VŠB - Technical University of Ostrava

Applications of DynamicalSystems

Marek Lampart

Habilitation Thesis Summary October, 2013

AcknowledgementsThe research was supported by projects provided by the Grant Agency ofthe Czech Republic, the Ministry of Education of the Czech Republic andstructural funds from the European Union. The above mentioned projectswere carried out at the Mathematical Institute of the Silesian University atOpava, the Department of Applied Mathematics and IT4Innovations of theVŠB-Technical University of Ostrava, Czech Republic. The support of theabove mentioned institutions and suppliers are hereby gratefully acknowl-edged.

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AbstractThe main aim of the habilitation thesis is to quantify and describe dynamicalproperties of models that are motivated by real models. There are analyti-cally researched discrete dynamical models: Lotka-Volterra biological models,Cournot economy models describing oligopoly, CML systems related to theBelousov-Zhabotinskii chemical reaction and finally dynamical systems onhyperspace that were motivated by set-valued functions used by Kakutani inhis fixed point theorem. Moreover, simulation results concerning continuousmodels of the electromechanical systems are detailed with analysis of theirdynamical properties.

AbstraktHlavním cílem předložené habilitační práce je kvantifikovat a popsat vlast-nosti dynamických modelů, které jsou motivovány reálnými problémy. Ana-lyticky jsou zkoumány následující diskrétní dynamické systémy: Lotkovy-Volterrovy biologické modely, Cournotovy ekonomické modely popisující oli-gopolii, CML systémy spjaté s Bělousovova-Žabotinského chemickou reakcía konečně dynamické systémy hyperprostoru, které jsou motivovány množi-nohodnotovými funkcemi, jež byly použity při konstrukci Kakutaniho věty opevném bodě. Dále jsou v práci uvedeny simulační výsledky spojitých mo-delů elektromechanických systémů s analýzou jejich dynamických vlastností.

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Contents1 Introduction 4

2 Structure and Goals of the Habilitation 6

3 Predator Prey Model 8

4 Coupled Lattice Maps 10

5 Cournot Oligopoly Model 13

6 Set Valued Dynamical Systems 14

7 Electromechanical Model 16

References 23

List of publications of Marek Lampart 24

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1 IntroductionThe theory of dynamical systems and chaos penetrates into many disciplineslike philosophy, art, biology, genetics, economy and other engineering fieldsthrough periodicity, recurrence, sensitivity, entropy and finally chaos.

As a first step in the analysis of dynamical properties, the periodic struc-ture should be explored. Here, famous fixed points theorems (i.e. Banach,Brouwer, Kakutani) can be utilized. So there are effective tools to find equi-libria of the systems. They could be stable or not reflecting the property ofthe system.

The first and crucial development in the field of dynamical systems wasmade by H. Poincaré in 1890 [40] by recurrence, that is a point returns toitself arbitrarily close under the actions (iterations), or equivalently the pointbelongs to its omega limit set. Hence, a dynamical system preserves volume,all trajectories return arbitrarily close to their initial position and they dothis an infinite number of times. More precisely, H. Poincaré discovered: Ifa flow preserves volume and has only bounded orbits then for each open setthere are orbits that intersect the set infinitely often.

As a consequence of the orbit observation situations where the trajectoryis dense in the state space appear. Such a property is called transitivity andcould be defined equivalently for action F (under some assumptions of thestate space): for any two non-empty open sets U, V , that are subsets of thestate space, there is n ∈ N such that Fn(U)∩V 6= ∅. The notion of transitivitywas introduced by G.D. Birkhoff in 1920 for flows [7].

Consequently, a transitive dynamical system has points which eventu-ally move under iteration from one arbitrarily small open set to any other.Such a dynamical system cannot be decomposed into two disjointed sets withnonempty interiors which do not interact under the transformation. So, thenotion of transitivity is still too rough for the observation of a local dynamicsand moreover it is not possible to quantify (and compare) the complexity ofsystems. The topological entropy (defined by R.L. Adler, A.G. Konheim andM.H. McAndrew in 1965 [2]) measures the complexity of the dynamical sys-tem. Later on the notion of topological entropy was equivalently formulatedfor compact metric spaces by R. Bowen in 1971 [9], namely

h(f) = limε→∞

lim supn→∞

1

nlogN(n, ε)

where N(n, ε) is the maximum cardinality of an (n, ε)-separated set. The

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subset of the state space is (n, ε)-separated if each pair of distinct points is atleast ε away in the metric dn(x, y) = max{d(f j(x), f j(y)) : 0 ≤ j < n}, hered is a metric in the state space. The dn metric permits us to distinguish in aneighborhood of an orbit those points that move apart from each other duringthe iteration from the points that travel together. The setN(n, ε) is finite andrepresents the number of distinguishable orbit segments of length n, assumingthat we cannot distinguish points within ε from one another. Obviously,the limit defining topological entropy always exists (h(f) ∈ [0,∞)). Hence,the topological entropy may be interpreted as the measure of the averageexponential growth of the number of distinguishable orbit segments. Finally,whether h(f) is positive or not is observed as standard. Positive topologicalentropy corresponds to the topological chaos.

