From first sight, it might seem to be far away from your question but it can be explained on a simple equation mathematically. We all know that there are existing equations of the type

X_(n+1) = r * X_(n) * (1 - X_(n))

When the parameter 'r' is changed continuously, the equation expresses sudden changes in its behavior. As the parameter 'r' increases doubling of a period of present cycles occurs until the system enters the chaotic regime.

This simple nonlinear equation was originally developed to describe the evolution of populations with the scarcity of resources present in the case of overpopulation.

Just by tuning some parameters, the system can express abrupt changes in its behavior.

Another rich source of self-organization of biological systems that Prigogine surely had known is the periodic chemical system of the type of Belousov-Zhabotinski reaction. Basically, it is an example of how chemical oscillators arise from chaotic reactions.

Another very important source of his insights is the concept of entropy that has wide applications in chemistry, biochemistry, and biology. Basically, entropy serves as a driving force that leads to the preferential reaching of certain configurations and states.

Ask more when you want to go deeper.

References:

(1) https://en.m.wikipedia.org/wiki/Logistic_map

(2) https://en.m.wikipedia.org/wiki/Belousov%E2%80%93Zhabotinsky_reaction

Frpm what Karl Friston is writing about, the life is actually striving against the second law of termodynamics, hence self-organization, never in equilibrium, chaotic outcomes (deterministic chaos) and so forth. It is like we used dorations which are simplifying the reality, think of euclidean geometry versus fractals. It is how I understand Prigogin's seminal work

Sorry typo, instead of dorations, equations...

Thank you Jiri and Milena; I agree with what you say . Yes, I know that certain nonlinear equations can give rise to chaotic behavior as their parameter values change. But these are changes in state, affecting the behavior but not the structure of the equations. Prigogine uses examples of nonlinear equations moving among attractors (strange attractors or not) to talk about structural changes, changes in organization, self-organization. And this jump is what I do not follow. Structural change is understood as a change in the equations themselves, not merely in the values of its variables or parameters. Can you help me with that?

Dear Gilberto, the best that can be done with your question is to explain the equation you are speaking about.

Equations lead to solutions, some of them can have the form of (strange) attractors. I am afraid that equations are not moving along solutions. Equations are generating solutions. Please specify your question in more detail.

I would like to stress out that since the work of Prigogine we discovered the importance of complex systems that are not based on the concept of equations at all. They work with large numbers of simple, mutually interacting entities, that are communicating locally.

Self-organization, self-repair, self-replication, and emergence are the central terms and processes related to those fascinating systems. Those are the core of my own research and popularization.

When you want to understand self-organization and emergence in complexity, you can read the poster and review paper from my research outputs. My guess is that you will understand the core of this problem more easily.

Do not hesitate to ask, we are here to help each other. Complexity is offering unbelievably effective tools to the mathematical description of naturally observed phenomena :-)

Comments on the last answer. Those review papers explain complexity in biology and medicine. This is close to your own research field.

What I found out in my own research very useful is to enter a new field, in this case, mathematical formalisms, in the area of my own, active research.

We can try to find out some books together that explain ecology using complex systems. Luckily, my personal knowledge in this area is not zero. Hence, we can start somehow effectively to dive into your field using this type of mathematics.

Dear Jiri: I apologize for being out of reach for a time; I had a minor surgery and could not use my right arm until now.

Yes, of course equations do not move along solutions; it was a sloppy statement from my part. I tried to say that the values of state variables can move between attractors, sometimes abruptly.

I am also aware of the use of agent-based models (cellular automata) in the study of complexity and self-organization.

My problem is not with any particular equation, but is of a more conceptual nature.

What I am trying to find out (in an attempt to apply the theory of dissipative systems to socio-ecological systems), is whether Prigogine implied that, similarly to the changes in the state of the (dynamical) system, a dissipative system could also exhibit bifurcations and sudden shifts from one structure to another.

My problem is that he explicitly refers to structural changes, but illustrates with changes in state.

Besides Prigogine: is it legitimate to extend the concepts of bifurcation and abrupt change to structures (I am using “structure” as the set of equations that describe the dynamics of a system, and “state” as the set of values of the variables of the systems)

And thanks for your offer to explore together the application of complexity theory to ecology. I will follow up on this.

Before trying to understand Ilya Prigogine and his work on self-organization, we must turn toward his forming experiences & knowledge. A huge part of it is associated with entropy. It is very important in chemistry.

Entropy is what you are looking for without realizing it. I wrote recently a section of the paper, not available online yet, that explains in detail the very principles of entropy to newcomers.

Despite having two courses on thermodynamics, bits, and pieces of statistical physics (just my own interests), dynamic systems & chaos (self-study), knowing complexity and a large number of entropy measures used there, it took me one month to study the topic in-depth and to distill just about for or five pages of explanation.

At this moment, I have to tell you just one sentence: "The notion of entropy is the key to self-organization & emergence and other processes related to life!"

The notion of entropy does not allow us to explain all that is associated with life but it will dramatically boost our capabilities to mathematically describe life.

It takes time & quite a huge effort to tunnel through all of it but after this painful process, everyone becomes the master of entropy, self-organization, and emergence.

Thank you Jiri. I agree on the importance of entropy to understand life. While I cannot claim a deep understanding like yours,, I read Schrödinger about negentropy, and what I could follow from Prigigine. Entropy has been used a lot in theoretical ecology. I hope your paper will be available soon.

But now I limit my need for an answer to this basic question (it may be a question also for a mathematician):

"Is it legitimate to extend the concepts of bifurcation, abrupt change, and alternative attractors used to describe changes in the state of a system, to changes in their structure?" (I am using “structure” as the set of equations that describe the dynamics of a system, and “state” as the set of values adopted by the variables of the systes)

To restate your question. Are you asking whether is it possible to study changes of equations generating attractors? The answer is yes. There exist a branch of mathematics called functional analysis, which deals exactly with such operations.

The problems are that we have none or just a few explicit solutions describing attractors. The vast majority of such systems can be studied only numerically.

In this sense, we are on grounds of experimental mathematics and not a theoretical one. There are some ways to overcome such hurdles. Nevertheless, I do not know anyone who is working in this direction.

Thanks Jiri: one last question:

Some people speak of "stability landscapes" to denote a 3D representation of the state space of a nonlinear equation, representing the attractors as basins on a flat surface. This surface (the state space) represents of course all posible solutions of the equation.

Now, is it correct to assume that a structural change in the equation mayresult in a change of the landscape itself, and conversely that a change in the stability landscape shows that a structural change has taken place?

I see from your second paragraph where you aim. You have some equation and this defines its state space. The problem is that given state space can be produced by not only equations, it can be for example some kind of massively parallel environment.

For smooth equations, it is quite probable that they have unique 'landscapes' in state-space -- which is N-dimensional in general.

PDEs can be viewed as large ensembles of mutually linked ODEs as can be seen from their numerical discretizations.

Only under some conditions -- probably smoothness of coefficients in equations & their nice behavior can be reflected in the uniquenesses of the relevant state-space.

Going in the opposite direction, from state space to the equation is even more tricky. There are some counterexamples that intuitively tell us that we must be very cautious with one to one relationship between equation and it's 'landscape' in the state space.