Using MatContM in the study of a nonlinear map in economics

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1 Journal of Phsics: Conference Series PAPER OPEN ACCESS Using MatContM in the stud of a nonlinear map in economics To cite this article: Niels Neirnck et al 016 J. Phs.: Conf. Ser Related content - European Conference - Workshop Nonlinear Maps and Applications - Averaging of Random Flows of Linear and Nonlinear Maps Vsevolod Zh. Sakbaev - Relativistic quantum equation in nonstationar space-time geometr L Gaffour View the article online for updates and enhancements. Recent citations - Subharmonics and Chaos in Simple Periodicall Forced Biomolecular Models Evgeni V. Nikolaev et al This content was downloaded from IP address on 1/0/018 at 0:

2 doi: /17-696/69/1/0101 Using MatContM in the stud of a nonlinear map in economics Niels Neirnck 1, Bashir Al-Hdaibat 1, Will Govaerts 1, Yuri A. Kuznetsov, and Hil G E Meijer 1 Ghent Universit, Ghent, Belgium Utrecht Universit, Utrecht, Netherlands Universit of Twente, Twente, Netherlands Niels.Neirnck@UGent.be Abstract. MatContM is a MATLAB interactive toolbo for the numerical stud of iterated smooth maps, their Lapunov eponents, fied points, and periodic, homoclinic and heteroclinic orbits as well as their stable and unstable invariant manifolds. The bifurcation analsis is based on continuation methods, tracing out solution manifolds of various tpes of objects while some of the parameters of the map var. In particular, MatContM computes codimension 1 bifurcation curves of ccles and supports the computation of the normal form coefficients of their codimension two bifurcations, and allows branch switching from codimension points to secondar curves. MatContM builds on an earlier command-line MATLAB package Cl MatContM but provides new computational routines and functionalities, as well as a graphical user interface, enabling interactive control of all computations, data handling and archiving. We appl MatContM in our stud of the monopol model of T. Puu with cubic price and quadratic marginal cost functions. Using MatContM, we analze the fied points and their stabilit and we compute branches of solutions of period, 10, The chaotic and periodic behavior of the monopol model is further analzed b computing the largest Lapunov eponents. 1. Introduction We consider dnamical sstems generated b smooth nonlinear maps f(, α), R n,α R m (1) with state variable and parameter vector α. The understanding of the dnamics of the map (1) requires most often numerical bifurcation analsis with the use of a dedicated software package. For this purpose we developed MatContM, a MATLAB continuation toolbo available at MatContM builds on the command line code Cl MatContM described in [], [] and[6] but supports several new functionalities, as well as providing a uniform interface. A comprehensive tutorial is provided that illustrates the use of MatContM b investigating an eample model. The tutorial can be found on matcontmp/tutorial MatcontMGUI.pdf. Eperienced users can also use the code in MatContM via the MATLAB command line, which in some cases can etend the capabilities of the software. MatContM computes bifurcations of fied points and ccles in one parameter, continues bifurcation curves in two parameters and detects codimension bifurcation points using Content from this work ma be used under the terms of the Creative Commons Attribution.0 licence. An further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence b Ltd 1

3 doi: /17-696/69/1/0101 v v =(α 0, 0 ) h v 0 α 0 α Figure 1: Schematic representation of pseudo-arclength continuation. The represents the state space, the α represents the varing parameter, also referred to as the free parameter. MatContM uses a variant of Moore-Penrose continuation which builds upon pseudo-arclength continuation. numerical continuation. A variant of pseudo-arclength continuation is used to approimate these curves (Figure 1). Bifurcations are detected b encountering zeros of test functions during the continuation process. The video file animation1.mpg demonstrates the continuation process for a fied point continuation and animation.mpg demonstrates how test functions are used to detect bifurcations (see Appendi A). Critical normal form coefficients are computed at bifurcation points, using smbolic derivatives or if these are not available, finite differences or automatic differentiation is used. Automatic differentiation is usuall slow, but becomes more competitive for ccles with high iteration number. We note that there is another MATLAB interactive toolbo called MatCont [] that provides numerical continuation and bifurcation algorithms for the stud of parameterized continuoustime dnamical sstems (ODEs).. Features and functionalities of MatContM In a tpical use of MatContM, one starts with an initial fied point or ccle, which ma be obtained from analsis, simulations or previous continuations. One first computes curves of fied points or ccles under variation of one parameter, and ma detect bifurcation points on such curves. Starting from such bifurcation points, the continuer algorithm in MatContM can compute bifurcation curves. These curves are defined b a sstem of equations consisting of fied point and bifurcation conditions. With one free parameter we can also compute curves of connecting orbits. B varing two sstem parameters we can compute bifurcation curves of limit points, period-doubling and Neimark-Sacker points as well as tangencies of homoclinic and heteroclinic orbits. The sstems that define connecting orbits and their tangencies are fairl complicated as the involve the saddle equilibria at the endpoints of the orbits, their eigenspace structures and the whole connecting orbit. The following list contains functionalities that are provided b MatContM: Simulation (iteration) of maps, i.e. computation and visualization of orbits (trajectories).

