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Extension of Lorenz Unpredictability

It is found that Lorenz systems can be unidirectionally coupled such that the chaos expands from the drive system. This is true if the response system is not chaotic, but admits a global attractor, an equilibrium or a cycle. The extension of sensitivity and period-doubling cascade are theoretically proved, and the appearance of cyclic chaos as well as intermittency in interconnected Lorenz systems are demonstrated. A possible connection of our results with the global weather unpredictability is provided.


Introduction
In his famous study, to investigate the dynamics of the atmosphere, Lorenz [43] built a mathematical model consisting of a system of three differential equations in the following form, where σ, r and b are constants.
System (1.1) is a simplification of a model, derived by Saltzman [61], to study finite amplitude convection. The studies of Saltzman originate from the Rayleigh-Bénard convection, which describes heat flow through a fluid, like air or water. In this modelling, one considers a fluid between two horizontal plates where the gravity is assumed to be in the downward direction and the temperature of the lower plate is maintained at a higher value than the temperature of the upper one. Rayleigh [55] found that if the temperature difference is kept at a constant value, then the system possesses a steady-state solution in which there is no motion and convection should take place if this solution becomes unstable. In other words, depending on the temperature difference between the plates, heat can be transferred by conduction or by convection. Assuming variations in only x 1 − x 3 plane, Saltzman [61] considered the equations ∂ ∂t where ψ is a stream function for the two dimensional motion, θ is the departure of temperature from that occurring in the state of no convection and the constants ∆T 0 , H, g, ε, ν and κ denote, respectively, the temperature contrast between the lower and upper boundaries of the fluid, the height of the fluid under consideration, the acceleration of gravity, the coefficient of thermal expansion, the kinematic viscosity and the thermal conductivity [43,61]. In his study, Saltzman [61] achieved an infinite system by means of applying Fourier series methods to system (1.2), and then used the simplification procedure proposed by Lorenz [42] to obtain a system with finite number of terms. Lorenz [43] set all but three Fourier coefficients equal to zero and as a consequence attained system (1.1), which describes an idealized model of a fluid.
In system (1.1), the variable x 1 is proportional to the circulatory fluid flow velocity, while the variable x 2 is proportional to the temperature difference between the ascending and descending currents. Positive The variable x 3 , on the other hand, is proportional to the distortion of the vertical temperature profile from linearity, a positive value indicating that the strongest gradients occur near the boundaries. The parameters σ and r are called the Prandtl and Rayleigh numbers, respectively [11,43,65].
The dynamics of the Lorenz system (1.1) is very rich. For instance, with different values of the parameters σ, r and b, the system can exhibit stable periodic orbits, homoclinic explosions, perioddoubling bifurcations, and chaotic attractors [65].
The appearance of chaos in differential/discrete equations may be either endogenous or exogenous.
As the first type of chaos birth, one can take into account the irregular motions that occur in Lorenz, Rössler, Chua systems, the logistic map, Duffing and Van der Pol equations [18,19,29,40,58,66]. To indicate the endogenous irregularity, we use: (i) ingredients of Devaney and Li-Yorke chaos, (ii) perioddoubling route to chaos, (iii) intermittency, (iv) positive Lyapunov exponents. Symbolic dynamics and Smale horseshoes have been widely used for that purpose [19,21,36,37,40,54,64,69]. While the endogenous chaos production is widespread and historically unique, the exogenous chaos as generated by irregular perturbations has not been intensively investigated yet. In our study, we will appeal to endogenous chaos, but mostly to exogenous chaos.
In this paper, the main attention is given to the extension of chaos among interconnected Lorenz systems. We make use of unidirectionally coupled Lorenz systems such that the drive system is chaotic and the response system possesses a stable equilibrium or a limit cycle. We theoretically prove that the chaos of the drive system makes the response system behave also chaotically. Extension of sensitivity and period-doubling cascade are rigorously approved. The appearance of cyclic irregular behavior is discussed, and it is shown that the phenomenon cannot be explained by means of generalized synchronization.
Intermittency in coupled Lorenz systems is also demonstrated.
The principal novelty of our investigation is that we create exogenous chaotic perturbations by means of the solutions of a chaotic Lorenz system, plug it into a regular Lorenz system, and find that chaos is inherited by the solutions of the latter. Such an approach has been widely used for differential equations before, but for regular disturbance functions. That is, it has been shown that an (almost) periodic perturbation function implies the existence of an (almost) periodic solution of the system.
While the literature on chaos synchronization [1,25,33,35,38,47,53,60] has also produced methods of generating chaos in a system by plugging in terms that are chaotic, it relies on the asymptotic convergence between the chaotic exogenous terms and the solution of the response system for the proof of chaos creation. Instead, we provide a direct verification of the ingredients of chaos for the perturbed system [2]- [10]. Moreover, in Section 6 we represent the appearance of cyclic chaos, which cannot be reduced to generalized synchronization. Very interesting examples of applications of discrete dynamics to continuous chaos analysis were provided in the papers [14]- [17]. In these studies, the general technique of dynamical synthesis [14] was developed, and this technique was used in the paper [4].
There are many published papers which have results about chaos considering first of all its mathematical meaning. This is true either for differential equations [41,69] or data analysis [22]. Apparently there are still few articles with meteorological interpretation of chaos ingredients. We suppose that our rigorously approved idea for the extension of chaotic behavior from one Lorenz system to another will give a light for the justification of the erratic behavior observed in dynamical systems of meteorology.
The question "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" is very impressive and it has done a lot to popularize chaos for both mathematicians and non-mathematicians [45].
From this question one can immediately decide that the butterfly effect is a global phenomenon, and consequently, the underlying mathematics has to be investigated. Some of the authors say that the question relates sensitive dependence on initial conditions in dynamical systems considered as unpredictability for meteorological observations. Lorenz himself, in successive his talks and the book [45], was obsessed by the question and sincerely believed its possibility. He also supposed that his system can give a key for the positive answer of the question. Generally, analysis of chaotic dynamics in atmospheric models is rather numerical [23,34,39,46,50] or depend on the observation of time-series [26,27]. In Section 8, we describe how one can use the rigorously approved results of the present paper to investigate the global behavior of the weather unpredictability. Our suggestions are not about a modelling, but rather an effort to answer the question why the weather is unpredictable at each point of the Earth, on the basis of the Lorenz's meteorological model and other models. We should recognize that all our discussions can be considered as a "toy object" in the theory, and according to the complexity phenomenon in meteorological investigations one can say that the investigation of chaos in meteorology still remains as a "toy object" [28,43].

