Chimeras-States in the Distributed Systems in the Case of Multi-Valued

Solutions. To the Statement of Possible Research Problems

ALEXANDER MAKARENKO

Institute of Applied System Analysis

National Technical University of Ukraine

―Kiev Politechnic Institute‖

Prospect Peremogy 37, building 35, 03056, Kyiv-56,

UKRAINE

Abstract: - Nonlinear science has about a hundred years of rapid development. One of the new directions is the

study of chimera solutions, i.e. solutions that exhibit significantly different solution behaviour in different

regions of space or time. Just a few examples of such behavior have been studied in different systems.

However, all examples of chimeras have been single-valued solutions. Recently, however, many nonlinear

models of physical and social phenomena have multivalued solutions. It is conceivable that one of the possible

directions for further research in chimera science is to allow multivalued solutions to be considered.

The paper is devoted to describing some possible research settings in this area. Some modifications of existing

models are also proposed. The interesting class of such objects is the models with strong anticipation (models

with advanced arguments). Furthermore, some problems of computational theory and mind with such models

are discussed.

Keywords: - Chimeras, distributed systems, multi-valuednes strong anticipation, computation theory

Received: January 2, 2023. Revised: October 11, 2023. Accepted: November 14, 2023. Published: December 11, 2023.

1 Introduction

Complexity and synchronization phenomena in

technical and natural systems have recently become

one of the key topics for investigation in

philosophy, physics, humanities, and biological

sciences. The reason is the globalization of

processes, hierarchical structures with many levels,

interconnections between elements, existence of

many sub-processes with different time and space

scales.

Synchronization theory has a history of about two

centuries and a more or less well-defined list of

topics [1-4]. Usually, the objects of synchronization

research are collections of elements with internal

dynamics, with a set of boundaries between

elements and types of behavior in such systems.

The recent development of theory and practice in

the field of synchronization has many different

ways. Usually, the nature of a specific field of

investigation follows the specific new research

problems and new objects. One of the most recent

examples is

One of the new, rapidly developing

directions in nonlinear science is the study of so-

called chimeras as solutions of nonlinear equations.

Roughly speaking, these are solutions of distributed

systems with coherent and incoherent behavior of

solutions in different regions of space. This behavior

of solutions was first discovered in computer

experiments and described in the paper [5]. The

actual name of such solutions, "chimera", appeared

a little later in [6]. Since then, the flow of

publications on this topic has been steadily

increasing, e.g. in Physical Review, Nonlinearity,

Chaos, etc., as well as in the proceedings of many

conferences. The current state of the art is reflected

in the papers of the Dresden conference on chimeras

in May 2022 [7].

Much of the research on chimeras has been

based on the study of initial equations and various

aspects of their behavior. In addition, new sources

of chimera solutions have been explored, especially

in the context of new applications. Some (far from

complete) insights into new applications may be

proposed- for example in neuroscience [8-10]. One

of the research directions has also been the

transition from one-dimensional (1D) to two-

dimensional (2D) and then to three-dimensional

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E-ISSN: 2732-9976

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(3D) systems [11]. We should also mention the

studies on the system of oscillators, taking into

account the inertia [12]. Note that for further studies

of chimeras in systems with inertia, studies on

hydrodynamics with memory can be useful [13]. In

this case, the 'bursts' in the flows can be considered

as the chimeras.

In all cases of studies of chimeras so far, objects

with single-valued solutions (o.d.e., chains of

mappings, chains of oscillators, etc.) have been

considered. This is while that most models for many

different processes have only ONE-valued solutions.

However, it is now becoming increasingly clear that

one should also consider cases where model

solutions for processes have MULTIPLE-valued

solutions. Without giving a special survey here, let

us mention some areas where the study of

multivaluedness is important: mechanical systems

with friction, the Everett interpretation in quantum

mechanics, underdetermined systems, controlled

systems, some models of hydrodynamics, etc. Note

also the possibility of multivaluedness in systems

with strong anticipation - called hyperincursion [14,

15].

