1. Introduction
Uninformed (blind) search strategies present a wide class of search problems where no information about the search domain is available [
1]. Those strategies provide basic search techniques to explore problems in which additional knowledge cannot be obtained beyond the definition of the problem. Blind search algorithms solve a wide range of problems in artificial intelligence, such as pathfinding, puzzle solving, and state-space search; see, e.g., [
2,
3,
4]. Their applications also include various problems in astronomy [
5], image processing [
6], computational biology [
7], animal foraging [
8], etc.
While uninformed search refers to diversity of cases in different (i.e., spatial and non-spatial) search domains [
9,
10], various blind search techniques share several measures of their efficiency among which are completeness and time complexity. The completeness criterion asks the question: “Can we find a solution, if it exists?”, while the time complexity criterion asks: “How long does it take to find a solution?”. Search problems that have a deceptively simple formulation may demonstrate very complex properties when the above criteria of efficient search are applied, and one such problem is presented in this paper.
In our work, we consider an uninformed search problem in a spatial domain: a particle moves step by step along a unit circle to find the escape channel and exit through it. Discrete motion in a unit circle to find an escape channel can be classified as a blind search in confined space problem where an idealised setting we deal with implies a very simple and straightforward search algorithm which cannot be optimised, e.g., by making the particle’s step size variable. On the other hand, the particle’s motion can be considered as a discrete time linear dynamical system, and that system exhibits interesting and unusual properties when the completeness and time complexity criteria of the blind search are investigated. We demonstrate in the paper that while the exit time (that is, the search time) remains finite for any initial position of the particle, it experiences random jumps when an initial position of the particle is slightly changed. The incompatibility between the distance that the particle covers moving step by step and the circumference length generates noise responsible for unpredictable changes in the exit time. The noise in the system is inherent and cannot be suppressed because the number acts as a random number generator in the problem. Furthermore, narrowing the escape channel moves the system from discrete changes in the exit time to the state where the system becomes extremely sensitive to initial conditions as the channel width becomes infinitely small.
Since the problem can be studied in the framework of a discrete time linear dynamical system, its sensitivity to initial conditions is of particular interest, as this property is often considered the hallmark of chaotic dynamics [
11,
12]. Mathematicians are concerned with a rigorous definition of chaos, where sensitive dependence on initial conditions is a necessary but not sufficient requirement to conclude that a dynamical system is chaotic [
13,
14,
15,
16,
17]. Meanwhile, detection and quantification of chaos in applied problems is often based on investigation of sensitivity alone [
18,
19] where conventional analysis of sensitivity to initial conditions employs calculation of Lyapunov exponents to measure the distance between two nearby solutions. A dynamical system that has at least one positive Lyapunov exponent is considered chaotic as nearby trajectories diverge exponentially with time and their evolution becomes unpredictable [
20,
21,
22].
The analysis of Lyapunov exponents is widely employed by applied scientists (see, e.g., [
23,
24,
25,
26,
27,
28] among many other works), yet it does not allow us to estimate the divergence rate of trajectories in our problem since the maximal Lyapunov exponent is
in a linear dynamical system we deal with. The above result implies that if we have sensitive dependence on initial conditions, then nearby trajectories diverge slower than exponentially (cf. the discussion of ‘weak chaos’ in e.g., [
29,
30,
31]) and it will be argued in the paper that trajectories diverge linearly when their rate of divergence is measured in terms of random jumps in the exit time. In the extreme case of an infinitely narrow channel, the system has sensitive dependence on initial conditions, which will be called
countable chaos in the paper: any two trajectories separated initially by a small distance will be separated by an arbitrary large distance over time in the problem.
This paper is organised as follows. In
Section 2, we formulate a blind search problem when a particle moves in a unit circle in discrete time. We then explain how to analyse the exit time as a function of the initial position of the particle in
Section 3. Our analysis is essentially based on the concept of generating points, which is studied in detail in
Section 4. Then in
Section 5 it is argued that the analytical results obtained in the previous sections should be backed by computer simulation to efficiently address the question of the exit time, and we present a computational algorithm used to find the exit time for any initial condition taken from the domain of definition. In
Section 6, we explain the random nature of jumps in the exit time and demonstrate the increasing unpredictability of the system when the channel narrows. We then investigate how the system will respond when the channel width becomes infinitely small in
Section 7, where the concept of countable chaos is introduced. Conclusions, a brief analysis of the results, and suggestions for future work are provided in
Section 8.
2. Problem Statement
We consider a particle moving along a unit circle parametrised by the angle
. The circle has an ‘escape channel’ whose centre is positioned at
and the channel half-width is
; see
Figure 1a. In the rest of this paper, we assume that the channel half-width
is sufficiently small (see
Section 4.2 for a more accurate definition of
).
