The last several posts have been looking at the bifurcation diagram below in slices.

The first post looked at the simple part, corresponding to the horizontal axis r running from 0 to 3.

The next post looked at the first fork, for r between 3 and 3.4495.

The previous post looked at the period doubling region, for r between 3.4495 and 3.56995.

For r greater than about 3.56995 we enter the chaotic region. This post looks at an apparent island of stability inside the chaotic region. This island starts at r = 1 + √8 = 3.8284.

The whitespace indicates an absence of stable cycle points. There are unstable cycle points, infinitely many in fact, as we will soon see. For a small range of r there are three stable cycle points. When r = 3.83, for exampel, the three cycle points are 0.1561, 0.5047, and 0.9574.

This is significant because period three implies chaos. There is a remarkable theorem that says if a continuous map of a bounded interval to itself has a point with period 3, then it also has points with periods 4, 5, 6, … as well as an uncountable number of points with no period. If there are points with period 4, why can’t we see them? Because they’re unstable. You would have to land exactly on one of these points to go in a cycle. If you’re off by the tiniest amount, which you always are in computer arithmetic, you won’t stay in the cycle.

So even when we’re in the region with three stable points, things are still technically chaotic.

There seems to be something paradoxical about computer demonstrations of chaos. If you have extremely sensitive dependence on initial conditions, how can floating point operations, which are not exact, demonstrate chaos? I would say they can illustrate chaos rather than demonstrate chaos. And sometimes you can do computer calculations which do not have such sensitivity to show that other things are sensitive.

For example, I asserted above that there are points with period 4. Let f be the logistic map with r = 3.83

f(x) = 3.83 x(1 − x)

and define

p(x) = f(f(f(f(x)))),

i.e. four applications of f. Then you can see that there must be a point with period 4 because the graph of p(x) crosses the graph of x.

So while our region of whitespace appears to be empty except for three stable cycle points, there are infinitely many more cyclic points scattered like invisible dust in the region, a set of measure zero that we cannot see.

I recently ran into Recamán’s sequence. N. J. A. Sloane, the founder of the Online Encyclopedia of Integer Sequences calls Recamán’s sequence one of his favorites. It’s sequence A005132 in the OEIS.

This sequence starts at 0 and the nth number in the sequence is the result of moving forward or backward n steps from the previous number. You are allowed to move backward if the result is positive and a number you haven’t already visited. Otherwise you move forward.

Here’s Python code to generate the first N elements of the sequence.

    def recaman(N):
        a = [0]*N
        for n in range(1, N):
            proposal = a[n-1] - n
            if proposal > 0 and proposal not in set(a):
                a[n] = proposal
            else:
                a[n] = a[n-1] + n
        return a

For example, recaman(10) returns

    [0, 1, 3, 6, 2, 7, 13, 20, 12, 21]

There’s a Numberphile video that does two interesting things with the sequence: it visualizes the sequence and plays a representation of the sequence as music.

The rules for the visualization are that you connect consecutive elements of the sequence with circular arcs, alternating arcs above and below the numberline.

Here’s code to reproduce an image from the video.

    import numpy as np
    import matplotlib.pyplot as plt
    
    def draw_arc(a, b, n):
        c = (a + b)/2
        r = max(a, b) - c
        t = np.linspace(0, np.pi) * (-1)**(n+1)
        plt.plot(c + r*np.cos(t), r*np.sin(t), 'b')
           
    N = 62
    a = recaman(N)
    for n in range(N-1):
        draw_arc(a[n], a[n+1], n)
    plt.gca().set_aspect("equal")
    plt.show()

Here’s the output:

To create a music sequence, associate the number n with the nth note of the chromatic scale. You can listen to the music in the video; here’s sheet music made with Lilypond.

Update

Here is another Recamán visualization going further out in the sequence. I made a few aesthetic changes at the same time.

I’ve also made higher resolution versions of the images in this post. Here are SVG and PDF versions of the first visualization.

Here is a PDF of the sheet music.

And here are SVG and PDF versions of the new visualization.

