Chaotic patterns in the discrete-time dynamics of social foraging swarms with attractant–repellent profiles: an analysis

Abstract

This article investigates the chaotic dynamical characteristics of a discrete-time social foraging swarm model in which the individuals move in a d-dimensional space according to an attractant/repellent or a nutrient profile. The collective behavior of the swarm model results from the balance between inter-individual interactions and the simultaneous interactions of the swarm members with their environment. Analysis undertaken in the paper indicates that chaos can be introduced into the system by tuning the parameters of the mutual attraction-repulsion function and the attractant–repellent profiles of the swarm dynamics [as given in Eq. (1)]. Ranges of different parameters to ensure sufficient condition for chaotic characteristics have been determined. Apart from chaos, stable and limit cyclic behaviors are also demonstrated for various parameter ranges. Effect of different attractant/repellent gradient profiles on the mean swarm trajectory in the presence of chaos has also been studied. Presence of chaos is determined by means of the Lyapunov exponent. Results obtained from the theoretical analysis are validated through comprehensive simulations.

Appendix

1.1 Boundary values of $Z\big (\xi _p^{ij} \big )$

We have defined,

$Z\big (\xi _p^{ij} \big )\!=\!\left( {1\!-\!\frac{2}{c}\left( {x_p^i -x_p^j } \right) ^{2}} \right) \cdot \exp \left( {-\;\frac{\left\| {{\varvec{x}}^{i}-{\varvec{x}}^{j}} \right\| ^{2}}{c}} \right) .$ For each and every dimension we have a similar dynamical equation like (7) and we there is a function like $Z\big (\xi _p^{ij} \big )$. We have to find a vector ${\varvec{\xi }}^{ij} ={\varvec{x}}^{i}-{\varvec{x}}^{j}$, which gives the extreme value of $Z\big (\xi _p^{ij} \big )$. Indeed this is a multi-objective problem where we have d objective functions $\big (Z\big (\xi _1^{ij} \big ), \,Z\big (\xi _2^{ij} \big ), {\ldots },Z\big (\xi _d^{ij} \big )\big )$ for d dimensions. As for our analysis every dimension is alike so we are only interested for those solutions of the multi-objective problem which gives equal extreme value for each dimensional component. Now if the extreme values are equal for each dimension then we can write,

$$\begin{aligned} Z\big (\xi _1^{ij} \big )= Z\big (\xi _2^{ij} \big )=\cdots = Z\big (\xi _p^{ij} \big ) =\cdots = Z\big (\xi _d^{ij} \big ).\nonumber \\ \end{aligned}$$

(20)

The above condition will be satisfied if and only if

$$\begin{aligned} \xi _1^{ij} =\xi _2^{ij} =\cdots =\xi _p^{ij} =\cdots =\xi _d^{ij} . \end{aligned}$$

(21)

If condition (21) is satisfied then we can write,

$$\begin{aligned} Z\big (\xi _p^{ij} \big )= & {} \left( {1-\frac{2}{c}\left( {x_p^i -x_p^j } \right) ^{2}} \right) \nonumber \\&\times \, \exp \left( {-\;\frac{d\cdot \left( {x_p^i -x_p^j } \right) ^{2}}{c}} \right) .\nonumber \\ \Rightarrow Z\big (\xi _p^{ij} \big )= & {} \left( {1-\frac{2}{c}\cdot \xi _p^{ij2}} \right) \cdot \exp \left( {-\;\frac{d\cdot \xi _p^{ij2}}{c}} \right) .\nonumber \\ \end{aligned}$$

(22)

From (22) one can easily find out the maximum and minimum value of $Z\big (\xi _p^{ij} \big )$. And the maximum and minimum values are 1 and $-2\cdot \exp (-(1+d/2))/d$ and the corresponding values of $\xi _p^{ij}$ are 0 and $\pm \sqrt{(d+2)c/2}$.

1.2 Chaotic condition or Case II(A)

Case (i) For $k>0$.