While in the original problem statement [
32] the particle was positioned at
at the time
, here we assume that the particle starts its movement from some location
, where
The movement of the particle is discrete; i.e., the position of the particle has a constant increment
every next time
, where we require
. Hence, the angle
is changed with time as
and the problem can be considered as a discrete time dynamical system defined by the following linear map:
where
and
satisfies (
1).
It is worth noting here that the definition of a ‘particle’ we employ in our model of circular motion is generic as a simple setting (
1) and (
2) considered in the paper is not related to any specific problem in physics, engineering, biology, etc. Focusing on a specific application may require different terminology, e.g., there is a spatial ‘profitable patch’ instead of an ‘escape channel’ if an animal’s foraging problem is considered, yet the problem formulation (
1)–(
2) remains the same and we therefore use the term ‘particle’ throughout the paper for the sake of convenience (but see the discussion in
Section 8).
The discrete trajectory (
2) of the particle can be presented as a set of equidistant points positioned along straight lines with slope
. The position
of the particle is defined at the time
t as
where
is given by (
2); see
Figure 1b. The particle is said to exit the circle at the time
if it arrives at the escape channel at that time; i.e., the particle’s position is
after making
steps. The purpose of our study is to understand how the exit time
depends on the initial position
of the particle.
For the sake of our discussion, it is more convenient to consider an alternative representation of the circular motion (
3). That is, we consider a single straight line
and place the channel centre at the points
,
, where
is the channel number, as shown in
Figure 2. Also, while the number of channels can be infinitely large, in some cases we will consider a system of
N channels, where
is a finite number; i.e., we will assume that the particle stops moving forever if it cannot exit the circle through one of the first
N channels.
The exit condition (
4) is written for the system with many channels as follows:
In the next section, we will investigate the exit condition (
5) under the assumption that all but one channels are ‘closed’; i.e., the particle can only exit the domain through the
k-th channel located at
. The above assumption will allow us to find the solution
to inequality (
5), and the results obtained for the system with a single channel will then be generalised to a system where all channels are open.
4. Generating Points
In this section, we investigate the concept of generating points in more detail. We proceed with the example introduced in
Section 3 where we now want to compute the exit time for channels with the number
. Since the length of each interval
is entirely defined by the position of the left endpoint
(see
Section 3), it is more convenient to deal with the graph
instead of the graph
when a large number of channels are considered. Given the graph
, the graph
can be restored by defining
from (
18) and considering a constant exit time
over each interval
.
4.1. Example of Generating Points in the System
Let us increase the number of channels in the example in
Section 3. The graph
for
is presented in
Figure 5a, where the other parameters remain the same as in
Figure 4. The graph is further explained in
Figure 5b, where the point
is now shown as a closed green circle with the 1(1) label attached. The point
is a generating point which produces a regular grid
sketched as red and magenta dashed lines in
Figure 5b. The label attached to the generating point
shows its number
when a sequence of generating points is numbered and the channel number
in brackets.
The grid
is generated as follows. Consider the straight lines
and
starting at the generating point
in the
-plane and defined as
and
. The lines
and
have the direction vectors
and
, respectively, where
The increment
given to the variable
in the equation for the straight line
produces grid points
, along the line
, where the point
corresponds to
and
is the step size of the discrete movement of the particle. Those points define the exit through the same channel (i.e., the channel number is fixed) as explained in
Section 3. They are shown as black closed circles in
Figure 5b.
Similarly, the increment
given to the variable
starting from point
produces grid points
, along the straight line
. Those points are shown as closed magenta circles in
Figure 5b, where the label attached to each point indicates that it belongs to the grid
. The grid points along the line
define the exit through the next channel; i.e., the channel number increases by one when the next point
is generated. The grid
is then a tensor product of the one-dimensional grids in the
- and
- directions in the domain
; i.e., grid nodes are points of intersection between straight lines starting at points
and
and defined by the directed vectors
and
, respectively (see the red and magenta dashed lines in
Figure 5b).
If any
and
could be used in the definition of
, then an infinite grid
G would be generated that covers the entire
-plane. However, the values of
and
have to be chosen as explained above due to the requirement that all points on the grid belong to the domain
. Increasing the argument
, i.e., considering
, will result in a grid point outside the domain of definition as
. Furthermore, we have
for the channel number
(
) in (
17), so there are only three points positioned along the line
. Also, increasing the argument
, i.e., considering
, will produce a point that does not belong to the domain of definition as we have
. Using the definition (
11) gives
and we obtain by simple algebraic transformation that
where
. The direct calculation reveals that the point
is located within the same interval
as the point
. Hence, the point
is another generating point according to the definition (
22).