However, the notion of chaos was used for the first time by Li and Yorke in1975 [34]. Let us recall that a pair of points is called Li-Yorke chaotic if limessuperior of distances of their iterations is positive while limes inferior is zero.That means that there are time sections where two orbits are arbitrarilyclose following time sections where they are distant enough. The systemswere originally assumed to be chaotic (in the sense of Li and Yorke) if the setof those pairs is uncountable. Later on more sophisticated versions of chaoswere constructed and compared to discover which one is better. There arestill open problems in this area. The chaos in the sense of Li and Yorke was,and still is, extensively studied by many authors.

Now, the properties that lead to chaos were introduced. It remains togive a formal definition of the dynamical system that will be researched inthe next sections. A dynamical system consists of a phase space (state space)X (here normally X ⊂ Rn) and a family of transformations φ : X→ X wherethe “time” t may be either discrete (t ∈ N) or continuous (t ∈ R). The familyobeys properties of identity and group

1. φ0(x) = x for all x ∈ X, and

2. φt(φs(x)) = φt+s(x) for all s, t and x ∈ X.

Hence, a discrete dynamical system could be defined as an ordered pair (X, f),where X usually is taken as a compact metric space and f is a continuousmap on X (not neccesarily onto). Finally, a continuous dynamical system(or real-time dynamical system, or flow) is a triple (T,M,Φ), where T ⊂ Rstandardly is an open interval, M is a manifold (locally diffeomorphic to aBanach space) and Φ is a continuous (differentiable) function.

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2 Structure and Goals of the HabilitationThis thesis consists of a collection of eight papers and a commentary showingtheir role in the development of dynamical systems and their applications.Also two closely related papers that have been submitted recently are at-tached.

Papers included in the Habilitation

TP1 Balibrea F, Guirao JLG, Lampart M, Llibre J. Dynamics of a Lotka-Volterra map. Fundamenta Mathematicae 2006; 191(3): 265–279.

TP2 Guirao JLG, Lampart M. Transitivity of Lotka-Volterra map. Discreteand Continuous Dynanical Systems – Series B 2008; 9(1): 75–82.

TP3 Guirao JLG, Lampart M. Positive entropy of a coupled lattice systemrelated with Belusov-Zhabotinskii reaction. Journal of MathematicalChemistry 2010; 48(1): 66–71.

TP4 Lampart M. Stability of the Cournot equilibrium for a Cournot oligopolymodel with n competitors. Chaos, Solitons and Fractals 2012; 45(9-10):1081–1085.

TP5 Guirao JLG, Lampart M, Zhang GH. On the dynamics of a 4d localCournot model. Applied Mathematics and Information Sciences 2013;7(3): 857–865.

TP6 Lampart M, Zapoměl J. Dynamics of the electromechanical systemwith impact element. Journal of Sound and Vibration 2013; 332(4):701–713.

TP7 Guirao JLG, Kwietniak D, Lampart M, Oprocha P, Peris A. Chaos onhyperspaces. Nonlinear Analysis: Theory, Methods and Applications2009; 71(1-2): 1–8.

TP8 Lampart M, Raith P. Topological entropy for set valued maps. Non-linear Analysis: Theory, Methods and Applications 2010; 73(6): 1533–1537.

Papers related to the Habilitation

TR1 Lampart M, Oprocha P. Chaotic sub-dynamics in coupled logistic maps.submitted to Chaos.

TR2 Lampart M, Zapoměl J. Dynamical properties of the electromechan-ical system damped by impact element with soft stops. submitted toMechanism and Machine Theory.

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There are five main sections that reflect applications of dynamical sys-tems. Discrete dynamical systems are considered from Sections 3 to 6. Thelast Section 6 is devoted to the analysis of the dynamical properties of thecontinuous dynamical model.

When selecting papers the following criteria were taken into consideration.Firstly, to cover the main topics of author’s interest and secondly, to fulfillformal criterion of the quality of the journal (high repute, impact factor etc.)which is usually applied by the author of habilitation.

Goals of the thesis

The thesis covers several topics from the field of dynamical systems acrossengineering disciplines. The main aims are:

1. to solve problemsIn [TP1] authors found an interior periodic point of a discrete predatorprey model that disproved the generally assumed hypothesis of its non-existence (it was stated as an open problem by A.N. Sharkovskiı in[45] whether the set of all periodic points is dense in the state space).In [TP4] there is revised the Cournot oligopoly model introduced byT. Puu in [41], where the model was assumed to lead to the perfectcompetition, i.e. the equilibrium is unstable. It was proved in [TP4]that the model is stable in one direction, hence does not lead to theperfect competition.

