*3.3.1 Attractors*

Keeping parameters *a* and *r* fixed viz., *a* ¼ 2*:*0, *r* ¼ 2*:*4, attractors for different cases are obtained through numerical technique [16], and shown in **Figure 9**.

Looking plots of attractor of **Figure 9**, one finds a chaotic attractor, figure (a), when Allee effect is not in consideration, for *a* ¼ 2*:*0, *r* ¼ 2*:*4 . But, the application of Allee effect to either of the population or to both population, system returned to regularity, e.g. figures (b), (c) and (d) are no more chaotic. This also follow from the plots of LCEs given below.

### *3.3.2 Lyapunov exponents (LCEs)*

The phase space dynamics of a nonlinear chaotic physical system is very complex in general. One of the important feature of such a system is its sensitivity to initial conditions i.e., two very nearby trajectory in phase space show divergence exponentially. Such divergence are characterized by LCEs. To indicate chaotic and regular evolution, an appropriate measure is to find Lyapunov exponents (LCEs) which are obtained for different cases by using appropriate procedure. Plots of LCEs are shown in **Figure 10**.

### *3.3.3 Correlation dimension*

Lorenz attractor provides an example of a fractal object with noninteger dimension. The correlation dimension permits us to quantify the space filling property and provides the measure of dimensionality of the chaotic attractor. It is expressed as

$$D^\varepsilon = \frac{d\log C(R)}{d\log\left(R\right)}$$

where *C R*ð Þ is defined as

$$C(R) = \frac{1}{n(n-1)} \sum\_{i=1}^{n} \sum\_{j=1}^{n} \left[ \Theta(R - \|\boldsymbol{x}[i] - \boldsymbol{x}[j]\|) \right]^2$$

*3.3.4 Topological entropies*

**Figure 9.**

**41**

**Figure 8.**

entropy, it establishes a steady state situation.

*Allee effect on prey as well as on predator, ε* ¼ 4*:*5, *μ* ¼ 0*:*1*.*

As explained in the beginning, topological entropy measures the complexity of the system. More topological entropy implies system is more complex. Presence of complexity does not mean the system is chaotic and vice versa. In **Figure 12**, we have plots of topological entropy for different cases. In figure (a), topological entropy increases for *r*> 2 but bifurcation diagrams and calculations of LCEs indicate the system is regular within 2*:*0≤*r*≤ 2*:*2. Similar observation can be made looking at figures (b) and (c). In figure (d) one finds no fluctuations of topological

*Plots of regular and chaotic attractors for a* ¼ 2*:*0 *and r* ¼ 2*:*4*; (i) plot (a) without Allee effect, (ii) plot (b) with Allee effect on prey only, ε* ¼ 4*:*5 *, (iii) plot (c) Allee effect on predator only mu* ¼ 0*:*1*, and (iv) plot (d)*

*Bifurcation diagram of system (9), (a) Prey densities, (b) Predator densities for a* ¼ 2*:*0 *and* 1*:*8 ≤*r* ≤2*:*4*.*

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corresponds to the correlation sum and is a measure of total number of points contained within a hypersphere of radius *R* as a function of *R* normalized to the total number of points squared. Using the algorithm [17, 18], the correlation dimension can be determined from the scaling region found in the plot of log*C R*ð Þ as a function of log ð Þ *R* . For the Lorenz system with parameters *<sup>σ</sup>* <sup>¼</sup> 10, *<sup>ρ</sup>* <sup>¼</sup> 28 and *<sup>b</sup>* <sup>¼</sup> <sup>8</sup>*=*3, the correlation dimension *Dc* is found to be 2*:*069. The correlation dimension of the chaotic attractor **Figure 9(a)** is found to be *<sup>D</sup><sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*571 (**Figure 11**).