Oh :/

ALT

A recent New Scientist cartoon 🍄 #fungi

The Illustrated Flora of Burkina Faso and Mali is the first comprehensive documentation of the remarkable plant diversity in these two west African countries.

Written in French, the book is the outcome of decades of botanical research and scientific collaboration between institutions and botanists from Burkina Faso, Mali, France, Switzerland and Germany. For the first time, it provides a complete inventory of ferns and flowering plants in Burkina Faso and Mali. It catalogues 2,631 species – both native and introduced – with 2,115 identified in Burkina Faso, 1,952 in Mali, and 1,453 shared between both countries.

Featuring over 800 photographs, 2,631 scientific illustrations, detailed descriptions, distribution maps, and identification keys, it serves as an essential tool for scientific research and biodiversity conservation. It’s also useful for sustainable development in the region.

We are a team of botanists from Burkina Faso, Mali and Europe who worked on this guide. One of our team is the botanist Jean César, who has carried out botanical research in the region for over 30 years. We based the guide on his earlier work in researching the flora of West Africa, and training young botanists.

The guide shows how diverse the climate of west Africa is. From the Sahara Desert to the Sahelian zone and the savannas and open forests of the Sudanian region.

By identifying plant species – whether common, rare, overexploited, or invasive – this guide can play a crucial role in conservation efforts: one can only protect what one knows.

The publication lays the groundwork for conservation of Sahelian ecosystems, which face increasing degradation with direct consequences for rural communities.

How we came up with the guide

As a team, we’ve conducted more than 40 years of research in Burkina Faso and Mali, documenting different plants. We also studied herbarium collections in Paris, Montpellier, Frankfurt and Geneva in Europe and Ouagadougou and Bobo-Dioulasso in Burkina Faso.

We drew from online resources such as African Plants – A Photo Guide and the African Plant Database. These compile comprehensive data on African plant biology, distribution and taxonomy (the science of classifying and naming plants).

The book is written in French and includes an index of local plant names in the local languages of Bambara, Dogon, Sonrai, Sénoufo and Peulh. This makes it a valuable resource for local communities and researchers alike. There is an open access digital version to make sure that everyone can use the new illustrated guide.

Discovering new and rare species

The book highlights species previously known from only a few observations. These are both widely distributed species and plants that are rare, only found in unprotected areas facing heavy urbanisation.

About 330 of the plant species in the guide have only ever been seen once in Burkina Faso or Mali, although some are present in neighbouring countries.

Another 40 near-endemic species (mainly only found in Burkina Faso and Mali) have only been seen once 40 years ago. Most of those are aquatic plants, growing along the Niger River, or in small wetland environments.

Additionally, this research updates information on more than a hundred poorly understood species that require further study. Some of these are likely new to science and have not even been given formal names. For instance, we found a new type of Brachystegia tree in the Geneva Botanical Garden’s herbarium. It is new to science and will have to be described.

Many plants documented here hold ethnobotanical value. They are part of the indigenous knowledge of Burkina Faso and Mali and play roles in traditional medicine, agriculture and crafts.

We found more than 120 species that have medicinal uses. Identifying them with correct scientific names will be crucial for the study of how people can continue to use these plants, especially as medicine.

Collaboration in difficult times

The hospitality of Sahelian countries has fostered numerous collaborations over the years under different projects.

Unfortunately, the current insecurity in the region has made field studies extremely dangerous, threatening conservation projects. For instance, forest rangers can no longer travel freely, and some regions have become inaccessible.

Publishing this book at such a difficult time brings renewed momentum to scientists and serves as a positive sign of continued collaboration. It gives visibility to botanical studies in both countries and highlights the importance of collaborations among botanists from different continents.

By recording this biodiversity, this work not only preserves valuable ecological knowledge but also ensures that the knowledge of these species is not lost to conflict-driven environmental degradation. It sheds light on the importance of preserving plants for future generations.

Cyrille Chatelain receives funding from the Swiss Agency for Development and Cooperation (SDC).

Adjima Thiombiano, Blandine Marie Ivette Nacoulma, and Mamadou Lamine Diarra do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.

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I settled this years ago and it's pronounced hyeef.