Coupled Lorenz Systems
We couple Lorenz systems unidirectionally in such a way that the existing chaos propagates from one to another. We suppose that the coefficients σ, r and b are properly chosen in (1.1) so that the system is chaotic. In addition to system (1.1), we consider another Lorenz system, where the parameters σ, r and b are such that the system is non-chaotic. That is, the system does not possess chaotic motions such that, for example, it admits a global asymptotically stable equilibrium or a globally attracting limit cycle.

Extension of Sensitivity
In this section, we will demonstrate that the divergence of two initially nearby solutions (sensitivity) in the driving chaotic Lorenz system (1.1) leads to the presence of the same feature in system (2.4).
Additionally, a third Lorenz system will be considered in order to show the maintainability of the process.
The mathematical description of sensitivity and a theoretical proof for its extension are presented in the Appendix.
To reveal numerically the extension of sensitivity in system (3.5), we represent in Figure   Now, we consider the system System (3.6) is also in the form of (2.4), but this time the perturbations h 1 (y(t)) = 12y 1 (t), h 2 (y(t)) = 20[y 2 (t) + 2 arctan(y 2 (t)/5)] and h 3 (y(t)) = −8y 3 (t) are provided by the solutions of (3.5).   Figure 3 manifest not only the chaos extension, but also the existence of a chaotic attractor in the 9−dimensional phase space. It is worth noting that the projection on the x 1 − x 2 − x 3 space is the classical Lorenz attractor [43,65].
The next section is devoted to the extension of chaos obtained through period-doubling cascade.