4 doi: /17-696/69/1/0101 Computation of the Lapunov eponents of long trajectories. Continuation of fied points of maps and iterates of maps with respect to a control parameter. Detection of fold (limit point), flip (period-doubling point), Neimark-Sacker and branch points on curves of fied points. Computation of normal form coefficients for fold, flip and Neimark-Sacker bifurcations. Continuation of fold, flip and Neimark-Sacker bifurcations in two control parameters. See animation.mpg in Appendi A. Detection of all codimension fied point bifurcations on curves of fold, flip and Neimark- Sacker bifurcations. Computation of normal form coefficients for all codimension bifurcations of fied points. Switching to the period doubled branch in a flip point. Branch switching at branch points of fied points. Switching to branches of codimension 1 bifurcations rooted in codimension points. See animation.mpg in Appendi A. Automatic differentiation for normal form coefficients of codimension 1 and codimension bifurcations. Computation of one-dimensional invariant manifolds (stable and unstable) and in the twodimensional case computing their transversal intersections to obtain initial homoclinic and heteroclinic connections. Continuation of homoclinic and heteroclinic orbits with respect to a control parameter and the detection of tangencies on the curve of orbits. See animation.mpg in Appendi A. Continuation of homoclinic and heteroclinic tangencies in two control parameters. See animation.mpg in Appendi A.. Practical use of MatContM A simple but important feature is that MatContM has the abilit to simulate maps, which means that it can compute orbits (trajectories). These orbits can also be visualized and their Lapunov eponents can be computed. The advanced use of MatContM relies on the abilit to continue curves under variation of parameter(s) and appl advanced bifurcation theor on given eamples of maps. The aim of the package is to allow a user to perform a bifurcation analsis of a map, without deep knowledge of the workings of MatContM or even the continuation software. The user is able to enter a sstem, an initial fied point and start computing with most of the settings left on default. This should lower the entr barrier for researchers from different research fields who want to investigate their models but do not want to be confronted with specification and implementation details. The continuation curves can be visualized using the plot capabilities of MatContM; thiscan be done during and after the continuation. Tools are provided to help with managing sstems, diagrams and curves when generating large amount of data. Specific features have been implemented to make branch switching between continuation curves fast and eas. An initial point can be selected out of a list of special points or it can be selected b double-clicking on a graph of the computed curve. A list of available curves for computation is shown depending on the tpe of the initial point. An interface is also provided that allows the echange of information between MatContM and the MATLAB command line. This enables the user to combine MatContM with other MATLAB software or simpl to use computed data on the MATLAB command line.

5 doi: /17-696/69/1/0101 Figure : This action screenshot of MatContM presents the main windows that are opened during the computation of a cascade of period-doubling bifurcations in a predator-pre model [1]. Bottom right is the MatContM main window which contains general information on the model that is being used and the top-level commands. Top left is the Continuer window which shows the continuation variables. The fields of this window are independent of the particular curve that is being computed, though the numeric values can var. Bottom left is the Starter window whose fields strongl depend on the curve that is being computed. Top middle is the Data Browser which allows to inspect all introduced sstems, computed curves, etcetera. Top right is the Numeric window which during computation provides numerical values of computed quantities. Bottom middle is the D plot window which in this case plots several bifurcations curves; the horizontal ais presents a parameter of the sstem and the vertical ais is a state variable. The points are flip bifurcation points; the period-doubling cascade is clearl visible. Figure is an action screenshot of MatContM that gives a visual impression of the computation of a cascade of period-doubling bifurcations in a predator-pre model [1]. The video file animation.mpg shows a two-parameter bifurcation analsis of the same model (Appendi A). The bifurcation diagram in animation.mpg can be replicated using MatContM b following the tutorial available on the website.. Stud of a nonlinear map in economics We use MatContM to numericall analze a monopol model first suggested b Puu [7] which assumes a cubic price and quadratic marginal cost function when maimizing profits using as strategic variable the produced quantit. The price function p() =A B + C D