2. to describe dynamical properties of a concrete modelIn [TP2] the Lotka-Volterra model was shown, beside others to be tran-sitive. In [TP5] the Cournot model was constructed with the influenceof the neighboring competitors and its elementary dynamics were de-scribed. In [TP3] and [TR1] the Coupling Map Lattice (CML) mod-els related to the discrete chemical Belusov-Zhabotinskii reaction werestudied and the horseshoe was found. Finally, in [TP6] and [TR2] thecontinuous electromechanical model with impact element was simulatedand its dynamical behavior was discussed.

3. to add new pieces of knowledge to the theory of dynamical systemIn papers [TP7] and [TP8] a discrete dynamical system on a hyperspacewith naturally induced hyperspace map has been constructed. In thiscase the notion of chaos was investigated.

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3 Predator Prey ModelTwo-dimensional continuous maps of the plane

G(x, y) = (f(x, y), g(x, y))

are standardly constructed for the study of dynamical behavior of modelscoming from population dynamics, economy theory, social sciences and engi-neering.

The invariant subset, that is G(X) ⊂ X, of the state space is constructed;that define a dynamical system (X, G). The aim is to understand how thetrajectories evolve under the action G.

In applications the maps f and g are usually piecewise polynomials on X,e.g. the model of stability of synchronized states of Glendinning [22] or theDuffing transformation [50]. If the attention is restricted to the quadraticpolynomials of two variables the Lotka–Volterra transformation is derived inthe form

G(x, y) = (x(a1 + b1x+ c1y), y(a2 + b2x+ c2y))

where ai and bi ∈ R for any i. The special case of parameters was studiedby many authors, e.g. in [17] the situation for b1 = c1 = b2 = c2 = −1 wasdiscussed.

In papers [TP1] and [TP2] the investigated model was originally statedas an open problem in [45], where A.N. Sharkovskiı asked whether the setof all periodic points is dense in the state space. The mentioned systemwas obtained after some reductions that were done by A.N. Sharkovskiı on amodel given by Y. Avishai and D. Berend in [1]. In this case the model hasthe following parameters a1 = 4, b1 = −1, c1 = −1, a2 = 0, b2 = 0, c2 = 1,that is

F (x, y) = (x(4− x− y), xy)

here the invariant state space is the triangle 4 ⊂ R with vertices (0, 0), (4, 0)and (0, 4) which is strongly invariant, meaning F (4) = 4. It is easy to seethat the dynamics of (4|y=0, F ) equals the dynamics of the full parabolax(4 − x) on the unit closed interval, which is known, see e.g. [16], and thatother sides of4 are mapped into this base4|y=0. Hence, the most interestingpart is the interior of 4. To solve A.N. Sharkovskiı’s problem it is necessaryto find out the interior periodic points. In this direction there was, a myththat there are no other interior periodic points except the fixed one (1, 2). In[TP1] authors disproved this myth, the four cycle was found explicitly and

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the five cycle was described analytically. Motivated on these results in [36]it was proved that the set of interior points is infinite. The original problemstated by A.N. Sharkovskiı still remains open.

The crucial point in the study of the model was the observation that thereis a unique invariant curve joining points (0, 0) and (1, 2). It is spiral typewith focus at the point (1,2). It is strongly invariant and each of its pointsis eventually fixed, see [TP1].

For the understanding of the periodic structure of the map F the statespace 4 was decomposed into infinitely many pairwise disjoint regions ωnthat were ordered and union of their closure covers4. Each region is mappedby F onto its predecessor and the first one to the whole space 4. Theseregions help with detection of the periodic structure and the fact that thepoint (1, 2) is a global attractor.

As it was said above, it was proved in [TP1] that the map F has a uniqueperiodic trajectory of period four inside of 4. This trajectory is{(

2−√

2,1

2

),

(1 +

1√2, 1− 1√

2

),

(2 +√

2,1

2

),

(1− 1√

2, 1 +

1√2

)}.

Next, the map F has a unique periodic trajectory of period five inside of 4.The periodic point with period five is approximated by the point

(x, y) = (−0.7873282213706,−1.5245697755205).

Constructions that lead to the discovery of previous periodic points are basedon the procedure of the resultant. In the case of the forth cycle roots of thepolynomials of the fourth order have to be found and in the case of the fifthcycle roots of the polynomial of degree five, hence for computational reasonsit is very hard to analyze (using stated method) trajectories with a periodhigher then five.

Continuing in the analysis of the periodic structure it is worthy to note,that there are three fixed points for map F . Two of them (0, 0) and (3, 0) areon the boundary of region 4 and the last one (1, 2) is inside 4. It was provedin [47] that the set of all pre-images of point (0, 0) forms a dense subset of4. In this direction it is also proved in [TP2] that the set of all pre-imagesof point (1, 2) is a dense subset of 4.

