A Study of the Dynamics of a new Piecewise Smooth Map
.... ...; aop ...
Dhrubajyoti Biswas*, Soumyajit Seth, and Mita Bor
A Study of the Dynamics of a newPiecewise Smooth Map ... https://doi.org/..., Received ...; accepted ...
Abstract:
In this article, we have studied a 1D map, which is formed by combiningthe two well-known maps i.e. the tent and the logistic maps in the unit interval i.e.[0 , Keywords:
Piecewise Smooth and Non-Smooth Maps, Border Collision Bifurca-tions, Tent Map, Logistic Map, Boundary and Interior Crisis.
PACS: ...
Communicated by:
Dhrubajyoti Biswas
The well known logistic and the tent maps (traditionally defined on the unitinterval [0 , πΏ π ( π₯ ) = ππ₯ (1 β π₯ ) (1) *Corresponding author: Dhrubajyoti Biswas, Department of Physics, Indian Institute ofTechnology Madras, Chennai - 600036. Email: [email protected]
Soumyajit Seth,
Nonlinear Dynamics Laboratory, Department of Physical Sciences, IndianInstitute of Science Education and Research Kolkata, Mohanpur, West Bengal-741246.Email: [email protected]
Mita Bor,
Department of Physics, St. Xavierβs College (Autonomous), Kolkata-700016.Email: [email protected] a r X i v : . [ n li n . C D ] J un D. Biswas, S. Seth, M. Bor (a)
Bifurcation diagram of logistic map for π β [0 , . (b) Bifurcation diagram of tent map for π β [0 , . Fig. 1:
Bifurcation diagram for the Logistic and Tent maps as a function of π . and π π ( π₯ ) = π πππ { π₯, β π₯ } (2)respectively. The logisitc map is a quadratic continous map on the interval[0 , πΏ π β and β π π β have been used to represent them throughout thearticle. The control parameters π ( π β [0 ,
4] for the logistic map and π β [0 ,
2] forthe tent map) determine the dynamics exhibited by both of these. The Tent Mapgives a fixed point in the parameter interval [0 , , Study of the Dynamics of a new Piecewise Smooth Map (a) πΌ = 0 . (b) πΌ = 0 . (c) πΌ = 0 . Fig. 2:
Structure of π πΌ , for πΌ = 0 . , . , . This article deals with a map born out of an amalgamation of these two wellknown maps, aptly named as the βMixed Mapβ (MM), denoted by βMβ, and isdefined as follows: π₯ n+1 = π πΌπ π ( π₯ n ) = {οΈ π π ( π₯ n ) , β β€ π₯ n β€ πΌπΏ π ( π₯ n ) , β πΌ < π₯ n β€ π₯ n is the nth iteration. It is clear from the above expression that theMM is the tent map with the parameter value π in the region [0 , πΌ ] and it isthe logistic map with the parameter value π in the region [ πΌ, π₯ n = πΌ . Figure 2 shows the structure of the map forvarious values of πΌ for a fixed set of ( π , π ) = (2 , π₯ n = πΌ which is known as βBorderβ. As the systemis 1D, the border will be a point at π₯ n = πΌ . Depending upon the values of πΌ andparameters π and π , the map can behave as Piecewise Continuous or PiecewiseDiscontinuous. The map is continuous on each of the regions before and afterborder, but is discontinuous at border. In case of a Piecewise Discontinuous Map,there is a borderline in the Poincare section such that two arbitrarily close pointson the two sides of the border land far apart at the next observation instant[2, 8]. Otherwise, the map is known as the Piecewise Continuous Map [11, 3].The value of the discontinuity at border would be given by the difference in thefunctional values of π π and πΏ π . We can define a quantity πΏ π , π ( πΌ ) as follows: πΏ π ,π ( πΌ ) = ββ π π ( πΌ ) β πΏ π ( πΌ ) ββ (4)which can quantify the amount of discontinuity in the map at border, as afunction of the value of πΌ . Owing to the fact that π π ( π₯ n ) is defined differently D. Biswas, S. Seth, M. Bor
Fig. 3:
Plot of πΏ , ( πΌ ) vs πΌ . In this plot, we have shown the variation for the case of ( π , π ) = (2 , . The plot agrees with the structure of the maps obtained in Figure.(2)and Equations. 6 & 7. for π₯ > . π₯ β€ .
