This work is licensed under the Creative Commons Attribution 4.0 Public License.
The non-linear dynamical system is well known for its various applications such as in population growth, economics and so on. In the early 1963, Lorenz introduced the first weather forecasting dynamical system [1]. He showed that small changes in the initial points of the weather system could alter the outcome with significantly surprising results. This phenomena is known as the “butterfly effect” as the system is dependent on its initial conditions. There are plenty of research has been carried out in order to understand the chaotic behavior of the chaotic dynamical systems [2, 3, 4, 5, 6, 7, 8,9,10]. Edward Lorenz has published reports of a “strange attractor” where he discovered this attractor as a result of using computers to find the approximate numerical solutions to a system of differential equations in the weather model. Centering this classical model, there are several modifications of the model are introduced and studied their chaotic behaviors [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Some of the previous work by Zhou et al., Qi et al. and Yan have presented the modified Lorenz system and has discussed the system in terms of stability and dynamical behavior [21, 22].
Over the decades chaos theory has matured as a science that has given us an intense insights of some natural phenomena of the nonlinear systems. The term chaos is coined formally by Li and Yorke in 1975, where they established a simple criterion for chaos
in one-dimensional difference equations, the well-known “period three implies chaos” [23]. There are a lot of applications of chaos in circuit design and many other field including various biological systems. It is well accepted among modern mathematicians that a chaotic dynamical system is not only important but also useful from the very pure mathematics views [24].
In the present study, a variant model is defined which corresponds to a natural number n, we call it as the degree of the system and the dynamical system Da,r,b[n] is defined as follows:
where x, y and z are real variables and a, r and b are real parameters. Obviously the above system Eq.(1) is dissipative with an exponent contraction rate of
since the $\begin{array}{}
div V=\frac{\delta}{\delta x} (\frac{dx}{dt})+\frac{\delta}{\delta y} (\frac{dy}{dt})+\frac{\delta}{\delta z} (\frac{dz}{dt})=-a-b \lt 0
\end{array} $ where V is volume. It is worth mentioning that the modified model when n = 2 is studied in [25].
This new chaotic model is studied here by finding the fixed points, the eigenvalues of the Jacobian, and the Lyapunov exponents. The numerical simulations, the time series analysis, and the projections to the xy-plane, xz-plane, and yz-plane are conducted to highlight the chaotic behaviour. Also the role of the degree of system is taken into account to see the dominance behavior in characterizing the dynamical behavior of the system. This study fetches a new chaotic attractor, found by modification of the Lorenz system by a n-degree term. The detailed numerical and theoretical analysis reveals that the proposed system shows chaotic behaviour and the property of a two-scroll attractor like the Lorenz attractor.
It is has been seen in literatures that modified Lorenz system have stumbled upon or sought to discover the applications in reality. Tigan [26] shows a promising modified Lorenz system that has the potential application in secure communications. There are a lot of applications that chaos can be applied to other science and engineering problems [27, 28, 29, 30].
Stability of the Fixed Points of the Model Da,r,b[n]
Prior to the local stability analysis of the fixed points, let us see how does the 3D phase portrait looks like for different values of the degree (n) when the parameters a, r and b are fixed. The 3D phase portrait can be built by analogy with 2D as shown in the following Fig. 1. It is noted that the parameters are set to be a = 2, r = 4, b = 1 and the variable (x, y, z) ∈ [−10, 10] × [−10, 10] × [−10, 10] ⊂ ℝ3 with n = 1, 34, 67 and 100. In the first, second and third rows of the Fig.1 the x-stacked planes, y-stacked planes and z-stacked planes respectively are shown.
Fig. 1
First, second and third rows from top, x-stacked planes, y-stacked planes and z-stacked planes respectively are shown.
Now we shall describe the local stability of the fixed points of the model Eq.(1). The fixed points can be obtained by setting the derivatives Eq.(1) equal to zero.
The system Eq.(1) has three fixed points $\begin{array}{}
(0,0,0), \,(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)\text{ and }(\sqrt{b} \sqrt{(b r)^{1/n}}, \frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r).