If either $1-A_\sigma -a(M-1)(1-Z_m k)>1$ or $1-A_\sigma -a(M-1)(1-k)<-1$; then the swarm will be chaotic as per Eq. (9a). So we get,

$$\begin{aligned} k\!<\!\frac{1}{Z_m }\left( {1+\frac{A_\sigma }{a(M-1)}} \right) \hbox { or }\,k\!<\!\left( {1-\frac{2-A_\sigma }{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k\!<\!\min \left\{ {\frac{1}{Z_m }\left( {1\!+\!\frac{A_\sigma }{a(M-1)}} \right) ,\left( {1-\frac{2-A_\sigma }{a(M-1)}} \right) } \right\} . \end{aligned}$$

Case (ii) For $k<0$.

To have chaos either $1-A_\sigma -a(M-1)(1-k)>1$ or $1-A_\sigma -a(M-1)(1-Z_m k)<-1$ .

$$\begin{aligned} \Rightarrow k\!>\!\left( {1\!+\!\frac{A_\sigma }{a(M-1)}} \right) \hbox { or }k\!>\!\frac{1}{Z_m }\left( {1\!-\!\frac{2-A_\sigma }{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned}k>\max \left\{ {\left( {1+\frac{A_\sigma }{a(M-1)}} \right) ,\frac{1}{Z_m }\left( {1-\frac{2-A_\sigma }{a(M-1)}} \right) } \right\} . \end{aligned}$$

Thus the condition of chaos can be written by the following way,

$$\begin{aligned} k< & {} \min \left\{ {\frac{1}{Z_m }\left( {1\!+\!\frac{A_\sigma }{a(M-1)}} \right) ,\left( {1\!-\!\frac{2-A_\sigma }{a(M-1)}} \right) } \right\} .\nonumber \\ k> & {} \max \left\{ {\left( {1\!+\!\frac{A_\sigma }{a(M-1)}} \right) ,\frac{1}{Z_m }\left( {1\!-\!\frac{2-A_\sigma }{a(M-1)}} \right) } \right\} .\nonumber \\ \end{aligned}$$

(23)

Here we can easily notice that how the parameters of the gradient profile affect the range of k to ensure chaos. Case II(A) is a quadratic profile whose optima is at ${\varvec{c}}_\sigma$. Both ${\varvec{c}}_\sigma$ and $b_\sigma$ do not have any effect on the chaotic conditions. But their effect will be clear when we will be considering the mean swarm trajectory.

1.3 Chaotic condition or Case II(B)

Case (i) For $k>0$.

To ensure chaos, either $1-\sum _{l=1}^N {A_\sigma ^l } -a(M-1)(1-Z_m k)>1$ or $1-\sum _{l=1}^N {A_\sigma ^l } -a(M-1)(1-k)<-1$.

$$\begin{aligned} \Rightarrow k< & {} \frac{1}{Z_m }\left( {1+\frac{\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) \hbox { or}\\ k< & {} \left( {1-\frac{2-\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1\!+\!\frac{\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) ,\left( 1-\frac{2-\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)} \right) \right\} . \end{aligned}$$

Case (ii) For $k<0$.

Similarly for this case the chaos will be ensured if either $1-\sum _{l=1}^N {A_\sigma ^l } -a(M-1)(1-k)>1$ or $1-\sum _{l=1}^N {A_\sigma ^l } -a(M-1)(1-Z_m k)<-1$.

$$\begin{aligned}\Rightarrow & {} k>\left( {1+\frac{\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) \hbox { or}\\&\quad k>\frac{1}{Z_m }\left( {1-\frac{2-\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k> & {} \max \left\{ \left( {1\!+\!\frac{\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) ,\frac{1}{Z_m }\left( 1\!-\!\frac{2-\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)} \right) \right\} . \end{aligned}$$

Thus the condition of chaos can be written by the following way,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1\!+\!\frac{\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) ,\left( 1\!-\!\frac{2-\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)} \right) \right\} \nonumber \\ k> & {} \max \left\{ \left( {1\!+\!\frac{\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)}} \right) ,\frac{1}{Z_m }\left( 1\!-\!\frac{2-\sum _{l=1}^N {A_\sigma ^l } }{a(M-1)} \right) \right\} .\nonumber \\ \end{aligned}$$

(24)

In this case the gradient profile is a sum of N quadratic profiles; the effect of the parameters of the gradient profile is clearly observable from above expression. The effect of other parameters will be clear in Sect. 3.