Consequently, the natural question arises of whether system (4, F ) ischaotic. Contributions to this problem are the following. Firstly, it wasproved that the model is transitive. Secondly, the model is almost topologi-cally exact but not topologically exact. Here, a map is topologically exact if

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for any non-empty set U ⊂ 4 there is n ∈ N such that Fn(U) = 4, and amap is almost topologically exact if for any non-empty set U ⊂ 4 and ε > 0there is n0 ∈ N such that

Fn(U) ⊂ Fn+1(U) if n > n0 (1)Fn(U) ⊃ 4 \ Pε if n > n0 (2)

where Pε = {(x, y) ∈ 4 : y < ε} and ε → 0 for n → ∞. Let us point outthat topological exactness implies almost topological exactness that impliestransitivity. The converse implications are not valid in general.

Consequently, it was shown in [TP2] that system (4, F ) is transitive andalmost topologically exact but it is not topologically exact. Hence, naturalquestions arise:

Does system (4, F ) have positive topological entropy, specificationproperty or some type of shadowing property?

4 Coupled Lattice MapsWell-known models for spatio-temporal chaos are systems of partial differ-ential equations that are classically assumed to be the most physical models[14]. Unfortunately, these models are not consonant with numerical simula-tions and they are hard to understand from a dynamical point of view. Onthe other hand, cellular automata are numerically compliant but the theoryof dynamical systems is hard to use for the analysis of dynamical propertiessince the state space is discrete. The transitional approach on this settingis the so-called Coupled Lattice Maps (CML) which are defined over discretetime and continuous variables (see [25] and references therein).

CML systems were introduced in 1980’s through a series of papers relatedto the topic, see e.g. [15], [24], [48]. In [15] CML systems were used formodeling chemical spatial phenomena. Application of CML was used in [30]for electrical circuitry by developing a renormalization group approach. Amore general approach was introduced by [24] and it is still active, this areais developing in (see [27] and references therein): (1) spatio-temporal chaos,(2) statistical mechanics of an ensemble of chaotic elements, (3) turbulence,(4) pattern dynamics, (5) neural dynamics and applications for informationprocessing, and (6) biological and traffic network problems.

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The most tested discrete CMLmodel was introduced in 1983 by K. Kanekoin [24] where the difference equation introduced was as follows:

xt+1n = (1− ε)f(xtn) + ε/2(f(xtn+1) + f(xtn−1))

where t ∈ N corresponds to the discrete time, ε ∈ [0, 1] is the couplingconstant and f is a real map standardly chosen as a logistic map i.e. f(x) =4x(1− x).

In [TP3] it was shown that the CML system stated in [26], which is re-lated to the Belushov-Zhabotinskii chemical reaction, is topologically chaoticand that is has positive topological entropy. As the main result of [TP3] aconjugacy that maps a suitable subset of the state space to the shift on ksymbols where k is dependent on the size of the lattice was constructed. Thestudy was conducted for the easiest situation, the case of a zero couplingconstant. In this case the coupled map is equal to the Cartesian product ofthe base map. Hence, dynamics could be easily derived. Now, the desire isto analyze the situation for a non-zero coupling constant. In [33] there aretwo-ways of communication (so-called Laplacian-type coupling)

xin+1 = (1− ε)g(xin) +ε

2g(xi−1n ) +

ε

2g(xi+1

n ).

The above type of coupling is a restricted version of variable range coupling(e.g. see [38]):

xin+1 = (1− ε)g(xin) +ε

η(α)

N∑j=1

1

jα(g(xi−jn ) + g(xi+jn )).

Simply, we assume that α is very large (say α→∞) so coupling introducedby terms with j 6= 1 is not essential and can be ignored. Now, let us assumethat initial values on the lattice are periodic, say x0i = x0j provided that i =j (mod 2). Then we can view Laplacian-type coupling as a two dimensionalmap, after which

xin+1 = (1− ε)g(xin) +ε

2g(xi−1n ) +

ε

2g(xi+1

n )

= (1− ε)g(xin) + εg(xi+1n ) = (1− ε)g(xin) + εg(xi−1n ).

Moreover, the model was generalized towards the base map g, the familyof logistic maps was taken into consideration as fµ(x) = µx(1−x), hence theCML model is derived

F (x, y) = ((1− ε)fµ(x) + εfµ(y), (1− ε)fµ(y) + εfµ(x)). (3)

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The above introduced model (3) was studied in [29], some partial resultsand simulations were given as dependent on both parameters, coupling andµ. It was proved that the system is chaotic in the sense of Li and Yorke forsome ranges of parameters. The main result of [TR1] is to detect regionsof parameters for which the model shows a nontrivial chaotic behavior. Ob-viously, there will always be chaotic dynamics on the diagonal (i.e. pairs(x, x)), when fµ is chaotic itself, since if xin = xi+1

n then xin+1 = fµ(xin).Then one of our challenges is to detect regions outside the diagonal, wherechaotic dynamics can be supported.