\end{array} $
Here the Jacobian evaluated at the above fixed point say (x*,y*, z*) has the form
$$\begin{array}{}
\displaystyle \left(\begin{array}{ccc}-a & a b & 0 \\r-z^* & 0 & -x^* \\n y^* (x^* y^*)^{n-1} & n x^* (x^* y^*)^{n-1} & -b \\\end{array}\right)
\end{array} $$
In order to understand the local behaviour of the fixed points one needs to determine the signs of the real parts of the eigenvalues of the Jacobian evaluated at the fixed point. The following Theorem 1.1 is useful for checking the signs of the real parts of the eigenvalues of a 3 × 3 matrix.
Theorem 1.1
Routh-Hurwitz Criterion
The Routh-Hurwitz test [9] applied to a general third degree polynomial
states that the number of sign changes in the sequence {a3, a2, H, a0} whereH = a2a1 − a3a0is equal to the number of roots of the polynomial having positive real part, and if all entries in the sequence are nonzero and of the same sign, then all roots have negative real part.
Stability of the Fixed Point (0,0,0) of the System Da,r,b[n]
The Jacobian about the equilibrium point (0,0,0) of the system Eq.(1) is
and it has three eigenvalues $\begin{array}{}
-b,\,\frac{1}{2} \left(-\sqrt{a} \sqrt{a+4 b r}-a\right)\text{ and }\frac{1}{2} \left(\sqrt{a} \sqrt{a+4 b r}-a\right).
\end{array} $
Theorem 1.2
The equilibrium point (0,0,0) of the system Eq.(1) is locally asymptotically stable if
$$\begin{array}{}
\displaystyle a \gt 0, b \gt 0, -\frac{a}{4 b}\leq r \lt 0
\end{array} $$
Proof
It can be easily seen by doing a simple algebraic calculation that the eigenvalues are negative provided
$$\begin{array}{}
\displaystyle a \gt 0, b \gt 0, -\frac{a}{4 b}\leq r \lt 0
\end{array} $$
satisfied. Then the equilibrium point (0,0,0) is locally asymptotically stable.□
We have fetched a set of parameters (a,r,b) ∈ ℝ3 such that the origin is locally asymptotically stable. The 3D plot of the parameter subspace is given below in the Fig.2.
Fig. 2
Parameter Subspace of ℝ3 such that the origin is locally asymptotically stable.
It is importantly noted that there is no parameter (a,b,r) in the positive octant of ℝ3. If both a and b are positive then necessarily the parameter r has to be negative in order to get local asymptotically stability of the origin.
Here we illustrate some example of stability and instability of the equilibrium point (0,0,0). For a = 267, r = $\begin{array}{}
-\frac{267}{116}
\end{array} $ and b = 29, the equilibrium point becomes (0, 0, 0) which is locally asymptotically stable as shown in Fig.3.
Fig. 3
Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (0,0,0) where the initial values are taken from the neighbourhood of the origin.
For the parameters a = 56, r = 15 and b = 16 the equilibrium point (0,0,0) is unstable and forming strange attractor. The trajectories are shown in Fig.4.
Fig. 4
Unstable trajectories which are away from the origin in the system D56,15,16[6].
For the parameters a = 26, r = 93 and b = 36 the equilibrium point (0,0,0) is unstable and forming strange attractor. The trajectories are shown in Fig.5.
Fig. 5
Unstable trajectories (limit cycle) which are away from the origin in the system D26,93,36[5].
For the system D9,7,54[16], the eigenvalues of the Jacobian are −63, −54 and 54 where real part of the eigenvalues are not all negative and hence the equilibrium point (0,0,0) is unstable. The trajectories are shown in Fig.6 with many initial values taken from the neighbourhood of the equilibrium point (0,0,0).
Fig. 6
Unstable trajectories (limit cycle) which are away from the origin in the system D9,7,54[16].
Stability of the Fixed Point $\begin{array}{}
(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ of the System Da,r,b[n]
The Jacobian about the equilibrium point $\begin{array}{}
(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ of the system Eq.(1) is
$$\begin{array}{}
\left(\begin{array}{ccc}-a & a b & 0 \\0 & 0 & \sqrt{b} \sqrt{(b r)^{1/n}} \\-\frac{1.^{n-1} n \left((b r)^{1/n}\right)^{n-\frac{1}{2}}}{\sqrt{b}} & -1.^{n-1} \sqrt{b} n \left((b r)^{1/n}\right)^{n-\frac{1}{2}} & -b \\\end{array}\right)
\end{array} $$
Theorem 1.3
The equilibrium point$\begin{array}{}
(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $of the system Eq.(1) is locally asymptotically stable ifn > $\begin{array}{}
[\frac{a^2+a b}{3a b r+b^2 r}]
\end{array} $wherea, bandr ∈ ℕ. Here [] is the greatest integer function.