Extension of Period-Doubling Cascade
Consider the Lorenz system (1.1) in which σ = 10, b = 8/3 and r is a parameter [24,65]. For the values of r between 99.98 and 100.795 the system possesses two symmetric stable periodic orbits such that one of them spirals round twice in x 1 > 0 and once in x 1 < 0, whereas another spirals round twice in x 1 < 0 and once in x 1 > 0. The book [65] calls such periodic orbits as x 2 y and y 2 x, respectively. More precisely, "x" is written every time when the orbit spirals round in x 1 > 0, while "y" is written every time when it spirals round in x 1 < 0. As r decreases towards 99.98 a period-doubling bifurcation occurs in the system such that two new symmetric stable periodic orbits (x 2 yx 2 y and y 2 xy 2 x) appear and the previous periodic orbits lose their stability [24,65]. According to Franceschini [24], system (1.1) undergoes infinitely many period-doubling bifurcations at the parameter values 99.547, 99.529, 99.5255 and so on. The sequence of bifurcation parameter values accumulates at r ∞ = 99.524. For values of r smaller than r ∞ infinitely many unstable periodic orbits take place in the dynamics of (1.1) [24,65].
To extend the period-doubling cascade of (1.1), we take into account the system where ) is a solution of (1.1). Note that in the absence of driving, system (4.7) admits a stable equilibrium point, i.e., system (2.3) with σ = 10, r = 0.27 and b = 8/3 does not admit chaos.
By using Theorem 15.8 [70], one can verify that for each periodic x(t), there exists a periodic solution of (4.7) with the same period.
In Figure 4, we illustrate the stable periodic orbits of systems (1.1) and (4.7). Next, we continue with the extension of period-doubling cascade in Figure 5. The projection of the trajectory of system (1.1) with r = 99.51 corresponding to the initial data x 1 (0) = 10.58, x 2 (0) = 28.19, Figure 5, (a). Making use of the initial data y 1 (0) = 2.23, y 2 (0) = 1.26, y 3 (0) = 9.64, the projection of the corresponding trajectory of (4.7) on the y 1 − y 3 plane is depicted in Figure 5, (b). Moreover, the irregular behavior of the y 3 coordinate over time is illustrated in Figure 6. The simulation results reveal that the period-doubling cascade of (1.1) is extended to (4.7).
A theoretical investigation of the extension of period-doubling cascade is provided in the Appendix. In the next section, the extension of intermittency is considered.

Extension of Intermittency
Pomeau and Manneville [54] observed intermittency in the Lorenz system (1.1), where σ = 10, b = 8/3 and r is slightly larger than the critical value r c ≈ 166.06. Let us use r = 166.3 in system (1.1) such that intermittency is present. We perturb system (2.3), where σ = 10, r = 0.35, b = 8/3, with solutions of (1.1), and set up the following system, The graphs of the y 1 , y 2 and y 3 coordinates of (5.8) are shown in Figure 7 by making use of the initial For some parameter values, the Lorenz system can exhibit limit cycles [65]. In the next section, we will consider the Lorenz system (2.3) with a globally attracting limit cycle and verify numerically how to achieve a motion that behaves chaotically and cyclically in the same time.

Cyclic Chaos
In our previous illustrations, we considered system ( Making use of the solution of (1.1) corresponding to the initial data x 1 (0) = 5.71, x 2 (0) = 9.01, x 3 (0) = 17.06, we depict the trajectory of (6.9) with y 1 (0) = −21.67, y 2 (0) = 34.33, y 3 (0) = 346.38 in Figure 8, (a). The projection of the same trajectory on the y 1 − y 2 plane is shown in Figure 8, (b). It is seen in both figures that the trajectory behaves chaotically around the limit cycle of (2.3). To confirm one more time that the trajectory considered in Figure 8 is essentially chaotic, the graph of the y 2 coordinate of system (6.9) is illustrated in Figure 9. Although system (2.3) possesses a globally attracting limit cycle, the simulations seen in Figure 8 and Figure 9 indicate that the applied perturbation makes the system (6.9) behave chaotically. In other words, the chaotic behavior is seized by the limit cycle of system (2.3), and as a result a motion which behaves both chaotically and cyclically appears.  Figure 9: The time-series for the y 2 −coordinate of system (6.9). The picture confirms the chaotic behavior of the motion. The remaining coordinates of system (6.9), which are not just pictured here, behave also chaotically.
In order to compare our approach with that of generalized synchronization [1,25,33,38,60], let us apply the auxiliary system approach [1,25] to the couple (1.1) + (6.9). The corresponding auxiliary system is (6.10) The projection of the stroboscopic plot of the 9−dimensional system (1.1) + (6.9) + (6.10) on the y 2 − z 2 plane is depicted in Figure 10.  Figure 10 that the stroboscopic plot is not on the line z 2 = y 2 . Therefore, we conclude that generalized synchronization does not take place in the dynamics of the couple (1.1) + (6.9).  Figure 10: Application of the auxiliary system approach to system (1.1)+ (6.9) indicates that generalized synchronization does not exist for the couple.
Another approach to investigate the presence or absence of generalized synchronization is the evalu-ation of conditional Lyapunov exponents [25,38,53].