One of our goals for [TR1] was to split the region of parameters intoseveral areas. For the first area all points are attracted to the main invariantsubsystem which is always embedded in the diagonal and for the second casethe model shows chaotic motion outside the diagonal, i.e. there is a horseshoelocated outside the diagonal. The study is focussed on the two dimensionalcase (i.e. lattice with 2-periodic entries), since in higher dimension (i.e. largerperiod on the lattice) the problem could be dealt similarly by analogouscalculations and arguments.

In particular, there is z from the state space [0, 1] such that

limn→∞

||Fn(x, y)− Fn(z, z)|| = 0,

that is dynamics of every point is asymptotically reflected by the dynamicsof a point on the diagonal. When µ > 2 the situation becomes much morecomplex. If the point (x, y) approaches the diagonal under iterations thenit is easy to construct a sequence zn ⊂ Iµ whereby limn→∞ ||F (zn, zn) −(zn+1, zn+1)|| = 0 and limn→∞ ||Fn(x, y) − (zn, zn)|| = 0. If fµ has a so-called limit shadowing property in the core Iµ, then there is z ∈ Iµ =[f2µ(1/2), fµ(1/2)] such that

limn→∞

||Fn(x, y)− Fn(z, z)|| ≤ limn→∞

||Fn(x, y)− (zn, zn)||+

||Fn(z, z)− (zn, zn)|| = 0.

But it is known that there are numerous values of µ such that fµ is conjugatedon the core to a tent map without shadowing [11, 10] while it was recentlyproved that transitive maps with limit shadowing also posses a shadowingproperty [32]. In view of the above, the following question arises:

Let (µ, ε) ∈ {(µ, ε) ∈ [0, 4] × [0, 1] : 1 > µ|1 − 2ε|} and fix any(x, y) ∈ [0, 1]2 \∆. Does z ∈ [0, 1] exist, whereby limn→∞ ||Fn(x, y)−Fn(z, z)|| = 0?

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5 Cournot Oligopoly ModelThe main aim of papers [TP4] and [TP5] was to consider the Cournot pointsand discuss their stability while the number of players is increasing for themodel with an iso-elastic demand function and under the assumption thatthe firms’ costs are identical. The terminology of dynamical systems is used,that is the Cournot point is identified as fixed.

The following construction was inspired by the work of T. Puu [41] whoconstructed the model for two competitors (later in [42] for three players).This model could be extended for n firms.

Assuming that the level of demand is reciprocal to price, this representsan “iso-elastic” demand function reflecting a case where consumers alwaysspend a constant sum on the commodity, regardless of price. Inverting thedemand function gives the price that equals to the reciprocal of the sum ofthe supplies.

The revenues of these firms equal price times quantity. Assuming thatthe firms operate under constant unit costs ci, so ci > 0 for any i. Theirtotal costs are cixi where and xi are competitors. So, the profits becomerevenues subtracted by total costs. In order to maximize profits for the firmput their partial derivatives with respect to xi equal to zero. Hence, thesereaction functions are derived.

Introducing the adjustment process explicitly, the difference equations ofthe model are obtained. Thus, a dynamical system is deduced

(Xn, Fn) (4)

defined as follows for n ≥ 2.Firstly, set

Fn : Rn → Rn

defined by difference equations derived above.Secondly, the invariant set Xn is found. It is clear that the domain of Fn

isDn =

{(ix) ≥ 0, for any i

}.

Such a model is considered so that the process can be repeated, thus focussingon a subset of Dn for which F tn(x) ∈ Dn for any t ≥ 0. Such points in Dn areadmissible (see [3]). Unfortunately, not all admissible points are meaningful.Since economic interpretation of a point with negative value is not acceptablethe attention has to be restricted to feasible points. The first iteration needs

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to be taken into account. To do this, this system of equations that reflectthe image of points has to be solved. Repeating this process for all iterationsgives

Xn ⊂ Dn (5)

which is invariant under Fn. Unfortunately, it is not easy to ascertain the setXn explicitly.

It is also easy to see that a fixed point x? of Fn is the Cournot equilibrium(Nash equilibrium of the game) (see e.g. [3]).

Hence, the task is to discover the fixed points of system Xn ⊂ Dn and todetermine their stability.

In [TP4] it was proved that for n ≥ 2 the dynamical system (Xn, Fn) hastwo fixed points, one trivial (zero) f0 and one non-trivial f1 that correspondsto the Cournot point. Consequently, the fixed point f0 of Fn is a source forany n ≥ 2 and ci = c and the Cournot point f1 of Fn is a sink for any n = 2, 3and ci = c and it is a saddle for any n > 4 and ci = c which means that thedesired model does not lead to the perfect competition. In accordance withthe character of the Cournot point an open question can be formulated:

How to improve the given model to get perfect competition (an unsta-ble equilibrium)?