The generating point
is shown as a closed green circle along with the label 2(5) indicating the point number
and the channel number
in brackets in
Figure 5b. This point generates a regular grid
where the direction vectors are given by (
23) and the grid step sizes
and
are the same as they are on the grid
, i.e., both grids
and
can be considered as sub-grids of the same infinite grid
G generated when
and
. The straight lines
and
on the grid
start at the generating point
; see
Figure 5b. The grid points along the line
are shown as closed magenta circles and the label attached to each point indicates that they belong to the grid
(the other grid nodes on the grid
are not shown for the sake of visualisation).
The grid
contains again a finite number of nodes due to the restrictions imposed on the domain of definition. An analysis similar to that performed for the grid
leads us to the conclusion that another generating point will appear in the system when the number of channels increases further. The third generating point then produces a grid
and the number of generating points grows as the number of channels increases in the system; see
Figure 5b, where the first six generating points are shown as closed green circles in the graph.
Our study of the example in this subsection results in two important conclusions. First, the entire graph can be considered as a union of grid points belonging to grids , , where each grid is completely defined by its generating point. Second, each grid has a finite number of nodes, but that number is not the same. For example, the grid has 12 nodes, while the grid only has 9 nodes. Hence, we cannot assign some fixed numbers and to conclude that the transition from the current grid to the next grid will occur every time when and . In the next subsection, we provide a proof of the statement that, given the step size , the number of grid nodes on any grid , can be 16 at most.
4.2. Analysis of Grid
Let a grid
be produced by a generating point
,
. Consider straight lines
and
starting at the generating point
in the
-plane. The line
is defined as
by the definition of the exit through the same channel. Hence, the directed vector
is the same as on the grid
and is given by (
23). Since we have
and
, the minimum number
of grid points along the line
is
, while the maximum number
of grid points along the line
is
for any sufficiently small channel half-width
.
Let us now demonstrate that the line
is defined as
We first check that the grid points along the line
on the grid
are equidistant. Consider
, where
and
is an arbitrary channel number. We want to find the distance between the points
and
, where the point
corresponds to the escape through the next channel
. Taking into account (
11) and (
14), we have
and the distance between the two points is
We note that
, where the fractional part of the number
is
and
. Similarly,
, where the fractional part
and
. Hence, we introduce
where
and
. Gathering terms results in
Since and , we require . Given the bounds , , and , the value of can only be (A) or (B) .
Let us first consider the case (A). Substituting
into (
25) results in
Therefore, the point
belongs to the same straight line as the point
(see discussion in
Section 3) and
is not a generating point.
Consider now case (B). We substitute
into (
25) to obtain
Therefore,
does not belong to the same straight line as the point
. Furthermore, it follows from (
24) that
Since
, we have
and therefore
is another generating point in the system.
Let us now prove that there are at least
and at most
grid points positioned along the line
on the grid
. Consider a generating point
. We have by the definition of generating point
where
Consider now the exit through the next channel and let us check whether the point
belongs to the grid
generated by
. We require
for the point
to be on the grid
. The boundary condition gives
and therefore, we have
Since the generating length is
, we have
. Hence, the above condition holds for any
and the point
always belongs to the grid
generated by
.
Similarly, let us check whether the point
belongs to the grid
, i.e.,
. Implementing the boundary condition gives
Again, the above condition holds for any
.
Consider now
. Similar analysis gives
This condition holds for
, where
and is violated for any
. Hence, the point
can belong to the grid
generated by the point
or can be another generating point. In the first case, the distance between
and
is
, and
, while in the latter case the distance between
and
is
, and
.
Let , and then a new generating point appears at the interval . We have as by choice of . We also note that because always holds.
Finally, we check whether the point
belongs to the grid
. We have
This condition is violated for any
and we either conclude that
is another generating point in the system (which occurs if
is not a generating point) or
belongs to the grid
generated by point
.
We have shown that if is a generating point, then the points , , always belong to the line , while the point either belongs to the line or becomes the next generating point. Hence, the maximum number of grid points along the line is .
It is important to note that, while the number of points along the line
depends on the parameter
, the number of points along the line
is defined by the step size
and does not depend on the channel width. Furthermore, the condition
that defines the next generating point in the system depends on the value of
n, as
in (
27). We will argue in
Section 6 that the above condition cannot be checked beforehand for each new
n and therefore we cannot say a priori how the next grid
is shifted with respect to the previous grid.