Self-organization and Synergetics in Lorenz Systems
To illustrate the extension of chaos in large collections of interconnected Lorenz systems, let us introduce the following 27−dimensional system consisting of the subsystems S 1 , S 2 , . . . , S 9 : The coefficients of S 1 are chosen in such a way that the system is chaotic [43]. The systems S 2 , S 3 , . . . , S 9 are designed such that if the corresponding perturbations x(t), y(t), z(t), w(t) are chaotic, then the systems possess chaos. However, in the absence of the perturbations, S 2 , S 3 , . . . , S 9 admit stable equilibria and they are all non-chaotic. The connection topology of the systems S 1 , S 2 , . . . , S 9 is represented in Figure 11. On the other hand, Figure 12 depicts the chaotic attractors corresponding the each S i , i = 1, 2, . . . , 9, such that collectively the picture can be considered as the chaotic attractor of the whole 27−dimensional system. One can confirm that Figure 12 supports our ideas such that the chaos of S 1 generates chaos in the remaining subsystems even if they are non-chaotic in the absence of the perturbations.
The idea of the transition of chaos from one system to another as well as the arrangement of chaos in an ordered way can be considered as another level of self-organization [30,52]. Durrenmatt [20] described that "... a system is self-organizing if it acquires a spatial, temporal or functional structure without specific interference from the outside. By "specific" we mean that the structure of functioning is not impressed on the system, but the system is acted upon from the outside in a nonspecific fashion." There are three approaches to self-organization, namely thermodynamic (dissipative structures), synergetic and the autowave. For the theory of dynamical systems (e.g. differential equations) the phenomenon means that an autonomous system of equations admits a regular and stable motion (periodic, quasiperiodic, almost periodic). In the literature, this is called as autowaves processes [12] or self-excited oscillations [49]. We are inclined to add to the list another phenomenon, which is a consequence of the chaos extension. Consider the subsystems S 1 , S 2 , . . . , S 9 once again (in general, arbitrary finite number of systems can be considered). Because of the connections and the conditions discovered in our analysis (see Appendix), likewise S 1 all the other subsystems, S i , i = 2, 3, . . . , 9, are also chaotic. We suppose that this is a self-organization. This phenomenon can be restricted only for autonomous systems or it can be even interpreted for non-autonomous systems, too. So, we can say that the extension of unpredictability is an example of self-organization, that is a coherent behavior of a large number of systems [30].
In his fascinating paper, the German theoretical physicist Hermann Haken [30] introduced a new interdisciplinary field of science, synergetics, which deals with the origins and the evolution of spatio-temporal structures. The profound part of synergetics is based on the dynamical systems theory. Depending on the discussion of our manuscript, it is natural that we concentrate on the differential equations, and everything that will be mentioned below about synergetics concerns first of all dynamical systems with mathematical approach. One of the main features of systems in synergetics is self-organization, which has been discussed above, and we approved that the phenomenon is present in the extension of chaos among Lorenz systems. According to Haken [30], the central question in synergetics is whether there are general principles that govern the self-organized formation of structures and/or functions. The main principles by the founder of the theory are instability, order parameters, and slaving [30]. Instability is understood as the formation or collapse of structures (patterns). This is very common in fluid dynamics, lasers, chemistry and biology [30,31,32,51,68]. A number of examples of instability can be found in the The phenomenon is called as instability because the former state of fluid transforms to a new one, loses its ability to persist, and becomes unstable. We see instability in the chaos extension, as consecutive chaotification of systems S 2 , S 3 , . . . , S 9 joined to the source S 1 of chaos. The concepts of the order parameter and slaving are strongly connected in synergetics. For differential equations theory, order parameters mean those phase variables, whose behavior formate the main properties of a macroscopic structure, which dominate over all other variables in the formation such that they can even depend on the order parameters functionally. The dependence that is proved (discovered) mathematically is what we call as slaving. It is not difficult to see in the chaos extension mechanism that the variables of the system S 1 are order parameters, and they determine the chaotic behavior of the joined systems' variables.
That is, the slaving principle is present here.
The next section is devoted to the possible connections of our results about interconnected Lorenz systems with the global weather unpredictability.