5, the expression of πΏ π π ( πΌ ) would change for two regimes πΌ β€ . πΌ > .
5, which is as follows: πΏ π ,π ( πΌ ) = {οΈββ π πΌ + πΌ ( π β π ) ββ , β πΌ β€ . ββ π πΌ β πΌ ( π + π ) + π ββ , β πΌ > . πΌ β€ .
5, the map will have one discontinuity at π₯ n = πΌ . Therefore thesystem will have one border for πΌ β€ .
5. But when πΌ > .
5, the system willhave two borders i.e. one at π₯ n = 0 . π₯ n = πΌ where discontinuities willoccur. Therefore, the map, which we are taking here, can have single or multipleborders depending on the value of π₯ n = πΌ .Quite clearly, as seen in Figures. (2b) & (3), πΏ , ( 12 ) = 0 (6)At this time the map will be piecewise continuous map with discontinuity atborder π₯ n = πΌ = 0 . πΏ π ,π (0) = πΏ π ,π (1) = 0 (7)This is because πΌ = 1 or πΌ = 0 means that the existence of the Mixed Mapconcept is lost and it is either fully tent-like or fully logistic-like respectively.The MM, as defined in Equation. 3 has three parameters which determinesthe dynamics of the system. While this presents a very wide range of possibilities,it becomes a hard problem to keep track of various parameters and determine Study of the Dynamics of a new Piecewise Smooth Map (a) πΌ = 0 . (b) πΌ = 0 . (c) πΌ = 0 . Fig. 4:
Structure of π πΌπ ( π₯ ) for πΌ = 0 . , . , . their effect on the dynamics of the system when they are varied together. Thus,to simplify the problem, we define a subset of the MM, called the βReducedMixed Mapβ (RMM), in a reduced parameter space of two variables π and πΌ ,where the new parameter π is related to the old ones parameters π and π asfollows: π = π, π = 2 π (8)In the definition of the RMM, the value of πΌ remains the same as that ofthe original Mixed Map and it denotes the βBordersβ of the map. The value of π is bounded in the interval [0 ,
2] so as to keep the iterates of the map bounded to[0 , π₯ n+1 = π πΌπ ( π₯ n ) = {οΈ π π ( π₯ n ) , β β€ π₯ n β€ πΌπΏ π ( π₯ n ) , β πΌ < π₯ n β€ πΌ , in Figure. (4).We can similarly define the amount of discontinuity at π₯ = πΌ for the RMM byplugging in Equation. 8 into Equation. 5, which gives us: πΏ π ( πΌ ) = {οΈββ ππΌ β πΌπ ββ , β πΌ β€ . ββ ππΌ + π (1 β πΌ ) ββ , β πΌ > . πΏ if both π and πΌ are varied in diagram. D. Biswas, S. Seth, M. Bor
Fig. 5:
3D plot of πΏ as a function of π and πΌ . This shows how the discontinuity varies asthe parameters of the system is continously changed. The RMM can have atmost 3 fixed points whose existence, stability and expressiondepends on the values of π and πΌ , with one of them always being at π₯ β n = 0.The existence of the fixed point π₯ β n = 0 is always guaranteed because whateverbe the value of π and πΌ , the π¦ = π₯ line always intersects the map at the origin.The stability of the fixed point is determined by the value of both the parameters.If πΌ = 0, that means the map is totally logistic-like, the slope at π₯ β n = 0 is givenby 2 π . Then, by condition of stability, ββ π ββ < , β πΌ = 0 , π₯ * = 0 (11)Here, we are taking π₯ β n = π₯ β .But if the value of 1 β₯ πΌ >