5. The System with Channels Open
Knowledge of generating points allows one to find where the particle has to be located at the time
to leave the domain through the
k-th channel. This question can be investigated for any
and therefore the number of channels
N can be arbitrary in the problem. However, the analysis in
Section 3 and
Section 4 has been made under the assumption that only the
k-th channel is open for the particle’s escape through it, while the original problem statement demands that the particle can escape the domain through any channel. The definition of the number of channels
N requires careful consideration when all channels are open in the system. Based on the results obtained in
Section 3, in the following we explain how to obtain a solution to the escape problem with all channels open.
Given the number of channels N, we will say the total escape occurs in the system if the particle exits the domain through one of those channels, wherever the initial position of the particle is within the domain . Clearly, if the total escape occurs for some , then it will also occur for any . Hence, our next goal is to define the minimum number of channels for which the total escape is ensured.
Let us introduce the
escape range as follows:
where
is a subdomain in the domain
identified from the condition that the particle escapes through the
k-th channel if its initial position is
. We then define the minimum number of channels
required for the total escape from the following conditions:
Unlike the ‘single channel’ problem in
Section 3, the definition of
in (
28) now requires the analysis of intersection between intervals where the time
remains constant to avoid overlapping of those intervals. An example of such overlap is shown in
Figure 6a, where we sketch two hypothetical times
and
corresponding to the exit through the channels
and
, respectively (cf.
Figure 3). It is easily seen in the figure that we cannot consider the entire subdomain
as an interval where the particle has to be placed to leave the domain with the exit time
. The correct interval is now given by
and is highlighted in red along the
-axis in
Figure 6a. We also note that the subdomain
provides the exit time given by
or
.
The definition (
28) and (
29) can be illustrated by the baseline example introduced previously in
Section 3. We have
for the channel half-width
. For the number of channels
in the baseline example, there are nine intervals
(i.e., three intervals for each channel) where the escape is possible if the particle’s initial position belongs to any of those intervals; see
Figure 4. Since the length of each interval
is
in the example of
Figure 4, the size of the subdomain where escape is possible is
when
is considered in the definition (
28). The same result can be obtained by noting that we have three subdomains
,
, and
in the above example, where each subdomain
,
, consists of three equal intervals of length
; see the explanation of
in the definition (
28). Substitution of
into (
28) gives
and therefore escape through any of the first three channels is not the total escape. This statement is supported by the graph in
Figure 4 where we can see gaps in the domain of initial condition for which the exit time is not defined.
Consider now
in the same example (see
Section 4). Direct computation reveals that
for
, yet this number of channels is
as the total escape
can also be achieved for a smaller number of channels, e.g., when we have
or
. The accurate computation of
is then based on the Algorithm 1.
Algorithm 1 |
Consider the channel number . Let the domain and the subdomain . Define . Find all subintervals , where exit through the escape channel k is possible.
where , , and . Define the subdomain : Find and . Define the subdomain : Increase k by one and repeat steps 2–6 until . Then .
|
Application of the above algorithm to the baseline example gives
; that is, the particle initially positioned anywhere in the domain
always exits the circle through one of the first seven channels when the half-width of the channel is
. The graph
generated for
is shown in
Figure 6b where there are no gaps in the graph (cf. the graph in
Figure 4 generated for
). The results of
Figure 6b confirm the previous conclusion of
Figure 6a that shorter intervals of a constant exit time should be expected when the channel number
k increases. Furthermore, the number of subintervals
increases as we decrease the channel half-width
because it follows from (
10) that
. Hence, we have to increase the number of channels
in the system to ensure that the entire interval
is covered by subintervals
without leaving any gaps between them when the channel narrows. For the remainder of the paper, we will investigate how the graph
changes when the channel half-width decreases.
6. Random Jumps in the Exit Time
In
Section 4, we have shown that the graph
is entirely defined by generating points in the interval
, where
. The condition
imposed on the escape range in (
29) implies that we have to increase the number of channels
N when the channel width decreases. Consider an arbitrary interval of length
and assume that there are no generating points
within the interval
as shown in
Figure 7a. Since we need at least one generating point to define the exit time for the initial condition
, we have to add new generating points to the system until at least one of them will appear within the interval
. In other words, we should have a number of generating points sufficient to cover the whole interval
by subintervals
, otherwise the graph
,
will have gaps where the exit time is not defined and the escape range will be
. Producing new generating points requires us to increase the number of channels
N as follows from the discussion in
Section 4.
Conversely, increasing
N when the channel width decreases will produce new generating points. Thus, the following question arises from the above consideration: given the number of generating points, where will the next generating point be placed over the interval
? In the following, we demonstrate that the position (
26) of every next generating point cannot be predicted; that is, the generating points are randomly located in the interval
.
