Forcing the escape: Partial control of escaping orbits from a transient chaotic region
aa r X i v : . [ m a t h . D S ] F e b Noname manuscript No. (will be inserted by the editor)
Forcing the escape: Partial control of escaping orbits from atransient chaotic region
Gaspar Alfaro · Rub´en Cape´ans · Miguel A.F. Sanju´an
Received: date / Accepted: date February 25, 2021
Abstract
A new control algorithm based on the par-tial control method has been developed. The generalsituation we are considering is an orbit starting in acertain phase space region Q having a chaotic transientbehavior affected by noise, so that the orbit will def-initely escape from Q in an unpredictable number ofiterations. Thus, the goal of the algorithm is to controlin a predictable manner when to escape. While partialcontrol has been used as a way to avoid escapes, herewe want to adapt it to force the escape in a controlledmanner. We have introduced new tools such as escapefunctions and escape sets that once computed makesthe control of the orbit straightforward. We have ap-plied the new idea to three different cases in order to il-lustrate the various application possibilities of this newalgorithm. Keywords
Controlling chaos, partial control, tran-sient chaos, escaping orbits
Even though chaotic systems are difficult to deal withdue to its intrinsic unpredictability, there are nonethe-less methods that allow us to control them. Differenttechniques for controlling chaos have been developedin the past few years. A rough classification may di-vide them between feedback control and non-feedbackcontrol methods. Among the first, we can consider theOGY [1] or the Pyragas [2] control methods, while on
G. Alfaro · R. Cape´ans · M.A.F. Sanju´anNonlinear Dynamics, Chaos and Complex Systems Group,Departamento de F´ısica, Universidad Rey Juan Carlos,M´ostoles, Madrid, Tulip´an s/n, 28933, SpainE-mail: [email protected] the latter random, chaotic or periodical signals are usedas an appropriate mechanism to control the system.The partial control method, which is a feedbackmethod, has been used in previous works [3–9], andis applied to a map defined in a certain region Q wherethere is transient chaos and in absence of any controlthe orbit will eventually escape from the region after acertain number of iterations. Furthermore, the map issubjected to a disturbance which is always larger thanthe applied control. The goal is to use the minimumcontrol to keep the orbits inside Q in presence of thedisturbance. Precisely, one advantage of this method isthe capacity to keep small the amount of control. Asis well known, transient chaos is the physical manifes-tation of the the presence in phase space of a chaoticsaddle, which is a fractal set. Orbits starting close tothe chaotic saddle eventually escape in a highly unpre-dictable manner, and the escape times also depend onthe disturbance and the initial conditions. In any case,one key feature of the method is that the control usedis always smaller than the disturbance, which is rathersurprising and counterintuitive.In the present work, we face a new objective, whichcan be viewed as the converse of the previous one. Whilein the previous case the goal was to keep the orbit in Q for ever, now based on the same premises, our goalis to control the number of iterations necessary for theorbit to escape Q .We have analyzed three different scenarios to applythe new strategy. The first one corresponds to the casewhere the controller wants to force the escape of theorbits from Q in N or less iterations of the map. In thesecond case, we consider the situation when we wantthe orbit to escape in exactly N iterations, where thisstronger constraint would necessarily imply a higheramount of control. Finally, the third situation we con- Gaspar Alfaro et al. U N ( q ) U N ( q ) E N E N qqquu u Fig. 1