0, then near to π₯ = 0 the map is tent-like, andthus its slope at that point is given by π . Thus, by condition of stability, ββ π ββ < , β β₯ πΌ > , π₯ * = 0 (12)The other two fixed points do not always exist. If πΌ = 0, then the map isfully logistic-like, and therefore, the only other fixed point is π₯ * = 1 β π (13)The slope at this point is given by 2 β π , and therefore, the condition ofstability is Study of the Dynamics of a new Piecewise Smooth Map (a) πΌ = 0 . (b) πΌ = 0 . (c) πΌ = 0 . (d) πΌ = 0 . (e) πΌ = 0 . (f) πΌ = 0 . Fig. 6:
Structure of π πΌ r (x) for πΌ = 0 . , . , . and for π = 1 , . . We see that there is aline of fixed points from π₯ = 0 to π₯ = 0 . if π = 1 and πΌ β₯ . in Figures. (6a), (6b) and(6c). We also see that there are no other fixed points (other than π₯ * = 0 ) if the value of π < . and πΌ β₯ . , in Figures. (6d), (6e) and (6f). . > π > . , β πΌ = 0 , π₯ * = 1 β π (14)Quite clearly, the limiting case of π = 1 results in a line of fixed points from π₯ = 0 till π₯ = 0 . πΌ β₯ .
5. This is demonstrated in Figures. (6a), (6b) and(6c). But if the value of π is less than 1, and πΌ β₯ .
5, then there is no other fixedpoint other than π₯ * = 0, which is stable, having an attracting basin as that ofthe whole of the unit interval [0 , π < πΌ < . π₯ β = 0, given by π₯ * = 1 β π for πΌ β€ ( πΌ ππππ‘ππππ = 1 β π ) < . . > πΌ > πΌ ππππ‘ππππ , the fixed point again vanishes leaving behind only π₯ * = 0. This is clearly demonstrated in Figures. (7a), (7b) and (7c). This fixedpoint has the same stability range as discussed in Equation. 14. Again, similar D. Biswas, S. Seth, M. Bor (a) πΌ = = πΌ ππππ‘ππππ (b) πΌ = 0 . > πΌ ππππ‘ππππ (c) πΌ = 0 . < πΌ ππππ‘ππππ Fig. 7:
Structure of π πΌπ ( π₯ ) for πΌ = 3 / , . , . and for π = 0 . . We see that there is afixed point that appears at the value πΌ = πΌ ππππ‘ππππ and which remains β πΌ < πΌ ππππ‘ππππ anddisappears for β πΌ > πΌ ππππ‘ππππ (a) πΌ = 0 . (b) πΌ = 0 . (c) πΌ = 0 . Fig. 8:
Structure of π πΌπ ( π₯ ) for πΌ = 0 . , . , . and for π = 1 . to the previous discussion, if π = 1 and πΌ < .
5, there again exists a fixed pointgiven by π₯ * = 1 β π = 0 .
5, as shown in Figure. (8a), (8b) and (8c), which issuper-stable [1]. This is evident from the fact that ββ πππ₯ π πΌπ (0 . ββ = 0.The fixed points formed on the line π₯ = π¦ till π₯ = 0 . π = 1and πΌ > . π₯ > .
5, we see the iterates converge to a fixed point rapidly (in a single step),and the fixed point is given by: π₯ * π₯ ( π ) = {οΈ π (1 β π₯ ) , β π₯ β€ πΌ ππ₯ (1 β π₯ ) , β π₯ π > πΌ (15) Study of the Dynamics of a new Piecewise Smooth Map π β [0 , πΌ β [0 , Remarks π = 1 πΌ β₯ . β Line of fixed points for π₯ β [0 , . πΌ < . β Line of fixed points for π₯ β [0 , πΌ ] and the characteristic fixed point of the logistic map. π < πΌ > πΌ ππππ‘ππππ β Only one fixed point at π₯ = 0 πΌ β€ πΌ ππππ‘ππππ β Fixed point at π₯ = 0 and thecharacteristic fixed point of logistic map π > πΌ β [ πΌ πππ€ππ , πΌ π’ππππ ] β Two fixed points of characteristics tentand logistic map along with π₯ = 0 (3 fixed points overall) πΌ < πΌ πππ€ππ β Characteristic fixed point of logistic mapalong with π₯ = 0 πΌ > πΌ π’ππππ β Characteristic fixed point of tent mapalong with π₯ = 0 Tab. 1:
Summary of all the fixed point for various regimes.