Much of the research on chimeras was based on the

study of initial equations and various aspects of their

behavior. In addition, new sources of chimera

solutions have been explored, especially in the

context of new applications. Some insights (far from

complete) into new applications - for example in

neuroscience [8-10]. One of the research directions

has also been the transition from one-dimensional

(1D) to two-dimensional (2D) and then to three-

dimensional (3D) systems [11]. We should also

mention the studies on the system of oscillators,

taking into account the inertia [12]. Note that for

further studies of chimeras in systems with inertia,

studies on hydrodynamics with memory can be

useful [13]. In this case, the 'bursts' in the flows can

be considered as the chimeras.

In all cases of studies of chimeras so far, objects

with single-valued solutions (o.d.e., chains of

mappings, chains of oscillators, etc.) have been

considered. This is because the models for many

different processes have only ONE-valued solutions.

However, it is now becoming increasingly clear that

one should also consider cases where model

solutions for processes have MULTIPLE-valued

solutions. Without giving here a special review, let

us mention some areas where the study of

multivaluedness is important: mechanical systems

with friction, Everett interpretation in quantum

mechanics, underdetermined systems, controlled

systems, some models of hydrodynamics, etc. Also

note the possibility of multiple-valuedness in

systems with strong anticipation (anticipation) -

called hyperincursion [14, 15].

2. General Distributed Media With

Anticipation

Many more models can be proposed for the case of

considering continuous media with some kind of

anticipation. The simplest variant is when only the

nonlinear source

)(uf

in the equations of such

media (where

),( txu

is field value) has anticipatory

property, i.e.

)),(),,((



 txutxuff

in the simplest

case or











dtxKxuff )),(),,((

in more

complex case. An example of such a model is the

parabolic diffusion equation with such a non-linear

source. As an example of such a system, we can

propose models with continuous variables of

neuronal activation fields (counterparts of the well-

known models of Amari or H. Wilson & J. Cowan).

Note that much more complex and developed

models of such phenomena are the counterparts of

the well-known [13] rigorous models of media with

non-locality and memory in theoretical physics-

Note for illustration that in the case of discrete-time

equations with strong anticipation, they have the

form: (D. Dubois [14]) "The definition of a discrete

system with strong anticipation: it is a system which

calculates the current state at time t as a function of

past states,

1,2,3  ttt

, the present state and the

state in the future

,3,2,1  ttt

( 1) ( , ( 2), ( 1),

( ), ( 1), ( 2), )

x t f x t x t

x t x t x t

   



, (1)

where the variation in future time is computed

indirectly from the equation.

3. Towards 'Chimera' States In

Anticipatory Systems

Another promising topic for investigation in the

field of synchronization is the analog of 'chimera'

states in the case of anticipating systems. Recall that

'chimera' states are highly inhomogeneous transitive

solutions when different types of behavior coexist in

space [5-12].

The 'chimera' states are the solutions in chains of

elements or in distributed media that have different

behavior in different domains of space. For

example, such systems may have coexisting

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domains of chaotic and 'smooth' behavior at

different locations in space. One of the discrete

systems with putative 'chimeras' is the following [5-

12]







 Pi

Pij

izfzf

zfz )]()([

)(



(2)

The next example is the chain of oscillators:

() 1

sin[ ( ) ( ) ]

j k R

dt R



  









(3)

The next general examples with 'chimera' states are

the following: It is accepted that the main source for

the existence of 'chimera' states is the non-locality in

the equations. Note that the nonlocality described in

[5] essentially extends the cases with the presumed

origin of 'chimeras'. The possibilities of 'chimeras' in

systems with anticipation are new [15]. As

examples, we should mention the possibilities of

'multivalued chimeras', the coexistence of different

'chimeras' on different branches of multivalued

solutions, a coexistence of 'chimeras', and 'smooth'

behavior in different branches of the solution. As

one of the presumed basic systems with anticipation

for the study of 'chimeras', we can propose the next

one:

(1 )[ ( )

[ ( )

( )]]

[ ( ) 2

[ ( ) ( )]]

j i P t

ij i P

z f z

Pfz

fz P

f z f z

















  









(4)

The next example is the chains of anticipatory

oscillators:

() (1 )

sin[ ( ) ( ) ]

j k R

dt R





  



    













  



   



(5)

Note that more general examples with 'chimera' states

can also be proposed.