Connection with Global Weather Unpredictability
Lorenz [43] was the first who discovered sensitivity with the aid of system (1.1) and then made the conclusion on the butterfly effect. Nowadays, there is an agreement that the butterfly effect exists, if we mean sensitivity=unpredictability. It seems that Lorenz himself believed that sensitivity discovered in his equation is a strong indicator of the butterfly effect in its original meteorological sense. Possibly his intuition is based on the idea that the system of ordinary differential equations is derived from a system of partial differential equations. There should be a deeper interpretation for the effect of chaotic dynamics in the three dimensional system on the infinite dimensional one. We also believe that the opinion of Lorenz, who considered his results as an evidence of the meteorological butterfly effect, is very reasonable. Moreover, his claim has to be considered as a challenging problem for mathematicians. If one thinks positively on the subject, then by our opinion several next questions emerge. The first one is whether sensitivity in meteorological models is a reflection of the butterfly effect. Definitely, this question needs a thoroughly investigation. Possibly it requests a deep analysis on the basis of ordinary and partial differential equations. The problem is not solved in this section at all. We axiomatize somehow the state assuming that the butterfly effect is sensitivity in the mathematical sense or, more generally, chaos.
We can also reduce the question by considering the problem of unpredictability through sensitivity.
Consequently, the following questions are reasonable: Can one explain the global unpredictability of weather by applying models similar to the Lorenz system? How can Lorenz systems be utilized for a global description of the weather?
The physical properties of the atmosphere are not the same throughout the Earth. The tropical atmosphere possesses considerably different behavior from those in the temperate and polar latitudes, as if it were a different fluid [45]. Taking inspiration from its multifaceted structure, we propose to consider the atmosphere divided into subregions such that the dynamic properties of each region differs considerably from the others. In this case, one can suppose that the dynamics of each subregion of the atmosphere subjects to its own Lorenz system. That is, for different subregions the coefficients of the corresponding Lorenz system are different. Since for some parameter values chaos can take place in the Lorenz system and for some not, such chaotic or non-chaotic motions should have prolonged forever, conflicting the realistic dynamics of the atmosphere, where global unpredictability is present. To extend our attitude for the butterfly effect, we propose that instability, which may occur in a subregion, can be imported to neighbor subregions of the atmosphere, such that chaos occurs not only endogenously, but also exogenously. In other words, exterior perturbations influencing a part of the atmosphere may cause a chaotic behavior to occur in that region. In addition to this, we suppose that these perturbations most probably originate through the neighboring regions within the atmosphere, and the dynamics of coupled (iv) The Lorenz systems are connected unidirectionally.
Let us localize the global process by taking into account only two adjacent subregions of the atmosphere, labeled A and B. In the beginning, the subregion A is assumed to be chaotic, while the subregion B is non-chaotic. By the phrase "chaotic subregion" we mean that the coefficients of the corresponding Lorenz system are such that the system possesses a chaotic attractor. In a similar way, one should understand from the phrase "non-chaotic subregion" that the corresponding Lorenz system does not exhibit chaotic motions such that, for example, it admits a global asymptotically stable equilibrium or a globally attracting limit cycle.
Suppose that the dynamics of A is described by the chaotic Lorenz system (1.1). Besides, system (2.4) represents the dynamics of B after the transmission of chaos, and system (2.3) represents the dynamics of B before the process is carried out. According to the theoretical results of the present paper, the chaos of the subregion A influences the subregion B in such a way that the latter also becomes chaotic even if it is initially non-chaotic. The propagation mechanism is represented schematically in Figure 13.
By chaos propagation, we mean the process of unidirectional coupling of Lorenz systems. Figure 13, (a) illustrates the dynamics during the transmission of chaos. After the transmission of unpredictability is achieved, the dynamics of both subregions, A and B, exhibit chaotic behavior as shown in Figure 13, The mentioned local process can be maintained by considering more subregions, whose dynamics are also described by Lorenz systems. For instance, one can suppose that the systems S 1 , S 2 , . . . , S 9 defined in Section 7 reflect the dynamics of the weather in nine different subregions. The applied perturbations may not be in accordance with realistic air flows in the atmosphere. However, the exemplification reveals the propagation of unpredictability and indicate the possibility for the usage of different types of perturbations in the systems. We do not take into account changes which may happen because of the day light evolution, variety of land forms, seasonal differences in the region etc., but what we propose is to connect regional mathematical models into a global net so that understanding the unpredictability becomes possible. We make use of "toy" perturbations due to the lack of preexisting ones, which should be found through experimental investigations. What we propose in this section is a small step in the mathematical approach to the complexity of the weather.
It is worth noting that the chaotification principles proposed in this paper are not specific for the where X 1 represents the strength of a large scale westerly wind current, X 2 and X 3 represent the cosine and sine phases of a chain of superposed large-scale eddies, the parameter F represents the externalheating contrast, and G represents the heating contrast between oceans and continents. The coefficient b, if greater than unity, allows the displacement to occur more rapidly than the amplification, and the coefficientã, if less than unity, allows the westerly current to damp less rapidly than the eddies [13,44,48,57].