6 doi: /17-696/69/1/0101 is kept monotonicall decreasing b requiring that A, B, C and D are strictl positive and C < BD. The revenue of the monopolist is The marginal revenue is then given b The marginal cost is assumed to be R() =p(). MR = dr d = p + dp d. MC = E F+G, where the parameters E, F and G are positive constants. The profit is maimized when MR = MC using standard results of economic theor. The profit function is Π() =(A E) (B F ) +(C G) D. () The profit function () is assumed to be not eplicitl known b the monopolist. In [7] asimple Newton-Raphson-like algorithm is used to find the maimum of (). The last two visited points t and t are evaluated and using a step size δ we obtain the net point t+1 as t+1 = t + δ Π(t) Π(t) ( t t = t + δ (A E) (B F )( t + t )+ (C G)( t + t t + t ) D( t + t t + t t + t ) ) () The iteration of this procedure ma lead to convergence to a profit maimum, an oscillating process or chaos depending on the value of the step size δ and the parameters. Following [7] we assume A =.6, B =.7, C =0.6, D =0.0, E =,F =0. andg =0.0. Using these parameter values, the profit function () issmmetricaboutthepoint(, ).Wekeepthestep size δ as a parameter. The two step update process () can be interpreted as the two dimensional map: M : with one parameter δ, where ( t t ) ( t+1 ) ( = t+1 t t + δp( t, t ) ), () P (, ) =.6.( + )+0.6( + + ) 0.0( ). () We refer to [] for a detailed analsis of () which discusses the chaotic behavior of this monopol model and investigates solutions of period,, 10, 1 and 17. We implement () inmatcontm for numerical analsis. To start a fied point continuation we first need an initial fied point. Thefiedpointsof() are obtained b solving =, (6) = + δp(, ). (7) to get P (, ) = =0. (8)

7 doi: /17-696/69/1/ NS 1 NS δ Figure : Results of fied point continuations using the three fied points of (). ThefiedpointsofM are =± (maima of ()) and = (minimum of ()). Net, we investigate the stabilit of these fied points numericall using MatContM. We select a fied point continuation and enter = =+ inthestarter window and we select δ as the free sstem parameter with initial value 0.1. All other options remain on default. We select Compute Forward to start the continuation. We also enable the Numeric window for monitoring the multipliers. The results for all three fied points are shown in Figure. The fied points = =± are initiall stable for δ =0.1. The Numeric window indicates that the multipliers are within the unit circle. The fied points lose stabilit around δ = through a Neimark-Sacker (NS) bifurcation. This fied point = = is initiall unstable and remains unstable throughout the continuation. This confirms the theoretical results in of[]. One of the remarkable properties of () is that a stable ccle of period- is born. We can obtain this ccle using simulation. We set = = and δ =1.8. We select Point as an initial point tpe and select Orbit (simulation). After computing hundreds of points we see the convergence to a period- ccle, see Figure. When we start from = =wegetanother period- ccle. We compute over 000 points and select the last computed point. We can now continue the period- ccle b continuation of a fied point of the iterated map M ().Nearδ.6 we start to detect man branch points (BP), this is caused b coeistence with other ccles with higher periods. In [] it is also shown that the period- ccle loses stabilit at δ.867. Figure repeats the computation of orbits for different values of δ. Noticehowthefied points evolve into period- ccles when δ passes the Neimark-Sacker bifurcation (upper-right). The two lower images show chaos setting in and we observe the two stable objects merge into a strange attractor. MatContM allows to compute Lapunov eponents. Select Point as Point Tpe and then select compute Lapunov eponents as an initializer. The starter window allows to enter a range of parameter values. Figure 6 shows the largest Lapunov eponents for δ between 1. and. A bifurcation diagram, obtained through simulation, is also included. The negative values for σ in Figure 6 indicate the presence of stable ccles. Through simulations, we indeed find ccles of period, 10, 1 and 17 of (). See Table 1. We then perform a numerical stabilit analsis. Once a ccle has been discovered for certain parameter value, it can be used as a starting point for continuation. 6

