Abstract
Underlying the state feedback control, the complex dynamical disease of the hematopoietic stem cells model based on Mackey’s mathematical description is analyzed. The bifurcating periodical oscillation solutions of the system are continued by applying numerical simulation method. The limit point cycle bifurcation and period doubling bifurcation are observed frequently in the continuation process. The attraction basins of the positive equilibrium solution shrink as the differentiate rate is ascending and the observed Mobiüs strain is simulated with boundary as the period-2 solution. The period doubling bifurcation leads to period-2, period-4, and period-8 solutions which are simulated. Starting from period doubling bifurcation point, the continuation of the bifurcating solution routes to homoclinic solution is finished. The simulation results improve the comprehension related to the spontaneous dynamical character manifested in the hematopoietic stem cells model.
1. Introduction
Hematological diseases have attracted a significant amount of modelling attention because a number of them are periodic in nature. With the consideration of the periodic character manifested by different dynamical diseases, two principle mechanisms have been proposed to drive the oscillating dynamics in hematopoietic models [1], for example, the control mechanism in which the destabilization of the peripheral cells leads to disease such as autoimmune hemolytic anemia and so on. Whereas it is emphasized that other periodical hematopoietic diseases involving oscillation in all type of produced blood cells with long period and the examples include cyclical neutropenia [2, 3] and periodic chronic myelogenous leukemia [4–6]. Manifested by the complex dynamical characters, the mechanisms underlying periodical solution bifurcation are necessary analytical work, and the prior studies (and datasets) are cited at relevant places within the text as references [2, 7–9]. Our motivation is to investigate the rich periodical oscillation phenomena near bifurcation points.
We use DDE-Biftool software which is an on-hand tool with its artificial computation technique to complete the codimension-1 bifurcation related to equilibrium solution and periodical solution. Underlying state feedback control with multiple time delays, the spontaneous dynamical evolution behavior of the hematopoietic stem cell (HSC) model is complex. As a limitation of the chosen sets of parameters value, people have endeavored to analyze all sorts of bifurcation mechanism to explore bifurcation dynamical behavior in the HSC system which is based on Mackey’s hematological mathematical description. For example, the known periodical oscillation phenomena arise near Hopf threshold values.
We do fundamental computation with DDE-Biftool hand-tools, for example, the continuation of periodical solutions as varying free parameter continuously with Hopf singularity, the continuation branch of bifurcating doubly period solutions of periodical solutions underlying period doubling bifurcation, and so on.
Originated from hematopoietic stem cells compartment, blood cells are differentiated into leukocyte (white blood cells), erythrocyte (red blood cells), platelets, and so on. The human hematopoietic system produces about more than blood cells of various types per day, and its process is tightly regulated by a myriad of feedback loops. All the blood cells can either be in actively proliferating phase or in a resting phase, and the stem cells are capable of self-renewing process and mitosis into its daughter cells [4, 7, 8]. With the introduction rate, a cell is committed to undergo cell division at a fixed time later after entering into the proliferation phase. Hence, after division, the daughter cells population goes into resting phase and completes the cell cycle.
With the consideration of the classical model, the simplified DDEs version of the HSC system is set forth (Daniel et al. (2000)):where represents the concentration of hematopoietic stem cells in the bone marrow. is the introduction rate at which cell enters into division, and is the generating time delay during the cell division phase. is the rate that the stem cells differentiate into the progenitors of circulating blood cells.
We study the dynamical bifurcation behavior of system (1) undergoing state feedback control with time delay:with the introduction rate . System (2) contains multiple time delays, and the stability analysis of Hopf bifurcation is analyzed. DDE-Biftool is useful to compute the stability and bifurcation of equilibrium solution and periodical solutions (see [10–12]). With parameter varying, the periodical solutions bifurcate from Hopf bifurcation and then undergo period doubling bifurcation and limit point cycle bifurcation; therefore, the prominent work of bifurcation is undertaken. The other parameters are chosen as .