The randomness of generating points originates from the definition of the number
. We have
, where
. Let us present the fractional part
as
The decimal digits of the transcendental number
can be determined using digit-extraction algorithms where the
nth decimal digit can be computed without requiring the computation of earlier digits; e.g., see [
33]. Meanwhile, various studies [
34,
35,
36] have demonstrated that the sequence of decimal digits
in (
30) is random. In particular, it has been argued in [
36] that the number
can be used as a random number generator. Although randomness of the sequence
remains an open question that discussion is far beyond the scope of this paper, once the randomness of the decimal digits of
has been admitted it has far-reaching consequences in our problem. Consider
for an arbitrary
. Using expansion (
30) gives
where
is an integer number, and the new expansion coefficients
have been obtained by multiplying and adding terms in a random sequence
. Hence, the number
is
where
and a sequence of decimal digits in the fractional part
is random.
Let now
,
, be a new generating point added to the system. The point
is defined by (
26) for some
and
. On the other hand, we have
in (
11) when
and direct computation results in
Comparison of (
26) and (
32) gives
. Hence, the generating point is
where
. Since a sequence of decimal digits in the fractional part
of the number
is random, the position of the next generating point
appearing in the interval
cannot be predicted as the number of channels increases. In other words, the grids
produced by the points
,
are randomly shifted with respect to each other.
The distribution of generating points over the interval
is illustrated in
Figure 7b–d, where we show the position of the next generating point with respect to the previous generating point. The number of generating points
P increases from
in
Figure 7b to
in
Figure 7c, confirming that the generating points tend to be distributed over the entire interval
. However, their distribution is not uniform as strong clustering of generating points occurs at a finer spatial scale (see
Figure 7d) where the appearance of any new cluster of points cannot be predicted.
Consider any point
for which the exit time is
. Let
belong to the grid
defined by a generating point
. Let us now give a perturbation of the size
to the initial condition
, where we assume that
is small enough,
i.e.,
. Since the new point
is outside the interval
, the particle initially located at
will have a different exit time
. Furthermore, the point
does not belong to the same grid
to which the point
belongs because the distance between them is less than
(see the discussion in
Section 4).
Let us assume that
,
. Since the grids
and
are randomly shifted with respect to each other, the difference
between the exit times when the particle is initially positioned at the point
or at the point
cannot be determined a priori. Hence, any perturbation of the size
in the initial condition results in an unpredictable change in exit time. For the channel half-width
, the system goes to the state we refer to as
countable chaos in
Section 7 below.
The increasing unpredictability of the system as the channel becomes more narrow is shown in
Figure 8, where the channel half-width decreases from
in
Figure 8a to
in
Figure 8d. We want to emphasise here that although a visual inspection of the graphs in
Figure 8 may suggest their periodicity, those graphs are not periodic (see also the discussion in
Section 8). The positions of local minima and maxima in each graph in the figure are not equidistant along the
-axis: they are slightly yet randomly shifted from an equidistant distribution where a random shift of the maxima and minima points in the graph is a consequence of a random distribution of generating points over the interval
; cf.
Figure 7b–d.
It can be seen from
Figure 8 that oscillations in the exit time shown in the
-plane become more severe as
decreases. The analysis of the graphs in the figure also reveals that the maximum exit time increases as the channel narrows. Thus, our next aim is to investigate what conclusions can be drawn about the exit time
in the extreme case of an infinitely narrow channel
.
7. Countable Chaos
In this section, we check to what extent the exit time is sensitive to initial conditions in the system with an infinitely narrow channel. Sensitivity to initial conditions is a pivotal feature of chaos, and investigating this issue should help us to decide whether the system (
1)–(
2) has chaotic properties when the channel width becomes infinitely small. The sensitivity to initial conditions can be confirmed by calculating Lyapunov exponents, where a dynamical system that has at least one positive Lyapunov exponent is considered chaotic [
20,
21,
22]. However, the maximal Lyapunov exponent calculated for a simple linear map (
2a) is
. Therefore, we suggest an alternative approach to measure the divergence of the nearby trajectories as the channel half-width
.
Consider the total escape defined by the number of channels
for the given channel width
and let
be the number of subintervals where the exit time
remains constant. The number of jumps
between those subintervals is given by
Let us numerate the subintervals of constant exit time with the index
. The jump magnitude
is then defined for each jump between the subintervals as
where
is the left boundary of the
p-th sub-interval
.