Steps that summarize the partial control procedure. (Left) Escape function U N ( q ). For a given initial point q ∈ Q ,this function represents the minimum upper control bound u necessary to expel an orbit out of the region Q in N iterations.(Center) Escape set E N (blue boxes) obtained as a result of the intersection from the escape function with the control bound u (horizontal blue line). The points in E N will escape from Q by using controls | u n | ≤ u . (Right) A point outside the escapeset is controlled to insert it in the set. By doing so, each iteration guarantees that the orbit will escape in N iterations or less. sider is somehow different. We have a map defined intwo different regions in phase space, so that for a valueof the parameter two chaotic attractors coexist inde-pendently in each region, and after a certain parametervalue both attractors merge into a single global chaoticattractor. The idea here is to apply our control tech-nique to fix the precise number of iterations of the orbitto stay in each region. As a result, we will get a chaoticorbit that periodically oscillates between the regions.Obviously this scheme could be generalized to a largernumber of regions.The paper is organized as follows. In Sect. 2, weintroduce the partial control method and explain howto use it to control the escape from the chaotic region.Furthermore, we describe the escape functions and es-cape sets, adapted from previous work [10] that havebeen used for our objectives. In Sect. 3, we present twodifferent specific cases where we apply the algorithm tofix the number of iterations for the orbits to escape.In Sect. 4, we address the case where the algorithm isapplied for the goal of alternating the orbit betweentwo regions in a predictable manner. Finally, the mainconclusions are provided in the last section. We present here the partial control method [6, 9] thatis applied on maps in the following manner q n +1 = f ( q n ) + ξ n + u n | ξ n | ≤ ξ | u n | ≤ u < ξ , (1)where the map acts on values of q ∈ Q , Q is a regionin phase space, ξ n represents a bounded disturbanceaffecting the map at each iteration and u n is the applied control at each iteration, which is also bounded andimportantly, smaller than the disturbance.Our goal here is to perturb the orbit starting in Q byapplying a sequence of controls ( u , u , .., u N ) in orderto push the orbit out of Q in N iterations. Needless tosay, we can achieve this objective by using different se-quences, and the approach of partial control is to findthe strategy that minimizes the upper bound of thatsequence, that is, the min (cid:0) max ( | u | , | u | , .., | u N | ) (cid:1) = U N ( q ), where q is the initial point of the orbit and U N is the escape function . Once the escape functionis computed, we can choose an upper control boundvalue u and select the set of points q ∈ Q that sat-isfy U N ( q ) ≤ u . We name this set, the escape set E N .Any orbit starting in this set can be expelled from Q byusing a sequence of N controls u n ( n ≤ N ) with mag-nitude equal or smaller than u . The notions of escapefunctions and escape sets have been adapted from thesafe functions and safe sets defined in [10]. The steps toapply this control technique are summarized as follows1. Choose the phase space region Q where the controlmethod will be applied. We assume that we knowthe map and the upper disturbance bound ξ affect-ing it.2. Compute the escape functions U k with k = 1 : N in the region Q . Remind that N is the number ofiterations we need to expel the orbit out of Q .3. Set the value u and for every escape function U k ,compute the corresponding escape set E k .4. For every iteration of the map, we choose the ap-propriate u n , | u n | ≤ u , to bring the orbit to theescape set E k . Thus, we use the control so that thefirst iteration of the map brings the orbit to the es-cape set E N , the second iteration of the map to theescape set E N − and so on, until the orbit escapesout of the region Q . orcing the escape: Partial control of escaping orbits from a transient chaotic region 3 The steps 2, 3 and 4 of this procedure are illustratedin Fig. 1. The second step is the most computationallyexpensive, where the escape functions U k are computedby using an algorithm developed in [10]. The algorithmis based on the observation that this kind of controlproblems can be solved backwards, starting from thelast iteration. We will see that it is straightforward tocompute the first function U and through an iterativeprocedure obtain the rest of the escape functions U k ,since U k +1 = f ( U k ). Fig. 2