We see that there exists a region/basin of attraction in both cases wherethere exists fixed points. For the case π = 1 and πΌ < .
5, we see that the fixedpoint formed at π₯ * = 0 . π₯ β ( πΌ, + β β πΌ ). For the case π < πΌ < .
5, the stable fixed point π₯ * = 1 β π , which only exists for πΌ β€ πΌ ππππ‘ππππ , has a basin of attraction assame as the previous case (i.e. for π = 1 case).The last set of non-trivial fixed points are generated when π >
1. For the case πΌ β€ .
5, the map again has only two fixed points, π₯ * = 0 (which is unstable) and π₯ * = 1 β π , which is globally stable upto π = 1 . πΌ > .
5, whichresults in a peculiar case where three fixed points can exist for a certain rangeof πΌ β [ πΌ πππ€ππ , πΌ π’ππππ ]. For πΌ > πΌ π’ππππ , the fixed point is the characteristic tentmap fixed point and for πΌ < πΌ πππ€ππ , the fixed point is the characteristic logisticmap fixed point. This has been demonstrated in Figures. (10a) to (10f). Thevalues of πΌ πππ€ππ and πΌ π’πππππ is given by πΌ πππ€ππ ( π ) = π π (16) πΌ π’ππππ ( π ) = 1 β π (17)These can be obtained easily by finding the first intersection point of thetent map and the logistic map respectively with the π¦ = π₯ line. The fixed pointsthemselves are the characteristic fixed points of the tent and the logistic mapsand their stability has been discussed before for various values of π . The existanceand stabilities of all the fixed points have been shown in Table 1. D. Biswas, S. Seth, M. Bor (a) πΌ = 0 . (b) πΌ = 0 . (c) πΌ = 0 . (d) πΌ = 0 . (e) πΌ = 0 . (f) πΌ = 0 . Fig. 9:
Structure of π πΌπ ( π₯ ) for πΌ = 0 . and for π = 1 . , . , . , in Figures. (9a),(9b) and(9c). Structure of π πΌπ ( π₯ ) for πΌ = 0 . , . , . and for π = 1 . , in Figures. (9d),(9e) and(9f). In this section, we will present the various numerical results obtained for theRMM for various values of πΌ and π . Bifurcation diagrams form the main interest Study of the Dynamics of a new Piecewise Smooth Map (a) πΌ < πΌ πππ€ππ (b) πΌ = πΌ πππ€ππ . (c) πΌ β [ πΌ πππ€ππ , πΌ π’ππππ ] . (d) πΌ = πΌ π’ππππ . (e) πΌ > πΌ π’ππππ (f) πΌ >> πΌ π’ππππ Fig. 10:
Structure of π πΌπ ( π₯ ) for πΌ > . and for π = 1 . . Figure. (10a) shows the case πΌ < πΌ πππ€ππ , where the fixed point is the characteristic logistic map fixed point (apart from π₯ * = 0 ). Figure. (10b) shows the birth of the third fixed point. Figure. (10c) shows thesituation where all three fixed points exist. Figure. (10d) shows the situation for which thea fixed point vanishes - the fixed point which vanished is the one which existed before thethird one was created in Figure. (10b). Figures. (10e) and (10f) shows the case where onlytwo fixed points remain (i.e. the trivial fixed point at π₯ * = 0 and the characteristic fixedpoint of the tent map). D. Biswas, S. Seth, M. Bor (a)
Bifurcation diagram of logistic map for π β [0 , . (b) Bifurcation diagram of tent map for π β [0 , . (c) Lyapunov Exponent of Logistic map for π β [0 , . (d) Lyapunov Exponent of Tent map for π β [0 , . Fig. 11:
Bifurcation diagram and Lyapunov Exponents for the Logistic and Tent maps as afunction of π . in this section because it clearly shows how the state variable of the map changeswith the change of parameters.The bifurcation diagrams of the parent maps of the RMM is shown in Figures.