Consider two trajectories starting at points
and
separated by the distance
in the domain
, where we assume that
for the sake of convenience. Let
and
; i.e., there is a jump
located within the interval
. Let also
, and then the length
of the first trajectory remains the same for any time
as the particle has already left the domain. It follows from (
2) that the distance between the two trajectories accumulated over time
is
for
. Substituting
and
into the above expression, we obtain
i.e., the distance
between the trajectories grows linearly with time. The maximum distance between the trajectories is
=
and we have
where the jump magnitude
remains unknown until the solution
is obtained in the entire domain of definition
.
Given the results (
36) and (
37), we note that separation of trajectories (
36) occurs over a finite time only. In addition, many trajectories that have the distance
between them at time
will have the same distance
between them at any time
. Since the function
is piecewise constant, the size of the subdomain of the initial condition where trajectories that are initially a very small distance apart will diverge over time depends on the number of jumps and can be evaluated as
. Consider, for example,
(see the baseline case in
Section 3) and
. We have the number of jumps
, as seen in
Figure 6, which results in
, while the total size of the domain
where the trajectories start is
.
Meanwhile, it has been proved in
Section 3 that the maximum length of each subinterval
where the exit time
remains constant is
. Hence, for any given
there exists a channel half-width
such that any two trajectories initially separated by the distance
will have a different exit time for any
, no matter where the interval
is located within the domain
.
The threshold value
can be chosen as
and any two trajectories that are initially a distance
apart will then be separated as (
36) in the system with a channel half-width
. The above conclusion also implies that the number of jumps (
34) depends on the parameter
, and
increases as
decreases. For
in the above example, we require
to provide further separation of all trajectories separated initially by
and the number of jumps increases from
to
when the channel half-width decreases from
to
.
Let us demonstrate that the separation of trajectories becomes stronger in the entire domain
as the channel half-width
decreases. Since the solution to the problem (
2)–(
4) is available for any
, we find the maximum distance between the trajectories (
37) in the whole domain of the initial condition
by direct computation of the jumps (
35) in the function
. For any given channel half-width
, we introduce the maximum jump
,
The maximum jump
depends on
, because it follows from the analysis in
Section 3 and
Section 5 that the maximum exit time
can be evaluated as
for the given channel half-width
. We then conclude from (
39) that
We also compute the average jump
in the domain
,
The average jump
depends on
, since the number of jumps
defined by (
34) increases when the channel half-width
decreases; see the discussion in
Section 6.
The increase in the number of jumps
as
decreases is illustrated by
Figure 8, where we also provide the value of
in the caption of the figure for each
considered in the figure. We then show the graphs
and
on a logarithmic scale in
Figure 9a, where the channel half-width varies from
to
. It can be readily noticed from the figure that the maximum distance between the trajectories
(
38) and the average distance between the trajectories
(
41) increase when the parameter
decreases.
An important observation about the graphs in
Figure 9a is that intervals of very rapid growth are interspersed with intervals where the functions (
38) and (
41) remain constant. A slight change in the parameter
does not necessarily result in an increase in the number of channels required for the total escape. For example, the number of jumps is
for
in
Figure 6b. If we decrease the channel half-width from
to
, the position of each jump in the exit time
in
Figure 6b will change, but the number of channels required for the total escape will remain
. Consequently, we will have the same number of jumps
and the same maximum jump amplitude
as the graphs in
Figure 9a show.
We now define the minimum jump
in the domain
,
If the minimum jump is a monotone function of
and we have
then sensitive dependence on initial conditions can be defined in the system when
. Consider an arbitrary distance
. Under the above assumptions about
, for any two trajectories initially separated by the distance
there exists the channel half-width
such that the final distance (
37) between these trajectories will be
. We note that the channel half-width
can be found from condition
.
The graph
is shown on a logarithmic scale in
Figure 9b for the same values of the parameter
as the graphs in
Figure 9a. Given
, consider an arbitrary distance
(solid red line in the figure). For any channel half-width
(vertical black dashed line), any two trajectories separated initially by the distance
will have the distance
between them, but the condition
will not be held for all such trajectories. Consider now
(vertical red dashed line). For any
, the distance
(
37) between any two trajectories initially separated by the distance
is
.
Let us note that we have only used the term ‘chaos’ in this section to indicate the sensitivity to initial conditions in the problem as
. Furthermore, the definition of sensitive dependence above is different from the conventional definition of sensitive dependence on initial conditions (see e.g., [
37,
38]) since it is based essentially on the asymptotic behaviour of the system when
. The sensitivity to initial conditions in the problem manifests itself through an infinitely large number of singular points (jumps)
in the function
whose locations cannot be predicted a priori. We have
as
because the exit time
is different for any two points initially separated by the distance
in the domain
. As the number of jumps increases, so does their magnitude. The maximum jump (
38) in the entire domain of the initial condition
is
as
. The numerical results of
Figure 9b suggest that the minimum jump
(
42) in the domain
is
as
, although we do not provide a rigorous mathematical proof of this statement. Since the maximum distance
between the two trajectories is measured by the jump magnitude (
37), any two trajectories initially separated by given
will be separated by an arbitrarily large distance
J in the system with an infinitely narrow channel
under assumption (
43).