Scheme of the escape function (red) and the terminol-ogy used in the control procedure. Here the logistic map with µ = 4 . ,
1] will escape the interval [0 ,
1] after a few iterations. Thedashed arrows are the mapping of an initial point q [ i ] for oneof the possible values of the perturbation ξ [ s ]. The horizontalblack arrow represents the applied control that correspondsto the distance from the mapped point f ( q [ i ]) + ξ [ s ] to thearrival point q [ j ] = f ( q [ i ]) + ξ [ s ] + u [ i, s, j ]. The indices i , j refer to points in the discretization of the interval [0 , s refers to values of the discretization of ξ n . Since we are doing numerical simulations, we mustuse a grid on Q so that the map becomes q n +1 = f ( q n )+ ξ n + u n → q [ j ] = f ( q [ i ])+ ξ [ s ]+ u [ i, s, j ] , (2)where i = 1 : M denotes the number of the gridpoints in Q . The index s = 1 : W corresponds withthe number of possible disturbances ranging from − ξ to ξ . The index j = 1 : M denotes the arrival point q [ j ] = f ( q [ i ])+ ξ [ s ]+ u [ i, s, j ]. The term u [ i, s, j ] denotesthe control applied to the point f ( q [ i ] , ξ [ s ]) to put it inthe arrival point q [ j ]. All these terms are illustrated inFig. 2 for clarity. We will use the well-known logistic map x n +1 = f ( x n ) = µx n (1 − x n ) as an example to illustrate theapplication of the algorithm described earlier.When we consider values of µ >
4, the logistic mappresents transient chaos in the region Q = [0 , Q without control depends on the initialcondition and the sequence of disturbances affecting it.As a consequence the lifetime is highly unpredictableas shown in Fig. 3(b).The goal of the algorithm based on the partial con-trol method used in this work is to fix the number N of the iterations after which the orbit escapes from Q .Three different cases will be explored. In each case, wewill show the algorithm to compute the correspondingescape functions with some examples.3.1 Case A: Escape in N or less iterationsThis situation is justified in the case when we want anorbit to leave the chaotic region as quickly as possible.To do that, we choose the value N and design the con-trol algorithm so that the controlled orbit will abandon Q in N or less iterations. The lesser the value of N , thequicker the orbits will escape, though at a high price ofcontrol.As already commented, the escape functions U k arecalculated from the first escape function U througha recursive algorithm, since U k +1 = f ( U k ). Then, thevalue U ( q [ i ]) is defined to be the minimum controlnecessary to escape in the next iteration. This con-trol corresponds to the distance from the mapped point f ( q [ i ])+ ξ [ s ] to the nearest end points of Q (In this case0 or 1). However, since there is a control u [ i, s, j ] asso-ciated to each different value of the disturbance ξ [ s ], weneed to choose the maximum control among them all.Proceeding similarly for every initial condition q [ i ], weobtain the escape function U . This function representsthe minimum control bound necessary to escape in oneiteration.The next function U will correspond to the min-imum control bound necessary to force the escape of Gaspar Alfaro et al. Fig. 3 (a) Logistic map with µ = 4 . Q = [0 , the orbit within 2 iterations. There are two possibil-ities here, given an initial condition q [ i ], the image f ( q [ i ]) + ξ [ s ] can fall directly outside Q , in which caseno control is needed, or can fall in a point q [ j ] ∈ Q .In the latter case, we need to compute the suitablecontrol that minimizes the maximum between the val-ues u [ i, s, j ] (the control applied in this iteration ofthe map) and U ( q [ j ]) (the maximum control that wewill apply in the next iteration of the map), that is,the min j (max( u [ i, s, j ] , U [ j ])). Again, to take into ac-count all possible disturbances ξ [ s ] we must choose the maximum among all corresponding controls, thatis U ( q [ i ]) = max s (min j (max( u [ i, s, j ] , U ( q [ j ])))).The procedure to compute U is similar, where now U ( q [ i ]) = max s (min j (max( u [ i, s, j ] , U ( q [ j ])))). The al-gorithm is repeated until U N is obtained. The values U N ( q [ i ]) of this function represent the minimum con-trol