This paper presents continuation of oscillating periodical solutions based on the scheme of DDE-Biftool, which is an artificial work formulated by bifurcation analysis of DDEs. The common codimension-1 bifurcation including limit point cycle and period doubling bifurcation of the periodical solutions is observed in system (2) frequently. The period doubling bifurcation happens as the Floquet multiplier attains at in the unit circle. As varying free parameters further, the period doubling bifurcation manifests two different dynamical bifurcation mechanisms in this work. As common as often, one list example discovers the scenarios of doubly period oscillation bifurcation mechanism which leads to period-2 solution, period-4 solution, and period-8 oscillation and routes to chaos, whereas another example manifests the winding way to homoclinc solution through period doubling bifurcation. The homoclinic solution is found with the periodical solution attaining at the biggest period approximated to 550 days found, and the simulation solution is corrected by DDE-Biftool software.
The whole paper is organized as the follows. In Section 2, the attraction basin of chaos is drawn and Mobiüs strain of period-2 solutions is obtained. In Section 3, the bifurcating periodical solutions from Hopf point are continued as varying free parameter . By construction of the Poincare mapping, the period window of period-2 solutions and period-4 solutions is observed which also discovers period doubling bifurcation to chaos. In Section 4, until attaining at homoclinic solution, the limit point cycle bifurcation and period doubling bifurcation are successively observed while continuing period solution by varying differentiate rate. A discussion is given finally.
2. Attraction Basin of Stable Attractors
The most common bifurcation solutions of periodic oscillating phenomena appearing in system (2) are simulated. The period doubling bifurcation at which the Floquet multiplier lies on the unit circle with produces periodical oscillation solution with doubly period. As a comparison with system (1), we carry out scenarios of bifurcations continued as varying in system (2). With state feedback control, the long period oscillating solutions are observed which is the result of successively period doubling bifurcation and limit point cycle bifurcation.
The dimensional two unstable manifolds of the positive equilibrium solution of system (2) are simulated as varying free parameter . As becoming smaller, the attraction basin of system attractor is extended. The simulation result verifies the attractor lying in its neighborhood and wiggling around the unstable manifold. By fixing time delay and , the stable limit cycle near period doubling bifurcation point at is observed. The classical Mobiüs strain manifests in Figure 1(b) which is expanded out from the unstable manifold computation of the positive equilibrium solution. The unstable manifold is two-dimensional with definition as follows:which is plotted as shown in Figures 1(a)–1(d). As shown in Figures 1(c) and 1(d), the attraction basin of the stable attractor is bounded by chaos manifested at . Hence, the two-dimensional unstable manifolds of the positive equilibrium solution grow up into the neighborhood of chaos. As becoming smaller, the unstable manifold is toggled and embedded self-tangentially to become a bigger two-dimensional surface with view in axis space. The manifold computation is shown in [13].
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3. Bifurcating Solution of Hopf Bifurcation
By doing stability analysis of the positive equilibrium solution with the appropriate value of parameter , Hopf point is discovered at the threshold value and . As shown in Figure 2, the amplification with bigger amplitude of oscillating limit cycle is obtained at the subcritical Hopf point with . The logarithm scale of the amplitude is plotted with respect to free parameter , and the amplifying figure near subcritical Hopf point is viewed in the upright corner. As shown in Figure 2(a), the scenario bifurcation of the limit point cycle is observed as continuing the periodical solutions. The time series solutions and the phase portraits are drawn in Figures 2(b)–2(e). The periodical solutions with multiclustering oscillation are observed. It seems that the rhythm oscillation is triggered periodically and induced by limit point bifurcation.
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While continuing the bifurcating periodical solution along free parameter continuously, a successive bifurcation behavior as period doubling bifurcation and limit point cycle bifurcation points are also exploited. As shown in Figure 2, the bifurcating oscillation solution is continued and then ceased to exist at another super-critical Hopf point with . The dynamical disease of oscillating solutions expands the scenarios of bifurcation behavior.
DDE-Biftool is applied to carry out codimension-1 bifurcation analysis of DDEs by artificial computation technique. Both limit point cycle bifurcation and doubling bifurcation are observed while studying dynamical disease in system (2). Considering the period doubling bifurcation, the doubly period solutions are computed by applying the continuation algorithm developed by DDE-Biftool. Hence, after the period-2 solutions, period-4 solutions and even period-8 solutions are computed which are, respectively, related to period doubling bifurcation. The routes to chaotic attractor are given by mapping on the Poincare section, as shown in Figure 3(d).
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The prominent mechanisms undertaken bifurcation singularity of period-2 solutions explore the different routes to chaos, which manifests both the classical routes via period doubling bifurcation to chaos and specially via homoclinc bifurcation to chaos. The transition mechanism between periodical solutions is discussed in Section 3 and in Section 4 hereinafter.