Conclusion
In the present study, we investigate the dynamics of unidirectionally coupled Lorenz systems. It is rigorously proved that chaos can be extended from one Lorenz system to another. The extension of period-doubling cascade and sensitivity, which is the main ingredient of chaos, are shown both theoretically and numerically. Besides, the emergence of cyclic chaos, intermittency, and the concepts of self-organization and synergetics are considered for interconnected Lorenz systems. The results are valid if the drive Lorenz system is chaotic and the response system is non-chaotic, but admits a global asymp-totically stable equilibrium or a globally attracting limit cycle. Our approach can give a light on the question about how global weather processes have to be described through mathematical models. A possible connection of the presented results with the global weather unpredictability is provided in the paper. The usage of our approach for the investigation of global weather unpredictability is a one more small step in the mathematical approach to the complexity of the weather. This is not a modelling of the atmosphere, but rather an effort to explain how the weather unpredictability can be arranged over the Earth on the basis of the Lorenz's meteorological model. In fact, this is also true for other meteorological models, since mathematical properties of stability, attraction and chaotic attractors are common for all models. It is shown that our results can be used for the Lorenz model of general circulation of the atmosphere [44] too. The question whether the overlapping of two chaotic dynamics may produce regularity can also be considered in future investigations. We guess that it is not possible, but an analysis has to be made.

Acknowledgments
The authors wish to express their sincere gratitude to the referees for the helpful criticism and valuable suggestions, which helped to improve the paper significantly.
The second author is supported by the 2219 scholarship programme of TÜBİTAK, the Scientific and Technological Research Council of Turkey.