(11a) and (11b). We can see that the bifurcation diagram of the logistic map isdistinctively different from that of the tent map, in terms existence of differentdynamical phenomena like periodic doubling bifurcations, periodic windowsin between chaos, existance of chaos without any other periodic attractors orco-existing attractors. In the case of the RMM, there are two parameters, namely, π and πΌ , and both of them are suitable candidates for plotting the bifurcation Study of the Dynamics of a new Piecewise Smooth Map diagrams in the various domains as discussed in the previous section. In ourarticle, we will keep πΌ as a constant, and vary π and observe the bifurcationdiagrams. This would be done for various different values of πΌ to understand theoverall behaviour of the map for various values of πΌ and π . (a) Bifurcation diagram of RMM for π β [0 , . (b) Bifurcation diagram of RMM for π β [1 . , . (c) Lyapunov Exponent of RMM for π β [0 , . (d) Lyapunov Exponent of RMM for π β [1 . , . Fig. 12:
Bifurcation diagram and Lyapunov Exponents for the RMM as a function of π for πΌ = 0 . . If we keep the value of πΌ = 0 .
5, i.e. the map is piecewise smooth as well ascontinuous at border, the bifurcation diagram and corresponding itβs lyapunovexponents are shown in Figure. (12). We see the bifurcation diagram, is in a sense,a mixture of that of the logistic map and the tent mapβs bifurcation diagrams. D. Biswas, S. Seth, M. Bor
As seen in Figure. (11b) in the case of the tent map, zero is a stable fixed pointtill π <
1. After that, the fixed point jumps to the nontrivial value and thebifurcation diagram has a structure similar to that of a logistic map where aftera normal period doubling bifurcation periodic windows exist in between chaos.The corresponding lyapunov exponent shows the existence of the fixed points,periodic orbits and the values of which periodic windows exist in-between chaos.This behaviour is characterized by negative lyapunov exponent for the fixedpoints and the periodic orbits, zero for the quasi-periodic orbits and positivevalues for the chaotic orbits.Even more interesting bifurcation diagrams emerge when we break the conti-nuity of the map at border in case of πΌ ΜΈ = 0 .
5) i.e. the piecewise discontinuousmap with single or multiple borders. If we use πΌ = 0 .
6, i.e. there will be twoborders, one is at π₯ n = 0 . π₯ n = πΌ = 0 .
6, the bifurca-tion diagram shows a very interesting features. There exists many interestingdynamical structure - there are period-3, period-4 upto period-11 orbits in thiscase. The time series waveforms have been shown in Figure. (14). The period-11orbit occurs near the value of π β .
528 whereas the distinct period-3 orbitexists near π β .
22 (see Figure. (14a)). The period-11 orbit cannot be seen incase of both logistic and tent map bifurcation diagram. In Figure. (13a), thereexists a 2-piece chaotic orbits just after the bifurcation near π = 1. After thata period-3 orbit emerges, which gradually gives a periodic orbit as we changethe parameter further. The zoomed bifurcation diagram and corresponding itβslyapunov exponents in between 1 to 1 . π is varied more,another bifurcation occurs, where a normal period doubling bifurcation givesa period-11 orbit, which gradudally goes to 2- piece chaotic orbit. Also, in be-tween 1 . π parameter range, a interior crisis [4] happens where achaotic orbit suddenly expands itβs shape. This happens when a chaotic attractorjust overlaps with the co-existing unstable chaotic orbit and the main chaoticorbit suddenly expands. Apart from all these, this map also has the normalperiodic points and period-2 orbits as well. Figure. (14b) demonstrates βsensitivedependence on initial conditionsβ which is a strong indicator for chaos [6].Further changing the value of πΌ gives rise to other interesting phenomena.Also, πΌ β πΌ β πΌ = 0 .