Proof
The equilibrium point $\begin{array}{}
(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ of the system Eq.(1) is locally asymptotically stable if real part of all the eigenvalues are negative.
Here the characteristic polynomial of the above Jacobian is
Assuming all the parameters are positive, the coefficients are
$$\begin{array}{c}
\displaystyle a_3=-1 \lt 0\\\\\\\\a_2=-a-b \lt 0\\\\\\\\a_1=b n \left((b r)^{1/n}\right)^n-a b\\\\\\\\a_0=-2a b n \left((b r)^{1/n}\right)^n
\end{array} $$
and hence
$$\begin{array}{}
\displaystyle H=b n (-3 a-b) \left((b r)^{1/n}\right)^n+a b (a+b)
\end{array} $$
By the Theorem 1.1, if sign of H as well as a0 are both negative then the equilibrium point $\begin{array}{}
(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ would be locally asymptotically stable since already a3 and a2 are negative.
It is easy to reduce by simple algebra, the coefficient a0 and H will be negative only if n > $\begin{array}{}
[\frac{a^2+a b}{3a b r+b^2 r}]
\end{array} $ where a, b and r ∈ ℕ.
Hence the result follows.□
For the system D15,17,28[6], the eigenvalues of the Jacobian are −6.50559+282.764 i,−6.50559-282.764 i,−29.9888 where real part of the eigenvalues are all negative and hence the equilibrium point (−8.84531,−0.315904,17) is locally asymptotically stable. The trajectories are shown in Fig.7 with many initial values taken from the neighbourhood of the equilibrium point (−8.84531,−0.315904,17).
Fig. 7
Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (−8.84531,−0.315904,17) where the initial values are taken from the neighbourhood of the origin.
For the system D52,41,9[15], the eigenvalues are 17.7339 +231.063 i,17.7339 −231.063 i,−96.4679 where real part of the eigenvalues are not all negative and hence the equilibrium point (3.65333, 0.405925, 41) is unstable. The trajectories are shown in Fig.8 with many initial values taken from the neighbourhood of the equilibrium point (3.65333, 0.405925, 41).
Fig. 8
Unstable trajectories which are away from the equilibrium point (3.65333, 0.405925, 41) in the system D52,41,9[15].
Stability of the Fixed Point $\begin{array}{}
(\sqrt{b} \sqrt{(b r)^{1/n}}, \frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ of the System Da,r,b[n]
The Jacobian about the equilibrium point $\begin{array}{}
(\sqrt{b} \sqrt{(b r)^{1/n}}, \frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ of the system Eq.(1) is
$$\begin{array}{}
\left(\begin{array}{ccc}-a & a b & 0 \\0 & 0 & -\sqrt{b} \sqrt{(b r)^{1/n}} \\\frac{n \left((b r)^{1/n}\right)^{n-\frac{1}{2}}}{\sqrt{b}} & \sqrt{b} n \left((b r)^{1/n}\right)^{n-\frac{1}{2}} & -b \\\end{array}\right)
\end{array} $$
Theorem 1.4
The equilibrium point$\begin{array}{}
(\sqrt{b} \sqrt{(b r)^{1/n}}, \frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $of the system Eq.(1) is locally asymptotically stable ifn > $\begin{array}{}
[\frac{a^2+a b}{a b r-b^2 r}]
\end{array} $wherea, bandr ∈ ℕ. Here [] is the greatest integer function.
Proof
The equilibrium point $\begin{array}{}
(\sqrt{b} \sqrt{(b r)^{1/n}}, \frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ of the system Eq.(1) is locally asymptotically stable if real part of all the eigenvalues are negative.
Here the characteristic polynomial of the above Jacobian is
Assuming all the parameters are positive, the coefficients are
$$\begin{array}{c}
\displaystyle a_3=-1 \lt 0\\\\\\\\a_2=-a-b \lt 0\\\\\\\\a_1=-b \left(a+n \left((b r)^{1/n}\right)^n\right)\\\\\\\\a_0=-2a b n \left((b r)^{1/n}\right)^n
\end{array} $$
and hence
$$\begin{array}{}
\displaystyle H=b n ( b- a) \left((b r)^{1/n}\right)^n+a b (a+b)
\end{array} $$
By the Theorem 1.1, if sign of H as well as a0 are both negative then the equilibrium point $\begin{array}{}
(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ would be locally asymptotically stable since already a3 and a2 are negative.