On the other hand, it is possible to construct a model “à la Cournot”with the additional assumption that the firms compete with their closestcompetitors in either direction. The system (Yn, Gn) for four competitors wasintroduced in [21]. Such systems are equal to (Yn, Gn) where the constructionof state space Y exhibits the analogous difficulties as the construction of Xdescribed above. It was proved in [TP5] that system (Y4, G4), for suitablevalues of parameters, has any periodic point of period at most 2. Moreover,Per(G) = {(0, 0, 0, 0)} ∪ {Pi : i = 1, . . . , 6} where (0, 0, 0, 0) is the fixedpoint and points Pi form three two cycles. Furthemore, system (Y4, G4) haspositive topological entropy and is Li-Yorke chaotic for suitable parameters;has zero topological entropy and is not Li-Yorke chaotic for other values ofparameters.

6 Set Valued Dynamical SystemsIt is known that numerous real problems (see e.g. sections above and refer-ences therein) ranging from physics and chemistry to ecology and economics

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can be reduced to the study of the dynamical system (X, f) where all pointsxn from the compact metric space are mapped by a continuous transforma-tion T

xn+1 = T (xn). (6)

The main aim of the theory of discrete dynamical systems is to understandwhat all trajectories of points from the state space look like. Nevertheless,to understand many events that appear in fields such as biological species,demography, simulations etc. it is necessary to know how subsets of X aremoved. Hence, the system (6) induces in a natural way a set valued dynamicalsystem

Kn+1 = T (Kn) (7)

where map T represents the movement of individuals while T the movement ofcollectives. Here, T is defined as T (K) = T (K) for any non-empty compactsubset of X. So, a dynamical system is derived (K(X), T ), here the statespace K(X) is endowed with the Vietoris topology (or equivalently with theHausdorff metric).

It is worthy noting that the above introduced set valued dynamical systemis a generalization of a so-called set valued maps. The most famous resultin this field is the Kakutani fixed-point theorem from 1941 [28] that wasfamously used by J.F. Jr. Nash in his description of so-called Nash equilibria[37]. The Nash theory subsequently found widespread applications in gametheory and economics [8] and later in 1994 he received the Nobel MemorialPrize in Economic Sciences (along with J. Harsanyi and R. Selten) as a resultof his game theory work as a Princeton graduate student.

Returning to the research of dynamical systems the fundamental andnatural question arises. The dynamical system (X, T ) induces in a naturalway a set valued dynamical system (K(X), T ). Does (K(X), T ) have the samedynamical properties as (X, T )? Are the dynamics of (X, T ) preserved whilemoving to (K(X), T )? And conversely?

The first result on this setting, known to the author of this thesis, wasgiven in [6] and states that if a continuous map on a compact metric spacehas positive topological entropy then induced system (K(X), T ) has infinitetopological entropy, see also [31]. Motivated by this result authors in [TP8]distinguished if the induced map is defined on K(X) or C(X) the set of allnon-empty, compact and connected subsets of X. It was firstly proved in[TP8] that homeomorphisms on the unit closed interval or a circle havingzero topological entropy induces maps on C(X) that also have zero topological

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entropy. On the other hand it was also shown that a homeomorphism on theunit interval having second iteration non-identical with zero entropy induceson K(X) system with infinite topological entropy. Concluding observations ontopological entropy it was proved that there is a disconnected compact metricspace and a homeomorhpism on it with zero entropy whereas the induced mapon C(X) is again null and induced map K(X) has infinite topological entropy.Consequently, an open question was formulated (remains unsolved):

Which topological spaces X satisfy that h(T |K(X)) ∈ {0,∞} for allcontinuous maps T (or at least homeomorphisms)?

In [43] the transmission of transitivity was discussed. More precisely, if acollective system is transitive then the individual is also. A counter examplefor the opposite implication was also given, see also [39]. On the other handsome partial results on “chaos" were also published: in [51] on Devaney’schaos, in [23] on Block-Coppel’s chaos, in [44] on Robinson’s chaos, in [19]on Kato’s chaos and in [5], [20] on mixing properties.

The main contribution of [TP7] is the characterization of the movementof notions of chaos between individual and collective dynamical systems. Be-side others, the transition of the distributional, Li-Yorke, Devaney (and itsvariants) chaos and specification properties were discussed. Definitions ofall notions could be found in [TP7] (see also references therein). Closingthe section, in [TP7] it was proved that Li-Yorke chaos is transmitted fromindividual systems to the collective system, but not conversely.

7 Electromechanical ModelThe impact of solid bodies are important mechanical phenomena that couldbe observed in natural and technological processes. The experience and theo-retical analyses demonstrate that the behavior of the impact systems is highlynon-linear, sensitive to initial conditions and it is hard to predict their move-ments. The vibro-impact systems are governed by the momentum transferand mechanical energy dissipation through the body collision. These systemsare utilized for the attenuation of high oscillation amplitude, such as thoseappearing in subharmonic, self-excited and chaotic vibrations. The prob-lem of impacts is very old (the absorber was patented by H. Frahm in 1911[18]), the new prospective studies are possible due to new effective compu-tational simulations. Dynamical behavior of an impact damper for vibration

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yz

yt

y

k b

ϕ

kt

4

3

2

1

eT

Figure 1: Model of vibrating system

attenuation was studied by many authors from different points of view andcontrasting purposes, see e.g. [12], [13], [49], [46] and [35].