4. Possible Problems With

Chimeras In The Multivalued Case

As mentioned above, an example of systems with

multivalued solutions are systems with strong

anticipation. Therefore, here we will illustrate some

of the possible problem formulations in the field of

chimera research. Such systems have been studied

previously for models of cellular automata [16],

neural networks [17], and discrete dynamical

systems [18]. The figure below shows a schematic

case of multivalued solutions in the one-dimensional

case in space. On the horizontal axis the discrete

time moments are plotted (for discrete dynamical

systems). On the vertical axis, the states of

considered system X at discrete time steps are

conditionally plotted. For example, for a system of

N-connected one-dimensional elements, a point

corresponds to an N-dimensional vector. For

cellular automata, a values corresponds to the so-

called configuration of the automaton at given

moment of time (configuration is the representation

of states of all cells of cellular automata).

FIGURE 1: Plurality of system states during

evolution

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The thin lines correspond to the branches of the

multi-valued solution. The thick line corresponds to

the selection of a single-valued solution. For

systems described by differential equations, the

point can correspond to an infinite dimensional

object.

Because of the possible multivaluedness of the

solutions, we can propose the new problem

formulations in the field of chimera research.

1. Computational studies of models with

multivalued solutions, where the appearance of one-

dimensional analog can be expected. It should be

assumed that such studies are the first necessary

steps in the study of chimeras in the multivalued

case (let us call them multivalued chimeras). Even

in this case, many difficult problems arise, even for

one-dimensional systems, and, for example, in the

visualization of computational results.

Theoretically, there is the problem of a strict

definition of multivalued chimeras. This also raises

the problem of introducing indices that can be used

to check multivalued solutions for the presence of

chimeras. The inverse problem may also be of

interest, i.e. finding the models and initial conditions

that can generate chimeras with specific

deactivation properties.

3. The next set of problems arises in the study of the

peculiarities of multi-valued solutions, namely the

existence of different branches of the solution. One

of the first interesting problems is whether different

chimeras can occur on different branches of a

multivalued solution. For example, this leads to the

question of can the chimeras exist on some branches

and not on others.

4. The possible multivaluedness of chimeras leads to

unexpected problems at the interface of nonlinear

science and computational theory. Namely, can

chimeras be used in computational theory to

emulate logical operations, and independently on

different branches?

5. Conclusion

Thus, in the given paper we propose to consider the

problem of complex behavior and synchronization

for a new class of objects, namely for chains and

networks with presumed multivaluedness (e.g. for

systems with strong anticipation). The assumed

multivaluedness of the solutions leads to new

interesting properties within the framework of

existing concepts. However, new properties may

also emerge (e.g. non-heterogeneous multivalued

synchronization) which are very promising for

further research. There is also a need for

applications of multivalued analogs of groups,

geometries, and symmetries of multivalued objects,

including multivalued flows and semi-flows, and

methods of operator theory. Also new may be the

consideration of blow-up solutions in multivalued

cases, including blow-up of selected branches of the

solution.

We have described only the first results of

investigations and only some new presumed forms

of research problems. However, the obvious

mathematical novelty of the proposed problems and

the probable great importance in applications (e.g. in

social systems, computation, signal processing

theory, consciousness research, etc.) lead to the need

for further development of the investigations.

Acknowledgements.

The author thanks Yu. Maistrenko for the detailed

description of the investigations in the field of

'chimera' states.

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Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

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Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

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