bound necessary to force the orbit starting in q [ i ]to escape from Q within N or less iterations.In order to define the algorithm to compute the es-cape functions, we define u out [ i, s ] as the control appliedto the image f ( q [ i ] , ξ [ s ]) to move it outside the region Q . Then, U is calculated as follows. U ( q [ i ]) = max s ( u out [ i, s ]) (3)If q [ i ] + ξ [ s ] is beyond Q , then u out [ i, s ] = 0. Then,given U as the seed function, we can calculate the nextescape functions with the following recursive algorithm U ink +1 ( q [ i ]) = max s (cid:16) min j (cid:0) max( u [ i, s, j ] , U k ( q [ j ])) (cid:1)(cid:17) (4) U k +1 ( q [ i ]) = min (cid:18) U ( q [ i ]) , U ink +1 ( q [ i ]) (cid:19) , (5)where the intermediate function U ink +1 was introducedto allow the orbit to escape from Q before N iterations.Notice that U ink +1 only takes into account images inside Q to control the orbit, while U only takes into accountimages outside Q . Between these two possibilities, theone that minimizes the control will be chosen. By do-ing so, the orbit can be expelled in any of the k ≤ N iterations.As an example, we have chosen N = 3 so that con-trolled orbits will escape from Q in 3 or less iterations.The upper disturbance bound affecting the logistic mapwas set to ξ = 0 . U k are shown in Fig. 4, which as it can be ob-served take zero values in some intervals. This meansthat points in these intervals where U k = 0 will escapefrom Q within k iterations for any ξ [ s ], without ap-plying any control. It can be also observed that as theindex k increases, the escape function decreases sincethe orbit has more iterations to escape.Once we compute the escape functions, we have toselect the control value u to compute the correspond-ing escape sets E k . No tall u values are allowed. Escapesets only exist for values u ≥ min ( U N ). The bigger u ,the bigger the escape sets. These sets consist of points q [ i ] satisfying the condition U k ( q [ i ]) ≤ u . The escapesets E k for u = 0 .
022 are represented in Fig. 4. Forthis case, the set E represents the set of points q [ i ]that can escape from Q within 3 iterations of the map, orcing the escape: Partial control of escaping orbits from a transient chaotic region 5 by applying a control u n ≤ u at each iteration. In thefirst iteration, the control u will be applied to put theorbit in the closest point of E or outside Q if possible.In the second iteration, the control u will be appliedto put the orbit in the closest point of E or outside Q if possible. In the third and last iteration the control u will be applied to put the orbit outside Q . This isillustrated in Fig. 5.We have built in Fig. 6 a colormap plot showing theminimum number of iterations N needed for every or-bit to escape from Q = [0 ,
1] in a ( ξ , u ) parameterplane. In this figure the white points correspond to thecase where some initial conditions cannot escape from Q using these values of ( ξ , u ). A red line is plottedrepresenting the ratio u = ξ , so that values belowthis line are the ones we are interested due to the par-tial control method. Furthermore, we have chosen twopoints marked in red. The (+) point corresponds to N = 4, while the (*) point to N = 19.3.2 Case B: Escape exactly in N iterationsHere we analyze the case when the number of itera-tions for an orbit to escape from Q is exactly N . Arelevant observation here is that we need to control theorbit inside Q for N − N . The algorithm to obtainthe escape functions is now simpler, because we do notneed to consider the possibility that the orbit abandons Q before the iteration N , and is described next U ( q [ i ]) = max s ( u out [ i, s ]) (6) U k ( q [ i ]) = max s (cid:16) min j (cid:0) max( u [ i, s, j ] , U k ( q [ j ])) (cid:1)(cid:17) . (7)To show how to control orbits to escape from Q in exactly 3 iterations, we use the logistic map with µ = 4 .
7, and ξ = 0 . N iterations” (case B) is stronger than thecondition “expelling the orbit in N or less iterations”(case A). Therefore, we need bigger controls for the caseB than the case A.To get the escape sets E k , we have chosen the con-trol bound u = 0 . E escapes Q = [0 ,
1] in exactly 3 iterations is shown in Fig. 8.