1.4 Chaotic condition or Case III(A)

Case (a) When $A_\sigma$ is positive:

Case (i) For $k>0$.

From (11a) we can ensure chaos if either

$$\begin{aligned}&1-\frac{A_\sigma }{l_\sigma }-a(M-1)(1-Z_m k)>1\hbox { or}\\&1+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))\\&\quad -a(M-1)(1-k)<-1.\\&\Rightarrow k<\frac{1}{Z_m }\left( {1+\frac{A_\sigma }{l_\sigma \cdot a(M-1)}} \right) \hbox { or}\\&k<\left( {1-\frac{2+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))}{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1+\frac{A_\sigma }{l_\sigma \cdot a(M-1)}} \right) ,\right. \\&\left. \qquad \quad \quad \left( 1-\,\frac{2+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))}{a(M-1)} \right) \right\} . \end{aligned}$$

Case (ii) For $k<0$.

Similarly from (11b) we can write, either $1-\frac{A_\sigma }{l_\sigma }-a(M-1)(1-k)>1$ or $1+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))-a(M-1)(1-Z_m k)<-1$.

$$\begin{aligned} \Rightarrow&k>\left( {1+\frac{A_\sigma }{l_\sigma \cdot a(M-1)}} \right) \hbox { or}\\&k>\frac{1}{Z_m }\left( {1-\frac{2+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))}{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k> & {} \max \left\{ \left( {1+\frac{A_\sigma }{l_\sigma \cdot a(M-1)}} \right) ,\right. \\&\left. \quad \frac{1}{Z_m }\left( 1-\,\frac{2+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))}{a(M-1)} \right) \right\} . \end{aligned}$$

Thus the sufficient condition or chaos for positive $A_\sigma$ is,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1+\frac{A_\sigma }{l_\sigma \cdot a(M-1)}} \right) ,\right. \nonumber \\&\left. \quad \quad \left( 1-\frac{2+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))}{a(M-1)} \right) \right\} .\nonumber \\ k> & {} \max \left\{ \left( {1+\frac{A_\sigma }{l_\sigma \cdot a(M-1)}} \right) ,\right. \nonumber \\&\left. \quad \frac{1}{Z_m }\left( 1-\frac{2+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))}{a(M-1)} \right) \right\} .\nonumber \\ \end{aligned}$$

(25a)

Case (b) When $A_\sigma$ is negative:

Case (i) For $k>0$.

For this case chaos will be found if either, $1+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))-a(M-1)(1-Z_m k)>1$ or $1-\frac{A_\sigma }{l_\sigma }-a(M-1)(1-k)<-1$.

$$\begin{aligned} \Rightarrow&k<\frac{1}{Z_m }\left( {1-\frac{2A_\sigma \cdot \exp (-(1+d/2))}{d\cdot l_\sigma \cdot a(M-1)}} \right) \hbox { or}\\&k<\left( {1-\frac{2-\frac{A_\sigma }{l_\sigma }}{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1-\frac{2A_\sigma \cdot \exp (-(1+d/2))}{d\cdot l_\sigma \cdot a(M-1)}} \right) ,\right. \\&\left. \quad \quad \left( 1-\frac{2-\frac{A_\sigma }{l_\sigma }}{a(M-1)} \right) \right\} . \end{aligned}$$

Case (ii) For $k<0.$

The swarm will exhibit a chaotic phenomenon for either,

$$\begin{aligned}&1+\frac{2A_\sigma }{d\cdot l_\sigma }\cdot \exp (-(1+d/2))-a(M-1)(1-k)\\&\quad >1\hbox { or } 1-\frac{A_\sigma }{l_\sigma }-a(M-1)(1-Z_m k)<-1.\\ \Rightarrow&k>\left( {1-\frac{2A_\sigma \cdot \exp (-(1+d/2))}{d\cdot l_\sigma \cdot a(M-1)}} \right) \hbox { or}\\&k>\frac{1}{Z_m }\left( {1-\frac{2-\frac{A_\sigma }{l_\sigma }}{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k> & {} \max \left\{ \left( {1-\frac{2A_\sigma \cdot \exp (-(1+d/2))}{d\cdot l_\sigma \cdot a(M-1)}} \right) ,\right. \\&\left. \quad \quad \frac{1}{Z_m }\left( 1-\frac{2-\frac{A_\sigma }{l_\sigma }}{a(M-1)} \right) \right\} . \end{aligned}$$