8 doi: /17-696/69/1/ Figure : An orbit starting from (, ) converges to a period- ccle when δ =1.8. We obtain ((.10,.66), (.66,.), (.,.68), (.68,.10)) Figure : Orbits for δ =1.0 (upper-left), δ =1.8 (upper-right), δ =.6 (lower-left) and δ =.6 (lowerright). When δ =1.0 there is convergence to the stable fied points. When δ =1.8 there is convergence to the stable period- ccles. For δ =.6 and δ =.6 we notice chaotic behavior due to coeistence with other ccles of different periods. The stable period- ccles persist but have a ver small stabilit region. 7

9 NOMA 1 International Workshop on Nonlinear Maps and Applications doi: /17-696/69/1/ σ δ. Figure 6: Top: the largest Lapunov eponent, σ, as a function of δ. Negative values correspond to stable ccles, zero corresponds to -ccles with one multiplier equal to one, and positive values indicate chaotic behaviour. Bottom: bifurcation diagram obtained b simulation. For the ccles of period, 10, 1 and 17, and using the initial data in Table 1, we continue each ccle with free parameter δ. The continuation of each ccle leads to a closed curve of ccles. Limit point (), Branch point (BP) and period-doubling () bifurcations are found along these curves. Figure 7 shows the ccle of period 1 for δ =.8, A part of the branch of 1-ccles is shown in Figure 8. -Ccle 10-Ccle 1-Ccle 17-Ccle δ Table 1: One point on the,10,1,17-ccles. During continuation, MatContM can also compute the multipliers which determine the stabilit of the, 10, 1 and 17-ccles. Thus, the stabilit regions of the, 10, 1 and 17-ccles are bounded b the and points. The 10-ccles are stable in the regions bounded b the BP and points. The, 10, 1 and 17-ccles are unstable between two successive points or two successive points. Table shows the stabilit regions for each ccle. The eistence of bifurcations on the 10, 1, and 17-ccles indicates the eistence of ccles of higher periods as well. In animation6.mpg we show the continuation process of a -ccle in the state space and in 8

10 doi: /17-696/69/1/ Figure 7: The period-1 ccle represented in Table 1 and used as initial data for computing Figure δ Figure 8: Zoom of a bifurcation diagram of period-1 ccles in (δ, )-plane the parameter space. We obtain a closed curve similar to Figure 8. We select a Period-Doubling bifurcation point and start a continuation of the doubled ccle of period 10. This continuation curve also has Period-Doubling bifurcations which allows us to start the continuation of a 0-9

11 doi: /17-696/69/1/0101 Stable for δ in -Ccle (.681,.69111) (.,.7) 10-Ccle (.69817,.708) (.801,.801) (.170,.17) (.19,.) 1-Ccle (.786,.816) (.8987,.800) 17-Ccle (.977,.0) (.76,.76) Table : Stabilit regions of the,10,1 and 17-ccles of the monopol model. ccle. Appendi A. Animations The following video files were submitted: (i) animation1.mpg: Continuation of a fied point (ii) animation.mpg: Detection of bifurcations (iii) animation.mpg: Generating a bifurcation diagram (iv) animation.mpg: Continuation of connecting orbits (v) animation.mpg: Continuation of tangencies (vi) animation6.mpg: Period-doubling of ccles These files are also available on Documentation/MatcontM/files/NOMA1/. References [1] H.N. Agiza, E.M. ELabbas, H. EL-Metwall, and A.A. Elsadan. Chaotic dnamics of a discrete prepredator model with Holling tpe II. Nonlinear Analsis: Real World Applications, 10(1):116 19, 009. [] B. Al-Hdaibat, W. Govaerts, and N. Neirnck. On periodic and chaotic behavior in a two-dimensional monopol model. Chaos, Solitons & Fractals, 70:7 7, 01. [] A. Dhooge, W. Govaerts, and Yu. A. Kuznetsov. MATCONT: A matlab package for numerical bifurcation analsis of ODEs. ACM Trans. Math. Softw., 9():11 16, June 00. [] W. Govaerts, R. Khoshsiar Ghaziani, Yu. A. Kuznetsov, and H. G. E. Meijer. Numerical methods for twoparameter local bifurcation analsis of maps. SIAM J. Sci. Comput., 9(6):6 667, October 007. [] W. Govaerts, Yu. A. Kuznetsov, R. Khoshsiar Ghaziani, and H. G. E. Meijer. Cl MatContM: A toolbo for continuation and bifurcation of ccles of maps, [6] R. Khoshsiar Ghaziani. Bifurcations of Maps: Numerical Algorithms and Applications. PhD thesis, Ghent Universit, 008. [7] T. Puu. The chaotic monopolist. Chaos, Solitons & Fractals, (1):, Januar