The investigated system consists of a rotor (body 1, Figure 1), of its casing(body 2, Figure 1) and of a base plate (body 3, Figure 1), with which the rotorcasing is coupled by a spring and damping element. The casing and the baseplate can move vertically and the rotor can rotate and slide together with itscasing. Vibration of the base plate and unbalance of the rotor are the mainsources of casing excitation. To attenuate its oscillation an impact damperhas been proposed. It consists of a housing fixed to the rotor casing (body 2,Figure 1) and of an impact body (body 4, Figure 1), which is coupled withthe housing by a linear spring. The impact body can only move verticallyand is separated from the housing by lower and upper clearances that limitits vibration amplitude. The rotor is loaded by an external moment producedby a DC electric motor.

The task was to investigate influence of the upper and lower clearancesbetween the rotor casing and the impact body respectively on attenuation ofthe rotor frames oscillation and the character of its motion.

The mathematical model of the system has three degrees of freedom. Itsinstantaneous position is defined by three generalized coordinates: y-verticaldisplacement of the rotor casing, yt-vertical displacement of the impact bodyand ϕ-angular rotation of the rotor. The system vibration is governed by the

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equations of motion that have been derived by application of the Lagrangeequations of the second kind. It is assumed that there is no vibration in thebeginning of the simulation, the system is at rest and assumes the equilibriumposition that yields initial conditions.

Now, in [TP7] all bodies of the model are considered as absolutely rigid(Newton theory). The spring element coupling the rotor casing and the baseplate is nonlinear showing a cubic characteristic. Dynamics of the model areforced, not only by the rotor unbalance but also by vibration of the baseplate that plays the key role here. The computational simulations provedthat application of the impact body resulted in a significant decrease in vi-bration amplitude of the rotor frame. It was observed that there are valuesof parameters for which the vibration of the rotor casing is close to periodic(quasi-periodic), pre-chaotic or chaotic respectively, showing the existence ofbifurcation borders. The dynamical behavior of the movement was studiedusing the Fourier spectra and phase trajectories that are formed by a num-ber of harmonic components having the basic, super-harmonic, sub-harmonicand combination frequencies on which there further motions are superposedwith frequencies forming the sided bands of dominant frequencies. Theirmutual ratio indicates the irregularity of the vibration or regularity respec-tively. Consequently, the resonance frequency was found, and the best choiceof clearance for the given weight of the impact element was simulated for thepurpose of maximal attenuation of the rotor frame.

In addition to [TP7] it was assumed in [TP8] that the model exhibits:(1) soft stops, the Hertz theory has been accepted, the contact stiffness anddamping are considered as linear; (2) the current of the electric circuit. Thesimulation analysis of the model again shows a variety of dynamical behavior,periodicity, quasi-periodicity and chaos for suitable values of parameters. Thevalue of clearance were the attenuation is significant and the minimum totalvibration of the rotor frame was discovered.

In conclusion, let us note that the method for the detection of chaos usedhere was based on detection of the Fourier spectra as a classical and standardtool. The tool of the Lyapunov exponents (again classical) was not appliedhere due to the fact that standard numerical methods do not fit with themodel. In this direction a question appear:

Is it possible to introduce a new and effective procedure for the com-putation of Lyapunov exponents? How to verify chaos like Melnikov[4] (or other) using simulations?

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[TP7] Guirao JLG, Kwietniak D, Lampart M, Oprocha P, Peris A. Chaos onhyperspaces. Nonlinear Analysis: Theory, Methods and Applications2009; 71(1-2): 1–8.

[TP2] Guirao JLG, Lampart M. Transitivity of Lotka-Volterra map. Discreteand Continuous Dynanical Systems – Series B 2008; 9(1): 75–82.

[TP3] Guirao JLG, Lampart M. Positive entropy of a coupled lattice systemrelated with Belusov-Zhabotinskii reaction. Journal of MathematicalChemistry 2010; 48(1): 66–71.

[TP5] Guirao JLG, Lampart M, Zhang GH. On the dynamics of a 4d localCournot model. Applied Mathematics and Information Sciences 2013;7(3): 857–865.

[19] Gu R. Katos chaos in set-valued discrete systems. Chaos, Solitons andFractals 2007; 31(3): 765–771.

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[20] Gu R, Guo W. On mixing property in set-valued discrete systems.Chaos, Solitons and Fractals 2006; 28(3): 747–754.

[21] Guirao JLG, Rubio RG. Extensions of Cournot duopoly: An appliedmathematical view. Applied Mathematical Letters 2010; 23(8): 836–838.