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**Figure 8.** *Bifurcation diagram of system (9), (a) Prey densities, (b) Predator densities for a* ¼ 2*:*0 *and* 1*:*8 ≤*r* ≤2*:*4*.*

#### **Figure 9.**

tool to analyze the regular, chaotic as well as complexity within the system. For *a* ¼ 2*:*0 and 1*:*8≤*r*≤2*:*4 , **Figure 8** shows bifurcation of system (9), where some interesting phenomena observed that the system is not producing a period doubling

Keeping parameters *a* and *r* fixed viz., *a* ¼ 2*:*0, *r* ¼ 2*:*4, attractors for different

Looking plots of attractor of **Figure 9**, one finds a chaotic attractor, figure (a), when Allee effect is not in consideration, for *a* ¼ 2*:*0, *r* ¼ 2*:*4 . But, the application of Allee effect to either of the population or to both population, system returned to regularity, e.g. figures (b), (c) and (d) are no more chaotic. This also follow from

The phase space dynamics of a nonlinear chaotic physical system is very complex in general. One of the important feature of such a system is its sensitivity to initial conditions i.e., two very nearby trajectory in phase space show divergence exponentially. Such divergence are characterized by LCEs. To indicate chaotic and regular evolution, an appropriate measure is to find Lyapunov exponents (LCEs) which are obtained for different cases by using appropriate procedure. Plots of

Lorenz attractor provides an example of a fractal object with noninteger dimension. The correlation dimension permits us to quantify the space filling property and provides the measure of dimensionality of the chaotic attractor. It is

> *<sup>D</sup><sup>c</sup>* <sup>¼</sup> *<sup>d</sup>* log*C R*ð Þ *d* log ð Þ *R*

> > X*n j*¼1

corresponds to the correlation sum and is a measure of total number of points contained within a hypersphere of radius *R* as a function of *R* normalized to the total number of points squared. Using the algorithm [17, 18], the correlation dimension can be determined from the scaling region found in the plot of log*C R*ð Þ as a function of log ð Þ *R* . For the Lorenz system with parameters *<sup>σ</sup>* <sup>¼</sup> 10, *<sup>ρ</sup>* <sup>¼</sup> 28 and *<sup>b</sup>* <sup>¼</sup> <sup>8</sup>*=*3, the correlation dimension *Dc* is found to be 2*:*069. The correlation dimension of the chaotic attractor **Figure 9(a)** is found to be

½ � Θð Þ *R* � ∥*x i*½�� *x j* ½ �∥

X*n i*¼1

cases are obtained through numerical technique [16], and shown in **Figure 9**.

bifurcation scenario which is very common for many nonlinear systems.

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Simulation of the foregoing models provides,

**3.3 Numerical simulations**

the plots of LCEs given below.

*3.3.2 Lyapunov exponents (LCEs)*

LCEs are shown in **Figure 10**.

where *C R*ð Þ is defined as

*<sup>D</sup><sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*571 (**Figure 11**).

**40**

*C R*ð Þ¼ <sup>1</sup>

*n n*ð Þ � 1

*3.3.3 Correlation dimension*

expressed as

*3.3.1 Attractors*

*Plots of regular and chaotic attractors for a* ¼ 2*:*0 *and r* ¼ 2*:*4*; (i) plot (a) without Allee effect, (ii) plot (b) with Allee effect on prey only, ε* ¼ 4*:*5 *, (iii) plot (c) Allee effect on predator only mu* ¼ 0*:*1*, and (iv) plot (d) Allee effect on prey as well as on predator, ε* ¼ 4*:*5, *μ* ¼ 0*:*1*.*

#### *3.3.4 Topological entropies*

As explained in the beginning, topological entropy measures the complexity of the system. More topological entropy implies system is more complex. Presence of complexity does not mean the system is chaotic and vice versa. In **Figure 12**, we have plots of topological entropy for different cases. In figure (a), topological entropy increases for *r*> 2 but bifurcation diagrams and calculations of LCEs indicate the system is regular within 2*:*0≤*r*≤ 2*:*2. Similar observation can be made looking at figures (b) and (c). In figure (d) one finds no fluctuations of topological entropy, it establishes a steady state situation.