2, 0 . . Study of the Dynamics of a new Piecewise Smooth Map (a) Bifurcation diagram of RMM for π β [0 , . (b) Bifurcation diagram of RMM for π β [1 . , . . (c) Lyapunov Exponent of RMM for π β [0 , . (d) Lyapunov Exponent of RMM for π β [1 . , . . (e) Bifurcation diagram of RMM for π β [1 , . (f) Lyapunov Exponent for RMM for π β [1 , . Fig. 13:
Bifurcation diagrams and Lyapunov Exponents for the RMM for πΌ = 0 . . D. Biswas, S. Seth, M. Bor (a) π = 1 . (period-11 orbit, purple) and π = 1 . (period-3 orbit, green). (b) Time series for changes in initial condi-tions for π = 2 . The orbits show SIC. Fig. 14:
Illustrations of Period and Period along with Sensitive dependence of InitialConditions (SIC). of the two lyapunov exponents in the parameter values, but if one can look closelyin the two diagrams, it can be said that the values of the lyapunov exponentstoggle between the two lyapunov exponent values in that parameter ranges asthe parameter changes gradually and as they have plotted closely, it looks likethe existance of the two lyapunov exponents in the same parameter values. Thebifurcation diagrams for the cases πΌ = 0 . . π = 1 . . πΌ = 0 .
25 and πΌ = 0 .
75. But for the first case, as there is only oneborder which is at πΌ = 0 .
25, and for the next case, there are two borders, one isat 0 . .
75 although the amount of discontinuities are samefor the two cases, the bifurcations are different due to different borders. Thedifferent bifurcation diagrams have been shown in Figure.(21).Upto this, we have taken π = π and π = 2 π . In this consideration, the mapbecomes continuous across the border in case of πΌ = 0 .
5. Now, we are taking π = π = π , which makes the map piecewise discontinuous with discontinuity at πΌ = 0 . π = 1 .
6. This phenomenon is obversed generally in case of a 1D piecewisediscontinuous map [8]. As the parameter is evolved more. we get a normal perioddoubling bifurcations which goes to chaotic region. The periodic windows alsoexist in between chaos here as well.
Study of the Dynamics of a new Piecewise Smooth Map (a) Bifurcation diagram ofRMM for πΌ = 0 . . (b) Bifurcation diagram ofRMM for πΌ = 0 . . (c) Bifurcation diagram ofRMM for πΌ = 0 . . (d) Bifurcation diagram ofRMM for πΌ = 0 . . (e) Bifurcation diagram ofRMM for πΌ = 0 . (f) Bifurcation diagram ofRMM for πΌ = 0 . Fig. 15:
Approach to the parent map bifurcation structures in the RMM for πΌ β (ap-proaches logistic map structure, in Figures. (15a), (15b) and (15c)) and for πΌ β . (approaches tent map structure, in Figures. (15d), (15e) and (15f)). (a) Bifurcation diagram ofRMM for πΌ = 0 . . (b) Bifurcation diagram ofRMM for πΌ = 0 . . (c) Bifurcation diagram ofRMM for πΌ = 0 . . Fig. 16:
Zoomed Bifurcation Diagrams for the cases πΌ = 0 . , πΌ = 0 . and πΌ = 0 . . D. Biswas, S. Seth, M. Bor (a)
Lyapunov Exponent ofRMM for πΌ = 0 . . (b) Lyapunov Exponent ofRMM for πΌ = 0 . .. (c) Lyapunov Exponent ofRMM for πΌ = 0 . .. Fig. 17:
Lyapunov Exponents of RMM for πΌ = 0 . , πΌ = 0 . and πΌ = 0 . . (a) Bifurcation diagram ofRMM for πΌ = 0 . . (b) Bifurcation diagram ofRMM for πΌ = 0 . . (c) Bifurcation diagram ofRMM for πΌ = 0 . . Fig. 18:
Bifurcation diagrams for πΌ = 0 . , πΌ = 0 . and πΌ = 0 . (a) Lyapunov Exponent of RMM for πΌ = 0 . . (b) Lyapunov Exponent of RMM for πΌ = 0 . .. Fig. 19:
Lyapunov Exponents of RMM for πΌ = 0 . and πΌ = 0 . . Study of the Dynamics of a new Piecewise Smooth Map (a) Bifurcation diagram of RMM for πΌ = 0 . . (b) Bifurcation diagram of RMM for πΌ = 0 . . Fig. 20:
Zoomed Bifurcation Diagrams for the cases πΌ = 0 . , πΌ = 0 . . (a) πΌ = 0 . . (b) πΌ = 0 . . Fig. 21:
Bifurcation diagrams for πΌ = 0 . and πΌ = 0 . . Here, πΏ = 0 . , which is same forboth. D. Biswas, S. Seth, M. Bor
Fig. 22:
Bifurcation diagrams for πΌ = 0 . and π = π = π . (a) πΌ = 0 . , π β [1 , . (b) πΌ = 0 . , π β [1 . , (c) πΌ = 0 . , π β [1 . , . (d) LE of πΌ = 0 . , π β [1 , . (e) LE of πΌ = 0 . , π β [1 . , (f) LE of πΌ = 0 . , π β [1 . , . Fig. 23:
Bifurcation diagrams and Lyapunov Exponenets for the case of πΌ = 0 . and π = π = π of the MM. Study of the Dynamics of a new Piecewise Smooth Map Fig. 24:
Simulink based Implementation of the Mixed Map to generate time-series data.The initial condition, gains and the switching conditions are set arbitrarily.
The zoomed figures of all these bifurcations and corresponding itβs lyapunovexponents have been shown in Figure.(23).
In this section, we have shown a Simulink based implementation of the MM (not tobe confused with the RMM, which was discussed untill now). The implementationis shown below in Figure. (24).The implementation as shown in Figure. (24) assume the presence of variousblocks (which are readily available in Simulink) like the product (multiplier),amplifier, minimum finder, controllable switch and a delay. A basic circuit diagramfor implementing discrete maps is shown in [14]. Following a similar method, first,the individual maps are synthesized from the available blocks. The logistic map,which is defined continuously in the interval [0 ,
1] can be easily synthesized byusing a subtracter, product (multiplier) and an amplifier. The tent map, which ispiecewise continuous in [0 ,
1] is implemented by using a subtracter, a minimumfinder and an amplifier. The minimum finder takes two inputs and outputs thesignal which is lesser of the two. The switching condition is implemented byusing a controllable switch which is driven by the output of the circuit. Finally,a unit delay block is used to implement the discrete nature of our system. Aninitial condition block is used to set the initial condition i.e. π₯ , of our system.The values of the switching parameter, the initial condition and the gains of theamplifiers (which set the control parameters π and π as discussed in Equation.3) can be changed manually for each simulation run. D. Biswas, S. Seth, M. Bor (a)
Time series showing aperiodic orbit. (b)
Time series showing period-11 orbit for πΌ = 0 . . This has been detected previously inFigure. (14a). (c) Time series showing SIC for two very nearby initial conditions for a pair of identicalmaps.
Fig. 25:
Output from Simulink Implementation of MM showing Periodic orbits (Fig-ure.(25a)), Non-periodic orbits (Figure.(25b)) and Exhibition of SIC. (Figure.(25c)) Study of the Dynamics of a new Piecewise Smooth Map Some sample outputs are shown by running the simulation: Figures. (25a)and (25b) shows the period-11 orbit and a aperiodic orbit which were previouslyobserved using numerical techniques. Figure. (25c) exhibits Sensitive dependenceon Initial Conditions (SIC) for two very nearby initial conditions for a pair ofidentical maps.