Fig. 4
Case A. On top, the logistic map with µ = 4 .
7. Onthe bottom, the escape functions U k are computed for N = 3and shown in red. The escape sets E k are shown in blue.We have used ξ = 0 .
030 and u = 0 . E will escape from Q = [0 ,
1] in 3 or less iterationsby applying a control | u n | ≤ u at each iteration. The scalesused in the vertical axis are different for a better visualization. In the previous section, we mentioned three differentcases to explore the escaping of an orbit from Q in N iterations. We discussed earlier, cases A and B. Now,we focus our attention on the third case correspond-ing to multistable chaotic systems that merge into a Gaspar Alfaro et al.
Fig. 5
Case A. Example of a controlled orbit. An initial con-dition in E is controlled to E , then E , and finally it leavesthe chaotic region Q = [0 ,
1] under a suitable control. In gen-eral, orbits can escape in 3 or less iterations depending onthe initial condition and the disturbance ξ n . At each iter-ation, the applied control u n is the minimum between thenearest escape set and the end points of Q . We have fixedhere µ = 4 . ξ = 0 .
030 and u = 0 . Fig. 6
This color plot shows the minimum number of iter-ations N needed for every orbit to escape from Q = [0 , ξ , u ), while the whitepoints correspond to a situation where some initial conditionscannot escape. The red line represents the ratio u = ξ . Asan illustration we have marked two points in red. The (+)point corresponds to N = 4, while the (*) point to N = 19. larger chaotic attractor as a parameter is varied [11–13].This process occurs mainly when the basin boundary ofeach attractor collide, so that an orbit moves chaoticallyback and forth from one region to the other [14].Our goal here is to use the control algorithm to forcethe orbit to stay in each region for a fixed number ofiterations before moving to the other region. What wewant here is to maintain a perpetual periodical motion Fig. 7
Case B. On top, the logistic map with µ = 4 .
7. On thebottom, the escape functions U k are computed for N = 3 andshown in red. The escape sets E k are shown in blue. We haveused ξ = 0 .
030 and u = 0 . E willescape from Q = [0 ,
1] in exactly 3 iterations with a control | u n | ≤ u at each iteration. bouncing back and forth between regions. The initialchaotic orbit stays for N iterations in one region, thenit moves to the second region where it stays for N ′ iterations, and finally it comes back again to the firstregion. As we will see, the orbit will resemble a chaoticsignal modulated by a periodic one. orcing the escape: Partial control of escaping orbits from a transient chaotic region 7 Fig. 8
Case B. Example of a controlled orbit. An initial con-dition in E is mapped to E , then E , and finally it leavesthe chaotic region Q = [0 ,
1] after a suitable control. All orbitsstarting in E will escape Q = [0 ,
1] in exactly 3 iterations.
We have constructed a map that we name the doubleparabola map illustrated in Fig. 9 as an example of asimple map exhibiting the behavior described before, x n +1 = − µ ( x n + 12 x n ) si x < . µ ( x n − x n + 12 ) si x n ≥ . (8)This map is defined by a convex parabola at theregion of the left Q l = [0 , .
5) and a concave parabolaat the right region Q r = [0 . , µ .To show an example of the application of our loga-rithm to this case, we will focus on the behavior of themap for µ = 10, where we have a chaotic attractor thatexpands to all the interval [0 , Q l behave chaotically to eventually escapingto the region Q r , and vice versa as shown in the Fig. 9.Even though the map is well defined to map points of[0 ,
1] into itself, however this could not be so in pres-ence of a disturbance. In particular, for points close tothe end of interval, what it should be considered in thecontrol scheme.The main purpose now is to apply a control sothat the transition from the two regions would be pre-dictable. In other words, we want to control how manyiterations a given orbit stays on each region. As a con-sequence, our goal here is to keep the orbit N l iter-ations on region Q l and N r iterations on region Q r ,where the values N l and N r are previously chosen bythe controller. As a result, we will get a chaotic orbitthat periodically oscillates between regions Q l and Q r . Fig. 9 Double parabola map . The map is defined in apiecewise manner in two different regions Q l = [0 , .