Thus the sufficient condition or chaos for negative $A_\sigma$ is,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1-\frac{2A_\sigma \cdot \exp (-(1+d/2))}{d\cdot l_\sigma \cdot a(M-1)}} \right) ,\right. \nonumber \\&\left. \quad \quad \left( 1-\frac{2-\frac{A_\sigma }{l_\sigma }}{a(M-1)} \right) \right\} . \nonumber \\ k> & {} \max \left\{ \left( {1-\frac{2A_\sigma \cdot \exp (-(1+d/2))}{d\cdot l_\sigma \cdot a(M-1)}} \right) ,\right. \nonumber \\&\left. \quad \quad \frac{1}{Z_m }\left( 1-\frac{2-\frac{A_\sigma }{l_\sigma }}{a(M-1)} \right) \right\} \end{aligned}$$

(25b)

The above expressions clearly show that how chaotic phenomenon can be controlled by the parameters of the Gaussian profile.

1.5 Chaotic condition or Case III(B)

Case (a) When all the $A_\sigma ^l$ are positive:

Case (i) For $k>0$.

To have deterministic chaos in the swarm, either $1-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-Z_m k)>1$ or $1+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-k)<-1$.

$$\begin{aligned} \Rightarrow&k<\frac{1}{Z_m }\left( {1+\frac{\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) \hbox { or}\\&k<\left( {1-\frac{2+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum \limits _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1+\frac{\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\left. \quad \left( 1-\frac{2+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)} \right) \right\} . \end{aligned}$$

Case (ii) For $k<0.$

Similarly for here, to ensure chaos either $1-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-k)>1$ or $1+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-Z_m k)<-1$.

$$\begin{aligned} \Rightarrow&k>\left( {1+\frac{\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) \hbox { or }\\&k>\frac{1}{Z_m }\left( {1-\frac{2+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k> & {} \max \left\{ \left( {1+\frac{\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\left. \quad \frac{1}{Z_m }\left( 1\!-\!\frac{2+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)} \right) \right\} . \end{aligned}$$

Thus the sufficient condition or chaos for positive $A_\sigma$ is,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1+\frac{\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\left. \quad \left( 1-\frac{2+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)} \right) \right\} .\\ k> & {} \max \left\{ \left( {1+\frac{\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\left. \quad \frac{1}{Z_m }\left( 1-\frac{2+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)} \right) \right\} . \end{aligned}$$

Case (b) When all the $A_\sigma ^l$ are negative:

Case (i) For $k>0$.

Either $1+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-Z_m k)>1$ or $1-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-k)<-1$.

$$\begin{aligned} \Rightarrow&k<\frac{1}{Z_m }\left( {1-\frac{2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) \hbox { or}\\&k<\left( {1-\frac{2-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1-\frac{2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\left. \quad \quad \left( 1-\frac{2-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)} \right) \right\} . \end{aligned}$$

Case (ii) For $k<0.$

Either $1+2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-k)>1$ or $1-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} -a(M-1)(1-Z_m k)<-1$.

$$\begin{aligned} \Rightarrow&k>\left( {1-\frac{2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) \hbox { or }\\&k>\frac{1}{Z_m }\left( {1-\frac{2-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) . \end{aligned}$$

Combining these two we get,

$$\begin{aligned} k> & {} \max \left\{ \left( {1-\frac{2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\left. \quad \quad \frac{1}{Z_m }\left( 1-\frac{2-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)} \right) \right\} . \end{aligned}$$

Thus the sufficient condition or chaos for negative $A_\sigma$ is,

$$\begin{aligned} k< & {} \min \left\{ \frac{1}{Z_m }\left( {1-\frac{2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\quad \left. \left( {1-\frac{2-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) \right\} .\\ k> & {} \max \left\{ \left( {1-\frac{2\cdot \frac{\exp (-(1+d/2))}{d}\cdot \sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) ,\right. \\&\quad \quad \left. \frac{1}{Z_m }\left( {1-\frac{2-\sum _{l=1}^N {\frac{A_\sigma ^l }{l_\sigma ^l }} }{a(M-1)}} \right) \right\} . \end{aligned}$$