0 (green) to 5. Many blues ellipses, spread out in an orthogonal projection showing how the circle converges to the smallest ellipse. A zig zag black line shows how one points maps. A side view of the converging ellipses and the black zig zag path.
August 18, 2025 at 6:37 PM
mhender.bsky.social
We always used to call these locusts. I know they're cicadas and not grasshoppers, but that's what we called them.
August 17, 2025 at 9:44 PM
mhender.bsky.social
My first thought was "Wow". My second was "I really messed that up". But hey.
A very busy image. A blue sphere and a yellow ellipsoid-ish surface. Both have polygonal tilings with white edges. They also have short red normal vectors and the tile centers marked as white balls. The points on the yellow surface are well spaced, but those on the blue sphere cluster around a create circle, which shows up as a red blur because of the normals.
August 3, 2025 at 5:23 PM
mhender.bsky.social
Hello? What exactly are you?
A black beetle with white spots.
July 10, 2025 at 8:57 PM
mhender.bsky.social
I forgot to add the equations.
Three equations. The first says that the vector from the fixed point to the first iterate is normal to the unstable eigenspace (n). The second that the distance to the first iterate is less than R, and to the second is greater than R. The last equation is repeated, and say that the i+1 st iterate is the image of the i th under the map F.
July 8, 2025 at 1:11 PM
mhender.bsky.social
These are my first steps at computing invariant manifolds of maps. This is the stable manifold of the origin in discrete Lorenz. A "point" is five three-d pts, each the image of the other under the map, with the first in the stable eigenspace. The first pair straddles the small, thin red ellipse.
A blue oval, tiled y irregular polygons. A circular region is cut out of the enter, tiled in red polygons. Then a blue oval, then a white oval. Looks like an eye. An enlarged version of the previous image. The outer blue oval is almost enttrely off screen, and a smaller blue oval inside the red oval can be seen.
July 8, 2025 at 12:52 PM
mhender.bsky.social
If I'm using my library steps as extra bookshelf space, do I have a problem?
Brown wooden three step library steps, with books on them as extra shelf space
July 7, 2025 at 3:34 PM
mhender.bsky.social
I'm guessing that this takes way more coordination than I possess. The dog trots back and forth on the board. Looks slippery.

Ooo. Cool the way the reflections show on the waves.
A paddle board with a man and a dog. The dog is posed at the front of the board.
July 5, 2025 at 10:24 PM
mhender.bsky.social
I'm not going to try to explain these, but the gods of computation decided that they'd let my code work.

The curves/surfaces are points on all trajectories of a mapping (2d->2d and 3d->3d) starting on a manifold of initial conditions (a circle/sphere here).
A set of points on five curves with red normal vectors at the points. One curve is in green and is the first curve. The rest are iterates of the green curve and are in blue and black. The black is the fifth iterate. Four gray surfaces drawn using non-overlapping polygons. They begin on the left with a large sphere which is partially off the edge of the image. The surfaces are images of each other under a map, and get smaller and more sausage shaped from left to right. The equations for consecutive points on a boundary map trajectory that originate on a circle or sphere. The equations for a two dimensional map on the left and three dimensional on the right.
July 1, 2025 at 8:18 PM
mhender.bsky.social
Argh. I messed up the equations on the figure, and somehow managed to mess the alt text. I thought I'd got it set not to let me do that. Hmmm.
A bold curve which is a trajectory of a flow. The letmost piece between points u_a and u_b is labeled u_ab, and an curve offset from it in a direction phi_a is drawn. A second trajectory u_bc begins at u_b+e phi_b and ends at u_c. Equations and boundary conditions are written for the trajectory segments.
June 25, 2025 at 6:32 PM
mhender.bsky.social
Here's what I'm thinking about.

Invariant manifolds defined by smooth transverse sections are "easy". Those associated with a trajectory (like a periodic orbit) are collections of trajectories that approach or diverge from a curve tangent to the flow.
June 25, 2025 at 6:08 PM