Appendix: The Mathematical Background
In our theoretical discussions, we consider more general coupled systems, which are not necessarily Lorenz systems. We will denote by R and N the sets of real numbers and natural numbers, respectively, and we will make use of the usual Euclidean norm for vectors.
Let us consider the autonomous systems and where t ≥ 0 and the functions F : R m → R m and f : R n → R n are continuous in their arguments.
We perturb system (10.14) with the solutions of (10.13) and obtain the system in the form, dy dt = f (y) + µg(x(t)), (10.15) where the real number µ is nonzero and the function g : R m → R n is continuous. It is worth noting that the systems (1.1), (2.3) and (2.4) are in the form of (10.13), (10.14) and (10.15), respectively.
We mainly assume that system (10.13) possesses a chaotic attractor, let us say a set in R m . Fix x 0 from the attractor and take a solution x(t) of (10.13) with x(0) = x 0 . Since we use the solution x(t) as a perturbation in (10.15), we call it as chaotic function. Chaotic functions may be irregular as well as regular (periodic and unstable) [21,43,62,63,64,69].
Our purpose is the prove rigorously the extension of chaos from system (10.13) to system (10.15). In our theoretical discussions, we request the existence of a bounded positively invariant region for system (10.15). Such an invariant region can be achieved by different methods and one of them is mentioned in the next part. We will show the extension of sensitivity and the existence infinitely many unstable periodic solutions in Subsections 10.2 and 10.3, respectively.
In the following parts, for a given solution x(t) of system (10.13), we will denote by φ x(t) (t, t 0 , y 0 ) the unique solution of system (10.15) satisfying the initial condition φ x(t) (t 0 , t 0 , y 0 ) = y 0 .

Existence of a bounded positively invariant region
Making benefit of Lyapunov functions and uniform ultimate boundedness [59,70], we present a method in Theorem 10.1 for the existence of a bounded positively invariant set of system (10.15). Then, we will apply this technique to the Lorenz system.
Solutions of system (10.15) are uniformly ultimately bounded if there exists a number B 0 > 0 and corresponding to any number α > 0 there exists a number T (α) > 0 such that y 0 ≤ α implies that for each solution x(t) of system (10.13) and t 0 ≥ 0 we have φ x(t) (t, t 0 , y 0 ) < B 0 for all t ≥ t 0 + T (α).
The following condition is required: Then, for sufficiently small |µ| , the solutions of system (10.15) are uniformly ultimately bounded.
Proof. Fix arbitrary numbers t 0 ≥ 0, α > 0 and a solution x(t) of system (10.13). Take a number β satisfying 0 < β < b(B). We consider system (10.15) with a nonzero number µ which satisfies the Our aim is to show the existence of numbers B 0 > B and T (α) ≥ 0, independent of t 0 , such that if Consider an arbitrary y 0 ∈ R n such that y 0 ≤ α. For the sake of brevity, let us denote y(t) = φ x(t) (t, t 0 , y 0 ). In the proof, both of the possibilities y 0 < B and y 0 ≥ B will be considered. We start with the former.
Now, suppose that there exists a moment s 1 > t 0 such that y(s 1 ) ≥ B 0 . It is possible to find a moment s 2 satisfying t 0 < s 2 < s 1 such that y(s 2 ) = B and y(t) ≥ B for all t ∈ [s 2 , s 1 ].
The last inequality implies that V (y(s 1 )) < V (y(s 2 )). On the other hand, by the help of assumption ). This is a contradiction. Therefore, for all t ≥ t 0 the inequality y(t) < B 0 is valid.
Next, we consider the possibility y 0 ≥ B. Since the function V (x) is continuous and y 0 ≤ α, one can find a number K(α) > 0 such that V (y 0 ) ≤ K(α). By means of condition (i) used together with the inequality y 0 ≥ B, we have that K(α) ≥ a(B).
Assume that there exists a moment t > t 0 + K(α) − a(B) β such that y(t) ≥ B.
If there exists t 1 ∈ [t 0 , t] such that y(t 1 ) < B, then by means of uniqueness of solutions, using a similar discussion to the case y 0 < B considered above, one can show that for all t ≥ t 1 the inequality y(t) < B 0 holds. On the other hand, if for all t ∈ [t 0 , t] the inequality y(t) ≥ B is valid, then one can verify that the inequality holds. Under the circumstances we attain that This is a contradiction. Hence, for all Consequently, the solutions of system (10.15) are uniformly ultimately bounded.
In the next section, we will continue with the extension of sensitivity, which can be considered as the unique ingredient of chaos for a set of bounded solutions [43,56,69].