As before it is easy to reduce by simple algebra, the coefficient a0 and H will be negative only if n > $\begin{array}{}
[\frac{a^2+a b}{a b r-b^2 r}]
\end{array} $ where a, b and r ∈ ℕ.
Hence the result follows.
□
It is noted that both the Theorem 1.3 and Theorem 1.4 hold good for certain ranges of parameters a,r and b involved in the system. Here computationally we have fetched a set of parameters (a, r, b) ∈ R3 which is plotted in the following Fig.9.
Fig. 9
Parameter Plot
For the system D15,17,28[6], the eigenvalues are −6.50559 + 282.764i, −6.50559 − 282.764i, −29.9888 where real part of the eigenvalues are all negative and hence the equilibrium point (8.84531, 0.315904, 17) is locally asymptotically stable. The trajectories are shown in Fig.10 with many initial values taken from the neighbourhood of the equilibrium point (8.84531, 0.315904, 17).
Fig. 10
Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (8.84531, 0.315904, 17) where the initial values are taken from the neighbourhood of the origin.
For the system D52,41,9[13], the eigenvalues are 17.3142 +215.992 i,17.3142 −215.992 i,−95.6285 where real part of the eigenvalues are not all negative and hence the equilibrium point (3.76576, 0.418418, 41) is unstable. The trajectories are shown in Fig.11 with many initial values taken from the neighbourhood of the equilibrium point (3.76576, 0.418418, 41).
Fig. 11
Unstable trajectories which are away from the equilibrium point (3.76576, 0.418418, 41) in the system D52,41,9[13].
Trajectories Away from the Fixed points of the System Da,r,b[n]
Here we set parameters a = 52, r = 41 and b = 9 and also chosen three different degree n = 13, 21 and 14. For any initial values taken from the neighbourhood of the origin, the system D52,41,9[13] possesses to a limit cycle. Similarly, For any initial values taken from the neighbourhood of the other two equilibrium points $\begin{array}{}
(\sqrt{b} \sqrt{(b r)^{1/n}}, \frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $ and $\begin{array}{}
(-\sqrt{b} \sqrt{(b r)^{1/n}}, -\frac{\sqrt{(b r)^{1/n}}}{\sqrt{b}}, r)
\end{array} $, the system D52,41,9[21] and D52,41,9[14] possess limit cycles respectively. This has been depicted graphically in the following Fig.11. The trajectory plots are given too in the Fig.12.
Fig. 12
Unstable trajectories which are away from the all the equilibrium points of the system D52,41,9[n].
All the parameters a = 52, r = 41 and r = 9 keeps away the trajectories from all the three equilibrium points of the system and forming limit cycles, whereas the degree of the system is controlling the repelling behavior of those equilibrium points as observed.
Dominance of Parameters Involved in the system Da,r,b[n]
In this section, we would like to experience what happens by varying the parameters involved in the system. This would computationally able to understand the dominating governing parameters among all the parameters involved.
Transiting from Convergence Behavior to Periodic Behavior of the System Da,r,b[n]
Here we shall discover while changing the degree n of the system Da,r,b[n] takes the system behavior from convergence of the trajectories to the periodic trajectories (limit cycles).
Here we have equilibrium three parameters a = 51, r = 3 and b = 4 and five different set initial values are taken randomly from the neighbourhood of the origin. Different colors in the trajectory plots represents different trajectories corresponding to different set of initial values.
In the system D51,3,4[n], we vary the degree of the system n from 1 to 15 one by one and see their corresponding trajectories which are plotted in the Fig.12.
The system D51,3,4[n] behavior according as the degree of the system n varying from 1 to 15.
Sl. No.
System D51,3,4[n]
Character of the System
1
D51,3,4[1]
converges to the equilibrium point (6.9282, 1.7321, 3)
2
D51,3,4[2]
converges to the equilibrium point (3.7225, 0.9306, 3)
3
D51,3,4[3]
converges to the equilibrium point (3.0262, 0.7565, 3)
4
D51,3,4[4]
converges to the equilibrium point (2.7285, 0.6821, 3)
5
D51,3,4[5]
Periodic with high period.