**Figure 10.**

*Plots of Lyapunov exponents for a* ¼ 2*:*0, *r* ¼ 2*:*4 *and (i) figure (a) without Allee effect, (ii) figure (b) with ε* ¼ 4*:*5, *μ* ¼ 0 *, (iii) figure (c) Allee effect on predator only with μ* ¼ 0*:*1 *, (iv) Allee effect on both populations ε* ¼ 4*:*5, *μ* ¼ 0*:*1 *.*

**4. Recurrence plot**

**Figure 12.**

from the time series as:

We may write

**43**

**Xi** ¼ *xi*, *xi*þ*<sup>τ</sup>*, *xi*þ2*<sup>τ</sup>*, ⋯, *xi*þð Þ *<sup>m</sup>*�<sup>1</sup> *<sup>τ</sup>*

Natural system exhibits periodicities and also irregular cyclicities. Usually measures such as Lyapunov characteristic exponent (LCE), correlation dimension, Kolmogorov- Sinai (KS) entropy etc., have been used to characterize the complexity of observed nonlinear dynamical behavior of a system. But the analysis based on application of the foregoing tools inherently assumes the system to be noise free and stationary. An alternative framework based on the idea of recurrence plot was introduced in [19] for visualization of the dynamical behavior of a system in phase space and subsequently the formalism has been extended to quantify the recurrence plots to unravel the observed complexities i.e., regular, quasi-periodic, chaotic

*Plots of topological entropies for a* ¼ 2*:*0 *and* 1*:*8 ≤*r* ≤ 2*:*6 *: (i) figure (a) with no Allee effect, (ii) figure (b) when ε* ¼ 4*:*5, *μ* ¼ 0*, figure (c) with Allee effect on predator only μ* ¼ 0*:*1 *, (iv) when ε* ¼ 4*:*5, *μ* ¼ 01*.*

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where *xi*, *i* ¼ 1, 2, *::* … , *N* refers to observed values at time *t*1, *t*<sup>1</sup> þ Δ*t*, *:* … , *t*<sup>1</sup> þ *n*Δ*t*. If the system has true dimension *m* , a sequence of vectors may be constructed

where *τ* corresponds to time lag or delay and *m* - the embedding dimension of the phase space. By considering the distances in *m*- dimensional reconstructed points, we construct a recurrence plot (RP). In fact RP is an *n* � *n* symmetrical array where a *dot* is marked at a point ð Þ *i*, *j* if **Xi** is close to another point **Xj**.

; *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>n</sup>*; *<sup>n</sup>* <sup>¼</sup> *<sup>N</sup>* � ð Þ *<sup>m</sup>* � <sup>1</sup> *<sup>τ</sup>* (11)

*P* : f g *x*1, *x*2, *x*3, ⋯, *xN* , (10)

transition etc. For a discrete time series P with N data points such that

The results obtained through bifurcation plots, **Figure 8**, and those of LCEs plots, **Figure 10**, show that the Allee effect stabilize the motion from chaos to regularity. The correlation dimension of the chaotic attractor is obtained as *<sup>D</sup><sup>c</sup>* ffi 0*:*571. Through this study we find the existence of complexity within the system, even when system behavior is regular, we find significant amount of increase in topological entropy. This implies the fact that the system may be regular but may exhibit complexity.

**Figure 11.** *Plot of correlation integral data.*

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

**Figure 12.**

The results obtained through bifurcation plots, **Figure 8**, and those of LCEs plots, **Figure 10**, show that the Allee effect stabilize the motion from chaos to regularity. The correlation dimension of the chaotic attractor is obtained as *<sup>D</sup><sup>c</sup>* ffi 0*:*571. Through this study we find the existence of complexity within the system, even when system behavior is regular, we find significant amount of increase in topological entropy. This implies the fact that the system may be regular but may

*Plots of Lyapunov exponents for a* ¼ 2*:*0, *r* ¼ 2*:*4 *and (i) figure (a) without Allee effect, (ii) figure (b) with ε* ¼ 4*:*5, *μ* ¼ 0 *, (iii) figure (c) Allee effect on predator only with μ* ¼ 0*:*1 *, (iv) Allee effect on both*

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exhibit complexity.

*populations ε* ¼ 4*:*5, *μ* ¼ 0*:*1 *.*

**Figure 10.**

**Figure 11.**

**42**

*Plot of correlation integral data.*

*Plots of topological entropies for a* ¼ 2*:*0 *and* 1*:*8 ≤*r* ≤2*:*6 *: (i) figure (a) with no Allee effect, (ii) figure (b) when ε* ¼ 4*:*5, *μ* ¼ 0*, figure (c) with Allee effect on predator only μ* ¼ 0*:*1 *, (iv) when ε* ¼ 4*:*5, *μ* ¼ 01*.*