Unpredictability analysis
Extension of the sensitivity feature through system (10.15) will be handled in the present part. We shall begin with the meaning of the aforementioned property for systems (10.13) and ( Our main assumption is the existence of a bounded positively invariant set K for system (10.15).
The existence of such an invariant set can be shown, for example, by using Theorem 10.1.
The following assumptions are needed: (A2) There exists a positive number M F such that sup (A3) There exists a positive number L f such that f (y 1 ) − f (y 2 ) ≤ L f y 1 − y 2 for all y 1 , y 2 ∈ R n ; (A4) There exists a positive number L g such that g( In the next theorem, the extension of sensitivity from system (10.13) to system (10.15) is considered. Proof. Fix an arbitrary positive number δ 1 , y 0 ∈ K and a chaotic solution x(t) of (10.13). Since system (10.13) is sensitive, one can find ǫ 0 > 0 and ∆ > 0 such that for arbitrary δ 0 > 0 both of the inequalities x(0) − x(0) < δ 0 and x(t) − x(t) > ǫ 0 , t ∈ J, hold for some chaotic solution x(t) of (10.13) and for some interval J ⊂ [0, ∞), whose length is not less than ∆.
It is worth noting that there exist positive numbers K 0 and H 0 such that y(t) , y(t) ≤ K 0 for all t ≥ 0 and sup t≥0 x(t) ≤ H 0 for each chaotic solution x(t) of system (10.13).
Our aim is to determine positive numbers ǫ 1 , ∆ and an interval J 1 ⊂ [0, ∞) with length ∆ such that the inequality y(t) − y(t) > ǫ 1 holds for all t ∈ J 1 .
Since the derivative of each chaotic solution x(t) of (10.13) lies inside the tube with radius M F , the collection of chaotic solutions of system (10.13) is an equicontinuous family on [0, ∞). Suppose that g(x) = (g 1 (x), g 2 (x), . . . , g n (x)) , where each g j , 1 ≤ j ≤ n, is a real valued function. Making use of the uniform continuity of the function g : R m × R m → R n , defined as g(ν 1 , ν 2 ) = g(ν 1 ) − g(ν 2 ), on the compact region R = {(ν 1 , ν 2 ) ∈ R m × R m : ν 1 ≤ H 0 , ν 2 ≤ H 0 } together with the equicontinuity of the collection of chaotic solutions of (10.13), one can verify that the collection F consisting of the functions of the form g j (x 1 (t)) − g j (x 2 (t)), 1 ≤ j ≤ n, where x 1 (t) and x 2 (t) are chaotic solutions of system (10.13), is an equicontinuous family on [0, ∞).
Otherwise, if there exists s ∈ J such that for all 1 ≤ j ≤ n the inequality |g j (x (s)) − g j (x(s))| < L g n x(s) − x(s) holds, then one encounters with a contradiction since g(x(s)) − g(x(s)) ≤ n j=1 |g j (x(s)) − g j (x(s))| < L g x(s) − x(s) .

Existence of unstable periodic motions
Assume that system (10.13) admits a period-doubling cascade. That is, there exists an equation x ′ = G(x, λ), (10.19) where λ is a parameter and the function G : R m × R → R m is such that for some finite number λ ∞ , G(x, λ ∞ ) is equal to the function F (x) in the right hand side of system (10.13).
System (10.13) is said to admit a period-doubling cascade [21,62,63,71] if there exists a sequence of period-doubling bifurcation values {λ j } j∈N satisfying λ j → λ ∞ as j → ∞ such that as the parameter λ increases or decreases through λ j system (10.19) undergoes a period-doubling bifurcation for each j ∈ N.
As a consequence, at the parameter value λ = λ ∞ , there exist infinitely many unstable periodic solutions of system (10.19), and hence of system (10.13), all lying in a bounded region.
Now, let us introduce the following definition [70]. We say that the solutions of the non-autonomous system (10.15), with a fixed x(t), are ultimately bounded if there exists a number B > 0 such that for every solution y(t), y(t 0 ) = y 0 , of system (10.15), there exists a positive number R such that the inequality y(t) < B holds for all t ≥ t 0 + R.
We say that system (10.15) replicates the period-doubling cascade of system (10.13) if for each periodic solution x(t) of (10.13), system (10.15) admits a periodic solution with the same period.
The following condition is required in the next theorem, which can be verified by using Theorem 15.8 [70].
(A5) Solutions of system (10.15) are ultimately bounded by a bound common for all x(t).