6
D51,3,4[6]
Periodic with high period.
7
D51,3,4[7]
Periodic with high period.
8
D51,3,4[8]
Periodic with high period.
9
D51,3,4[9]
Periodic with high period.
10
D51,3,4[10]
Periodic with high period.
11
D51,3,4[11]
Periodic with high period.
12
D51,3,4[12]
Periodic with high period.
13
D51,3,4[13]
Periodic with high period.
14
D51,3,4[14]
Periodic with high period.
15
D51,3,4[15]
Periodic with high period.
Transiting from Periodic Behavior to Chaotic Behavior of the System Da,r,b[n]
Here in the system D76,36,31[n], we vary the degree of the system n from 1 to 15 one by one and see their corresponding trajectories which are plotted in the Fig.13. The chaotic behavior is detected through the positive Lyapunov exponents of the three dimensional trajectory as per the algorithm described in the article [31].
Fig. 13
Trajectories (Up) and three dimensional plots (Down) respectively for the system D51,3,4[n] where the degree of the system n varying from 1 to 15.
It is seen by computation that the system is possessing to a periodic trajectory only if n = 1 otherwise for all the other values of n ranging from 2 to 15 the system approaches either to the chaotic trajectories or it is unbounded. Now we consider a set of parameters a=3, r=12 and b=24 and varying n from 1 to 10. The trajectories and their corresponding 3D plots for each of the system with different degree ranging from 1 to 10 are given in the following Fig. 14.
Fig. 14
Trajectories are plotted for the system D76,36,31[n] where the degree of the system n varying from 1 to 15.
Fig. 15
Trajectories and 3D plots are plotted for the system D3,12,24[n] where the degree of the system n varying from 1 to 10.
Transition of the dynamical behavior of the system D51,3,4[n] according as the degree of the system n varying from 1 to 15.
Sl. No.
System D51,3,4[n]
Character of the System
1
D76,36,31[1]
Periodic with high period.
2
D76,36,31[2]
Chaotic with Positive Lyapunov exponent 0.6324
3
D76,36,31[3]
Chaotic with Positive Lyapunov exponent 0.7634
4
D76,36,31[4]
Chaotic with Positive Lyapunov exponent 0.4589
5
D76,36,31[5]
Chaotic with Positive Lyapunov exponent 0.3412
6
D76,36,31[6]
Chaotic with Positive Lyapunov exponent 0.2875
7
D76,36,31[7]
Chaotic with Positive Lyapunov exponent 0.2337
8
D76,36,31[8]
Chaotic with Positive Lyapunov exponent 0.5643
9
D76,36,31[9]
Chaotic with Positive Lyapunov exponent 0.2981
10
D76,36,31[10]
Chaotic with Positive Lyapunov exponent 0.6873
11
D76,36,31[11]
Chaotic with Positive Lyapunov exponent 0.5438
12
D76,36,31[12]
Chaotic with Positive Lyapunov exponent 0.9865
13
D76,36,31[13]
Chaotic with Positive Lyapunov exponent 0.7541
14
D76,36,31[14]
Chaotic with Positive Lyapunov exponent 0.6987
15
D76,36,31[15]
Chaotic with Positive Lyapunov exponent 0.3086
We have observed that for n = 1,2,3,4 and 5 the system D3,12,24[n] converges to the equilibrium point (83.1384, 3.4641, 12), (20.1815, 0.8409, 12), (12.5894, 0.5246, 12), ( 9.9433, 0.4143, 12) and ( 8.6307, 0.3596, 12) respectively. For any other values of n from 6 onwards (up to 10 are shown in Fig.14) the system possesses high periodic trajectories (limit cycles). Hence it is understood that the degree (n) of the system plays a governing role (one of the controlling parameters of the system) in transiting the behavior of the dynamics.
Concluding Remarks & Future Endeavours
In this article, we have found a new modified Lorenz system with some computational results and analysis. Also, we have studied the dynamic behaviour of the system and basic dynamic analysis. Throughout the study the vibrant role of the degree is adumbrated through by illustrating the transition from convergence behavior to limit cycles and from limit cycles to chaotic behavior. Different set of parameters are fetched which produce chaotic trajectories/attractors. Other classical chaotic models like Rosler system, Chen system also could be modified by introducing the degree and can be studied in the similar manners.