Meanwhile, one can say that for any finite channel half-width , there is no chaos in the system. The number of random jumps in the function is finite for any finite and there are a number of subintervals where the exit time remains constant. The number of jumps increases as the channel width decreases, yet the exit time can be accurately calculated by Algorithm 1 for any given and any to remove unpredictability. Since the function is entirely defined by generating points belonging to a countable set, a finite number of intervals are required to fill any gap along the -axis and therefore determine the exit time for any taken from the domain of definition. Hence, we also introduce the term ‘countable’ to emphasise the discrete nature of the process, as all exit times belong to a countable set and can be computed, no matter how small a finite channel half-width is.
8. Discussion and Conclusions
We have studied a linear dynamical system resulting from discrete time motion of a particle along a unit circle that has an escape channel of width . The time required for the particle to approach the escape channel and exit the circle through it depends on the initial position of the particle and it has been argued in the paper that the solution is ‘global’; that is, finding the exit time for any given requires reconstruction of the function in the entire domain of definition . The function is piecewise constant and the exit time remains the same at subintervals of length at most and has jumps between those subintervals. Hence, given the channel half-width , a slight change in the initial condition can leave the exit time the same or result in a significant change in the exit time. The system’s response to a slight change in the initial condition cannot be predicted unless the function is computed everywhere in the domain .
The dynamical system (
1)–(
2) studied in the paper presents a convenient formulation of a blind search in confined space problem where the results of our study allow one to conclude about the completeness and time complexity criteria of the search. Completeness of the search follows from randomness of generating points (see
Section 4 and
Section 6) since that property implies covering the whole interval
by subintervals
with no gaps between them as required to find the exit time for any given initial position of the particle. It also follows from the above requirement that the solution is
when the question of time complexity is addressed.
One important result of our study is that the number of random jumps in the function
becomes infinitely large resulting in arbitrary separation of nearby trajectories as the channel half-width
: the phenomenon of countable chaos arises in the problem with an infinitely narrow channel. The countable chaos is not a conventional chaos definition because it only reveals itself as
and also because the system has noise that induces sensitive dependence on initial conditions (cf. [
12] where it was argued that the system is not chaotic if there is external noise responsible for irregularity in the behaviour of the system). However, the noise is not external and is inherent in the system. The number
responsible for the definition of circular motion also acts as a random number generator which produces sensitivity to initial conditions when the channel half-width
.
The unpredictability of the exit time observed in the system for any finite channel half-width and the requirement to obtain a global solution , to find the exit time for a given initial position of the particle may be considered undesirable features of the system as they increase the time complexity in the blind search problem. However, the presence of inherent noise actually makes the blind search more successful because it increases the probability of escape. Removing the noise in the problem will result in the ‘the escape is/is not ever possible’ dichotomy as illustrated by the following example.
Let us make the system fully predictable by turning off the random number generator. That is, we want to consider a hypothetical system in which the first channel is located at and the distance between channels is . This hypothetical -system’ can be thought of as a setup where circular motion is replaced by linear motion; i.e., a unit circle is approximated by a polygon.
Generating points in the new
-system are periodic as the equation
has a solution
. For
and the generating length
, we find
and
. In other words, the residual
accumulated during the discrete time motion over the domain with
channels (see
Section 3) can be fully covered in the
additional steps of the particle. It is also obvious that Equation (
44) does not have any solution
when
in the original
-system’ because
is not a rational number.
We want to find the number of channels
required to make the total escape
as explained in
Section 5. Let us first consider the baseline case
,
in the original
-system where the particle moves along a circle. We have
for
channels; see
Figure 6b. Direct computation reveals that linear motion in the new
-system results in the same number of channels required for the total escape; i.e., we have
.
We now decrease the channel half-width as . For circular motion (the -system), the total escape will be achieved when there are channels in the system. However, the results are very different when linear motion (the -system) is considered. The escape range is established over the first channels and then does not change as due to the periodicity of the generating points. If we have an ensemble of particles statistically uniformly distributed throughout the interval at time , approximately half of them will escape from the domain, while another half will stay in the domain forever. Hence, the fully predictable -system can also be thought of as a half-degenerate system because the goal state is never achieved by approximately of the population.