The scenarios of oscillation bifurcating solutions are presented as varying . It is seen that the birth of the bifurcating period-2 solutions from starting point alike continues with free parameter lying inside the interval then vicinity at the terminal point with . Our principle method here is using DDE-Biftool software with solution perturbation along the direction dominated by the tangent vector at the proposed period doubling bifurcation point. As shown in Figure 3(a), the continuation of period-2 bifurcating solution is continued with varying free parameter , which losses its stability again via period doubling bifurcation at and which leads to period-4 solution. As shown in Figure 4, the period-8 solution also bifurcates from period doubling bifurcation point of period-4 solution. And the process can be continued further which induces the convergent of the Feigenbaum series number.
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The limit point cycle bifurcation occurs at differentiation rate and . The stable period-2 solution collides with the unstable period-2 solution underlying limit point cycle bifurcation with codimension-1 singularity. The hysteresis phenomena of period-2 solution bring forth the jump phenomena of the bifurcating period-2 solution as varying . As shown in Figures 3(b)–3(c), by varying , the continuation of period-2 solutions is finished with DDE-Biftool. The times series solution and phase portraits of period-2, period-4, are period-8 solutions are plotted, respectively, with , and , as shown in Figures 4(a)–4(f). The chaos solution is produced which exhibits discontinuous manifolds on Poincare section, as shown in Figure 5(c). The time series solution and phase portraits of chaos are plotted, as shown in Figures 5(a) and 5(b).
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4. The Construction of Homoclinic Solution
The homoclinic bifurcation happens with solution orbit if given thatwhere is the positive equilibrium solution. As shown in Figure 2(a), continuous periodic solutions arise at Hopf point and then bifurcates period-2 solution at since period doubling bifurcation. However, the successive limit point cycle bifurcation and period doubling bifurcation are frequently observed with the prolong period attained at 90 days. After that, the period increases steeply until gets to homoclinic solution with period almost attaining 550 days. The routes to homoclinic solution are shown in Figure 6. The period of oscillation solution with respect to free parameter varying inside the interval is shown in Figure 6(a). The amplitude of the oscillation phase with respect to free parameter is also plotted in Figure 6(b). All the above results have verified the existence of Shilnikov homoclinic connection orbit which implies the presence of complex dynamics in the system. The winding way to homoclinic solution is still seldom in the hematopoiesis system.
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At , it is easy to compute that the positive equilibrium solution is unstable with a pair of rightmost characteristic roots lying inside the right half plane. The unstable manifold of the equilibrium solution is two-dimensional. The corresponding time series solution and phase portraits of homoclinic solution are shown in Figures 7(a)–7(d), which perform the spontaneous bursting dynamical behavior when solution orbits close to the positive equilibrium solution.
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In the case of Shilnikov homoclinic happening, infinitely many periodic orbits of saddle type exist in a neighborhood of homoclinic solution. In addition, these periodic orbits are contained in suspended horseshoes that accumulate onto the homoclinic obtained; in this way, the occurrence of chaotic attractor is found [14]. We also simulate the corresponding chaos solution in system (2), and the simulation results are shown in Figures 8(a)–8(c).
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5. Discussion
The continuation work of periodical oscillation solutions, which arise from Hopf bifurcation point, was carried out in a hematopoietic disease model undergoing the state feedback control. The destabilization of dynamical periodical disease happens frequently as the Floquet multiplier attains at modulus 1 in the unit circle. Two kinds of interesting dynamical evolution behaviors are manifested by period doubling bifurcation phenomena. The doubly period oscillation solutions were observed through period-2, period-4, and period-8 solution of scenario bifurcation phenomena based upon the period doubling bifurcation mechanism. In addition, by continuation from period doubling bifurcation point, special homoclinic solutions were exploited with prolonged period about 550 days. The manifested dynamical character with codimension-1 periodical solution bifurcation is helpful to understand the complex dynamical disease in the hematopoietic stem cells model. In the future work, the more interesting dynamical phenomena alike bifurcation with codimension 2 singularity are challenging works in the hematological stem cells model. Underlying cells growth factor effects, the exciting bifurcation work with DDE-Biftool artificial technique will be finished.
Data Availability
The prior studies (and datasets) are cited at relevant places within the text as references [1–9].
Conflicts of Interest
The author declares that there are no conflicts of interest.