[22] Glendinning P. Milnor attractors and topological attractors of a piece-wise linear map. Nonlinearity 2001; 14(2): 239–257.

[23] Honary B, Jazaeri M. Block-Coppels chaos in set-valued discrete sys-tems. Iranaian Journal of Numerical Analysis and Optimization 2013;3(1): 9-12.

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[32] Kulczycki M, Kwietniak D, Oprocha P. On almost specification andaverage shadowing properties. [Preprint]

[TP4] Lampart M. Stability of the Cournot equilibrium for a Cournotoligopoly model with n competitors. Chaos, Solitons and Fractals2012; 45(9-10): 1081–1085.

[TR1] Lampart M, Oprocha P. Chaotic sub-dynamics in coupled logisticmaps. submitted to Chaos.

[TP8] Lampart M, Raith P. Topological entropy for set valued maps. Non-linear Analysis: Theory, Methods and Applications 2010; 73(6): 1533–1537.

[TP6] Lampart M, Zapoměl J. Dynamics of the electromechanical systemwith impact element. Journal of Sound and Vibration 2013; 332(4):701–713.

[TR2] Lampart M, Zapoměl J. Dynamical properties of the electromechan-ical system damped by impact element with soft stops. submitted toMechanism and Machine Theory.

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[42] Puu T. Complex dynamics with three oligopolists. Chaos, Solitons andFractals 1996; 7(12): 2075–2081.

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List of publications of Marek LampartComplete overview (up to October 31, 2013)

Journal papers (articles): 17 (14 of them published in impacted journals)Proceedings papers: 4

Textbooks for students: 6Books: 1

Database records overview

source H-index citations citations(total) (self-citations excluded)

SCOPUS 5 56 41Web of Knowledge 6 75 37Google Scholar 7 105 ×

Mathematical Reviews 3 × 29

List of publications

ML1 Lampart M. Scrambled sets for transitive maps. Real Analysis Ex-change 2001; 27(2): 801–808.

ML2 Lampart M. Two kinds of chaos and relations between them. ActaMathematica Universitatis Comenianae 2003; 72(1): 119–129.

ML3 Guirao JLG, Lampart M. Li and Yorke chaos with respect to the cardi-nality of the scrambled sets. Chaos, Solitons and Fractals 2005; 24(5):1203–1206.

ML4 Lampart M. Chaos, transitivity and recurrence. Grazer MathematischeBerichte 2006; 350: 169–174.

ML5 Guirao JLG, Lampart M. Relations between distributional, Li-Yorkeand ω chaos. Chaos, Solitons and Fractals 2006; 28(3): 788–792.

ML6 Balibrea F, Guirao JLG, Lampart M, Llibre J. Dynamics of a Lotka-Volterra map. Fundamenta Mathematicae 2006; 191(3): 265–279.

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ML7 Guirao JLG, Lampart M. Transitivity of Lotka-Volterra map. Discreteand Continuous Dynamical Systems-Series B 2008; 9(1): 75–82.

ML8 Guirao JLG, Kwietniak D, Lampart M, Oprocha P, Peris A. Chaos onhyperspaces. Nonlinear Analysis: Theory, Methods and Applications2009; 71(1-2): 1-8.

ML9 Lampart M, Oprocha P. On omega chaos and specification property.Topology and its Applications 2009; 156(18): 2979–2985.

ML10 Guirao JLG, Lampart M. Positive entropy of a coupled lattice systemrelated with Belusov-Zhabotinskii reaction. Journal of MathematicalChemistry 2010; 48(1): 66–71.

ML11 Guirao JLG, Lampart M. Chaos of a coupled lattice system relatedwith Belusov-Zhabotinskii reaction. Journal of Mathematical Chem-istry 2010; 48(1): 159–164.

ML12 Lampart M, Raith P. Topological entropy for set valued maps. Nonlin-ear Analysis: Theory, Methods and Applications 2010; 73(6): 1533–1537.

ML13 Balibrea F, Dvorníková G, Lampart M, Oprocha P. On negative limitsets for one-dimensional dynamics. Nonlinear Analysis: Theory, Meth-ods and Applications 2012; 75(6): 3262–3267.

ML14 Lampart M. Stability of the Cournot equilibrium for a Cournot oligopolymodel with n competitors. Chaos, Solitons and Fractals 2012; 45(9-10):1081–1085.

ML15 Lampart M, Zapoměl J. Dynamics of the electromechanical system withimpact element. Journal of Sound and Vibration 2013; 332(4): 701–713.

ML16 Guirao JLG, Lampart M, Zhang GH. On the dynamics of a 4d localCournot model. Applied Mathematics and Information Sciences 2013;7(3): 857–865.

ML17 Balibra F, Guirao JLG, Lampart M. A note on the definition of alphalimit set. Applied Mathematics and Information Sciences 2013; 7(5):1929–1932.

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