In the case of circular motion, all particles will sooner or later exit the domain, no matter how narrow the escape channel is. The definition of makes the system’s behaviour less predictable and more complex, yet the system is not degenerate. The complexity induced by should be taken into account when tracing the trajectories, but that complexity is an advantage rather than a drawback because it removes the escape dichotomy in the system.
In conclusion, we note that we have been concerned in this paper with reporting random exit times for any finite channel half-width
and sensitivity to initial conditions in the system (
1) and (
2) as
rather than investigating those phenomena in detail. Thus, our study leaves a number of open questions, some of them listed below.
Chaotic properties of the system: The divergence of nearby trajectories has been demonstrated in
Section 7, yet more thorough investigation of the condition (
43) is necessary to draw a rigorous conclusion about sensitive dependence on initial conditions in the system with an infinitely narrow channel. Furthermore, our study has been focused on sensitivity to initial conditions only as we have assumed that this is the most important property defining a chaotic system in line with the Experimentalists’ definition in [
11]. Although there is no universally accepted definition of chaos in a dynamical system so far [
19], other definitions of chaos provided, e.g., in [
13] or [
17], require additional properties of a dynamical system to conclude that the system is chaotic. Those properties will be verified in future work on the problem to see whether the definition of a chaotic regime in the system (
1) and (
2) can be made compatible with widely accepted definitions of chaos.
Weak chaos: It has been shown in
Section 7 that the separation of trajectories occurs weaker than exponentially as the distance between them increases linearly over time. The slower separation of chaotic trajectories is a feature of weak chaos reported, e.g., in [
29,
30,
31,
39], and comparison of the system’s asymptotical behaviour as
with systems where weak chaos has been detected is reserved as a topic of future work. Also, it is still unclear how quickly trajectories diverge on average when the parameter
decreases. A related question that arises here is whether the transition between domains where the functions
and
remain constant is continuous (albeit with a very steep gradient) or discontinuous in
Figure 9. The latter case may be related to the appearance of new generating points in the system when
decreases, and this issue requires further investigation because it may help us evaluate how fast the system moves to a chaotic regime as
.
System parameters: The analysis in the paper has been made for the step size of the particle
. If a different step size
is considered under the condition
, then Algorithm 1 in
Section 5 can be used to find the exit time for any finite channel half-width
. The conclusions made about sensitive dependence on initial conditions in
Section 7 will also remain the same if a different (yet finite) value
is chosen in the problem. Meanwhile, the asymptotic behaviour of the system when
and
is unclear and should be studied in future work.
Pseudo-random number generators: The randomness of the decimal digits of
has been the most important assumption made in the paper. The ability of irrational numbers to serve as pseudo-random number generators has been the focus of research for many years [
40], yet this topic remains a big and challenging problem. If we have a rigorous confirmation on the randomness of an irrational number
p, an algorithm similar to that presented in the paper can be designed for the number
p when a particle performs discrete time motion (
1) and (
2) along a closed curve of length
. On the other hand, if the number
p does not have a random sequence of decimal digits, then using it in the problem will result in different properties of the dynamical system (
1) and (
2). Generating points will not be located randomly and that, in turn, will result in only a limited number of trajectories escaping from the domain (cf. the conclusions for the rational number
in this section). Thus, we hope that the results of this paper may spark new interest in research on pseudo-random number generators.
Meanwhile, the solution algorithm in
Section 5 allows one to study more complex problems based on the results already obtained in this paper. The idealistic setting (
1)–(
2) employed in the problem can serve as an approximation to various realistic problems, including those in scattering theory [
41], animal foraging in a spatial domain [
8], cryptography [
42], and detection of pulsed signals [
43]. If the problem is considered in the framework of classical scattering, one question of interest could be to move from consideration of a single particle to an ensemble of particles. For instance, the approach developed in this paper allows for finding the probability of the event that all particles will leave the domain over a given time, the answer to the above question depending on the initial spatial distribution of particles. In addition, the channel width can be varied with time in order to maximise or minimise the exit time over which all particles will leave the domain.
Alternatively, the blind search problem can be thought of as a foraging problem in which an animal is looking for a profitable patch
. In the latter case, the setup (
1)–(
2) with a single animal can be further investigated if the condition of having a constant step size is relaxed. Although the requirement of discrete time is essential in animal movement models, more realistic movement rules will involve consideration of a stochastic step length defined by a Lévy walk or flight [
44]. Furthermore, even when the movement is not random, the step length
is likely to vary as the animal tries to optimise the search time. This brings the system to a new level of complexity, and the study of a variable step length is considered a topic of future work.