5) and Q r = [0 . , µ = 7 .
2, the orbits stay on oneside of the map, so there are clearly two different attractors.(Bottom) When µ = 10, a global chaotic attractor merges.Now, orbits starting in Q l have a transient chaotic behaviorbefore escaping to the region Q r where after another chaotictransient the orbit comes back to Q l . To compute the escape functions, we follow a similarmethodology as the one used on case B from the pre-vious section, though we must impose in the algorithmthe periodic condition. Here, we will have N l escapefunctions for orbits in Q l and N r escape functions fororbits in Q r . From now on, we will denote as U l and E l the escape functions and escape sets of the left region Gaspar Alfaro et al. Q l . Similarly U r and E r are defined in the right region Q r .Next, we briefly describe the algorithm to computethe escape functions. First, we start by computing thefunction U l ( q [ i ]) = max s (cid:16) min j (cid:0) u lin [ i, s, j ] (cid:1)(cid:17) . (9)Taking this function as a seed, we can compute therest of the escape functions with the following algorithm Loop until the functions U lk and U lk converges for k = 1 : N l − do U lk +1 ( q [ i ]) =max s (cid:16) min j (cid:0) max( u lin [ i, s, j ] , U lk ( q [ j ])) (cid:1)(cid:17) end U r ( q [ i ]) =max s (cid:16) min j (cid:0) max( u rout [ i, s, j ] , U lN l ( q [ j ])) (cid:1)(cid:17) for k = 1 : N r − do U rk +1 ( q [ i ]) =max s (cid:16) min j (cid:0) max( u rin [ i, s, j ] , U rk ( q [ j ])) (cid:1)(cid:17) end U l ( q [ i ]) =max s (cid:16) min j (cid:0) max( u lout [ i, s, j ] , U rN r ( q [ j ])) (cid:1)(cid:17) where u lin denotes the control applied to remain in Q l and u rin is the one to remain in Q r . On the other hand, u rout denotes the control needed to migrate from Q r to Q l and conversely u lout is the control to migrate from Q l to Q r .Now, we want to illustrate the computation of asimple example where we have chosen N l = 2, N r = 3and the upper disturbance bound ξ = 0 . U l and U r are shown in Fig. 10.For the computation of the escape sets E k , we choosethe control bound u = 0 . U lk ≤ u and U rk ≤ u so that we obtain E lk and E rk , respectively,as shown in Fig. 10. To control the orbit, we need tochoose an initial condition in one of the escape sets E lk or E rk , and then a suitable control is applied. Forexample, if we start with a point in E r , we applythe control in the next iterations to put the orbit in E r → E r → E l → E l → E r → E r → ... ( repeat ). Asa result, we obtain a chaotic motion modulated by aperiodic one. Q Q Fig. 10
On the top, the double parabola map with µ = 10.On the bottom, the escape functions computed for N l = 2(orange) and N r = 3 (cyan). Furthermore, the escape sets E r appear as blue boxes and E l as red boxes. We have used ξ = 0 .
015 as the disturbance bound and u = 0 .
014 as thecontrol bound, where this one corresponds to the minimumvalue of the escape functions.
To make this behavior even more clear, we showanother example considering now N l = 20, N r = 30, ξ = 0 .
015 and control bound u = 0 . Q l = [0 , .
5) and 30 iterations in Q r = orcing the escape: Partial control of escaping orbits from a transient chaotic region 9
50 100 150 200 25000.51
Typical orbit
50 100 150 200 25000.51
Controlled orbit
Used Control u =0.01350 50 100 150 200 250 Fig. 11
On the top, a typical orbit for the double parabolamap ( µ = 10). In the middle, we show an orbit affected by adisturbance ξ = 0 . u = 0 . N l = 20 iterations in Q l = [0 , .