In this article, we have proposed a novel chaotic map, called the Mixed Map(MM), which is formed from the amalgamation of two well known maps i.e. logisticmap and the tent map. It has two parameters π and πΌ , which are the controlparameter and the transition parameter respectively. This map was found to beboth piecewise discontinous and continous as well, depending upon the values ofthe parameters. The amount of discontinuity was investigated and plotted. Toreduce the complexity of the map, a Reduced Mixed Map (RMM) was definedfor further studies. All the fixed points for the RMM for the various cases of theparameter values was found and their stability was classified. The presence of athree fixed point map was observed for certain values of the parameter πΌ . Theresults have been summarized in Table. 1. Numerical studies were done on theproposed RMM and bifurcation diagrams, lyapunov exponents and orbits wereplotted for all the interesting cases. A stable period-11 orbit was observed for πΌ = 0 . πΌ β πΌ β
1, the structure becomes similar to that ofthe tent-map bifurcation structure. For the case of π = π = π and πΌ = 0 . πΌ with π = π = π . We can address these issues as ourfuture work. Finally, a simple simulink implementation was presented for theproposed MM, which was used to verify some of the time series diagrams andproperties we discussed in the previous sections. Acknowledgment:
DB acknowledges the HTRA scholarship provided by IndianInstitute of Technology Madras, Chennai. SS acknowledges the support of DST-INSPIRE, Government of India (Ref. No: IF150667). MB acknowledges thesupport provided by St. Xavierβs College (Autonomous), Kolkata. REFERENCES
References [1] Kathleen T Alligood, Tim D Sauer, and James A Yorke.
Chaos . Springer,1996.[2] Viktor Avrutin, Michael Schanz, and Soumitro Banerjee. Multi-parametric bi-furcations in a piecewiseβlinear discontinuous map.
Nonlinearity , 19(8):1875,2006.[3] Soumitro Banerjee, MS Karthik, Guohui Yuan, and James A Yorke. Bifur-cations in one-dimensional piecewise smooth maps-theory and applicationsin switching circuits.
IEEE Transactions on Circuits and Systems I: Funda-mental Theory and Applications , 47(3):389β394, 2000.[4] Soumitro Banerjee and George C Verghese.
Nonlinear phenomena in powerelectronics . IEEE, 1999.[5] Soumitro Banerjee, James A Yorke, and Celso Grebogi. Robust chaos.
Physical Review Letters , 80(14):3049, 1998.[6] Eli Glasner and Benjamin Weiss. Sensitive dependence on initial conditions.
Nonlinearity , 6(6):1067β1075, 1993.[7] Toshiki Habutsu, Yoshifumi Nishio, Iwao Sasase, and Shinsaku Mori. Asecret key cryptosystem by iterating a chaotic map. In
Workshop on theTheory and Application of of Cryptographic Techniques , pages 127β140.Springer, 1991.[8] Parag Jain and Soumitro Banerjee. Border-collision bifurcations in one-dimensional discontinuous maps.
International Journal of Bifurcation andChaos , 13(11):3341β3351, 2003.[9] Bruce E Kendall and Gordon A Fox. Spatial structure, environmentalheterogeneity, and population dynamics: analysis of the coupled logisticmap.
Theoretical population biology , 54(1):11β37, 1998.[10] LjupΔo Kocarev and Goce Jakimoski. Logistic map as a block encryptionalgorithm.
Physics Letters A , 289(4-5):199β206, 2001.[11] Helena E Nusse and James A Yorke. Border-collision bifurcations forpiecewise smooth one-dimensional maps.
International journal of bifurcationand chaos , 5(01):189β207, 1995.[12] Soumyajit Seth. Observation of robust chaos in 3d electronic system.
IETCircuits, Devices & Systems , 2019.[13] Steven H Strogatz.
Nonlinear Dynamics and Chaos with Student SolutionsManual: With Applications to Physics, Biology, Chemistry, and Engineering .CRC Press, 2018.[14] Madhekar Suneel. Electronic circuit realization of the logistic map.
Sadhana ,31(1):69β78, 2006.