5] and N r = 30 iterations in Q r = [0 . , u = 0 . [0 . , u = 0 . We have developed a new control algorithm based onthe partial control method aiming at keeping an orbiton a certain region Q for a given number of iterations N with a minimum control. For that purpose, we haveadapted known tools such as safe functions and safesets to the new escape functions and the escape sets.Once the latter are computed, it is straightforward tocontrol the orbit.We have considered three different possible scenar-ios. The first case, case A, where the main goal has beento force the escape of a given orbit of the phase space Q in N or less iterations. The second case, case B, wherethe goal has been to force the escape of the orbit tohappen in exactly N iterations.And the third scenario, case C, where we use thealgorithm in a situation where a system has chaotictransitions between two regions, with the goal to con-trol these transitions. As a consequence, this allows thecontroller to fix the number of iterations that the orbitwill stay in every particular region, making it to havea periodic sequence of transitions. Even though we have used one-dimensional mapsfor simplicity, we believe our control method is valid forhigher dimensions, and we hope that it can be appliedto different problems modeled with maps. Acknowledgements
This work was supported by the Span-ish State Research Agency (AEI) and the European RegionalDevelopment Fund (ERDF, EU) under Project No. PID2019-105554GB-I00.
Compliance with ethical standardsConflict of interest
The authors declare that theyhave no conflict of interest concerning the publicationof this manuscript.
References
1. E. Ott, C.Grebogi, and J. A. Yorke
Controlling chaos ,Phys. Rev. Lett. , 1196 (1990)2. K. Pyragas Continuous control of chaos by self-controllingfeedback , Phys. Lett. A , 421–428 (1992)3. S. Zambrano, M. A. F. Sanju´an, and J. Yorke
Partial con-trol of chaotic systems , Phys Rev E , 055201(R) (2008)4. S. Zambrano, and M. A. F. Sanju´an Exploring partial con-trol of chaotic systems , Phys Rev E , 026217 (2009)5. J. Sabuco, S. Zambrano, M. A. F. Sanju´an, and J. Yorke Finding safety in partially controllable chaotic systems ,Commun Nonlinear Sci Numer Simulat Dynamics ofpartial control , Chaos , 047507 (2012)7. R. Cape´ans, J. Sabuco, and M. A. F. Sanju´an, When lessis more: Partial control to avoid extinction of predators inan ecological model , Ecol. Complex. , 1–8 (2014)8. R. Cape´ans, J. Sabuco, and M. A. F. Sanju´an, Escapingfrom a chaotic saddle in the presence of noise , Int. J. Dyn.Control. , 78-86 (2018)9. R. Cape´ans, J. Sabuco, and M. A. F. Sanju´an, Partialcontrol of chaos: How to avoid undesirable behaviors withsmall controls in presence of noise , Discrete Cont Dyn-B , 3237–3274 (2018)10. R. Cape´ans, J. Sabuco, and M. A. F. Sanju´an, A newapproach of the partial control method in chaotic systems ,Nonlinear Dyn. , 873–887 (2019)11. E. L. Rempel and A. C.-L. Chian, Intermittency inducedby attractor-merging crisis in the kuramoto-sivashinskyequation , Phys Rev E , 016203 (2005)12. A. L. Livorati, I. L. Caldas, C. P. Dettmann, and E. D.Leonel, Crises in a dissipative bouncing ball model , PhysLett A , 2830–2838 (2015)13. S. Vaidyanathan, S. T. Kingni, A. Sambas, M. A. Mo-hamed, and M. Mamat,
A new chaotic jerk system withthree nonlinearities and synchronization via adaptive back-stepping control , Int. J. Eng. Technol. , 1936–1943 (2018)14. Y. F. Jin, Stochastic resonance in an under-dampedbistable system driven by harmonic mixing signal , ChinesePhys B27