The aim of this paper is to investigate the use of Pyragas control on the stability of stationary, localized coherent structures in a general class of two-component, singularly perturbed, reaction-diffusion systems. We use noninvasive Pyragas-like proportional feedback control to stabilize a singular pulse solution to a two-component, singularly perturbed reaction-diffusion system. We show that in a significant region of parameter space, the control can be adjusted to stabilize an otherwise unstable pulse.
Singularly perturbed pulses in two-component reaction-diffusion equations are generally unstable in large regions of parameter space. To address this instability, we utilize Pyragas control, which was originally developed for periodic solutions of ordinary differential equations, to create a noninvasive feedback control for singularly perturbed pulses in reaction-diffusion systems. We prove the effectiveness of this control technique by analyzing the spectral stability of the controlled singular pulse in a toy model, using Evans function techniques.
I. INTRODUCTION
Reaction-diffusion systems are mathematical models based on semi-linear parabolic partial differential equations.1 One of the most fascinating aspects of reaction-diffusion equations is the plethora of patterns that can emerge from their solutions. These patterns include such diverse phenomena spiral waves in a chemical oscillator,2 the different animal coat patterns,3 Faraday waves,4 or various patterns in Rayleigh–Bénard convection,5 geology,6 or fluid dynamics.7
These patterns, of which some can be classified as Turing patterns,8 exhibit a variety of behaviors, including the formation of traveling waves and wave-like phenomena, as well as self-organized structures such as stripes, hexagons, and dissipative solitons.9 The study of these patterns has significant implications for understanding natural phenomena and has led to new insights in various fields.10
The focus of this article is on the analysis and control of a specific spatially localized coherent structure in a two-component reaction-diffusion equation: a symmetric singular pulse.11–13,27,29 An illustration of this pulse is presented in Fig. 1. It is characterized by a noticeable scale separation, which is evident in the difference in pulse width between the two components.
A typical profile of a singular symmetric pulse, bi-asymptotic to the trivial background state. A spatial scale separation between the large scale component u (red) and the small scale component v (blue) is clearly visible.
A typical profile of a singular symmetric pulse, bi-asymptotic to the trivial background state. A spatial scale separation between the large scale component u (red) and the small scale component v (blue) is clearly visible.
It has been shown11,12 that such two-component pulses can only be stable when the nonlinear interaction between the two components is sufficiently strong. Even when this is the case, singular pulses are unstable for large regions in parameter space, see Ref. 12 (Lemmas 5.11, 5.12, and 5.14). As such, singular pulses are infrequently observed in nature or experiments. Often only the progression toward a stable steady state is observable, while unstable steady states remain largely imperceptible.
It is, therefore, our goal to introduce a Pyragas-like control term to make unstable pulses visible. Pyragas control14,15 is advantageous as it is noninvasive on the pulses, i.e., the control term vanishes and does not change the pulse itself. However, it changes the nearby solutions and thereby the stability properties of the pulse. Another benefit of this control approach is its model-independence and low implementation cost, as it does not require expensive calculations. Although originally designed for periodic solutions of ordinary differential equations, we adapt it for controlling singularly perturbed pulses in reaction-diffusion systems.
We aim to integrate control theory with the theory of pattern existence and stability in singularly perturbed reaction-diffusion systems. Specifically, we aim to control the stability of pulse solutions and develop a novel methodology to regulate the stability of diverse patterns in a broad class of singularly perturbed reaction-diffusion systems.
This paper is organized as follows: In Sec. II, we discuss our model system and introduce the singular pulse as the pattern of interest; we also introduce noninvasive control terms. Next, in Sec. III, we construct an Evans function to determine the spectral stability of the singular pulse. Our main result is presented in Sec. IV, where we demonstrate that noninvasive feedback stabilization can be achieved in a large region of parameter space. We provide a brief summary of the stability proof in this section, with a complete and detailed version of the proof available in the Appendix. We conclude with a short discussion in Sec. V.
II. MODEL, PATTERNS AND CONTROL
A. Model
B. Patterns
C. Control
Note that, consequently, on the linear level, the control is not a multiple of the identity matrix. Therefore, the influence of the control term is not as straightforward as shifting all eigenvalues to the left. Our goal is to derive conditions on the control function such that the singular pulse (5) is a stable solution to (7).
III. PULSE STABILITY AND THE EVANS FUNCTION
The essential spectrum of (9) is real, negative, and bounded away from the imaginary axis;12 hence, the pulse stability is determined by its discrete spectrum, i.e., the roots of the associated Evans function . In Ref 12, it is shown that the singularly perturbed structure of the pulse can be used to obtain an explicit characterization of the roots of to leading order in ; moreover, these roots perturb regularly in .
A typical configuration of the spectrum of (9) is shown in Fig. 2. The pulse can lose stability when a pair of eigenvalues crosses the imaginary axis (a Hopf bifurcation) or when a real eigenvalue passes through the origin. Note that the pulse spectrum consists of both discrete (point) spectrum and essential (continuous) spectrum, as the spatial domain is unbounded.
A typical configuration of the spectrum of the pulse (purple) in the complex plane. Spectrum to the left of the imaginary axis is stable. The essential spectrum is seen to be real, negative, and bounded away from the imaginary axis.
A typical configuration of the spectrum of the pulse (purple) in the complex plane. Spectrum to the left of the imaginary axis is stable. The essential spectrum is seen to be real, negative, and bounded away from the imaginary axis.
IV. PULSE STABILIZATION THROUGH PROPORTIONAL FEEDBACK CONTROL
The main research question that we address in this paper is: Given a singular pulse solution to (1), can we find control terms K, L such that this singular pulse is a stable solution to (6)?
As the introduction of control terms has a (potentially) significant influence on the stability analysis of the singular pulse, we try to answer the research question formulated above in the context of the toy problem (2), with proportional feedback control in v—Eq. (7). We first present the main outcome of our analysis in Theorem 1 and describe the main ideas of the proof. The full proof of Theorem 1 can be found in the Appendix.
Let be sufficiently small, and assume that is a nondegenerate solution to (4). Consider the symmetric singular pulse solution to (2), which is to leading order in given by (5), and introduce .
A visual representation of the statement of Theorem 1 is given in Fig. 3. A direct application of Theorem 1 for specific parameter values is shown in Figs. 4 and 5.
The stability of the pulse as determined in Theorem 1, with . In the blue region (numerically determined), the pulse is stable without control; stability is lost either through a Hopf bifurcation (red curve) or by a real eigenvalue passing through zero (blue curve). In the red region and on the dashed lines, the pulse is unstable and cannot be controlled, see Theorem 1 (1) and (3a). In the remainder of the parameter space, the pulse is unstable but can be stabilized by proportional control in the v-component, as implemented in (7).
The stability of the pulse as determined in Theorem 1, with . In the blue region (numerically determined), the pulse is stable without control; stability is lost either through a Hopf bifurcation (red curve) or by a real eigenvalue passing through zero (blue curve). In the red region and on the dashed lines, the pulse is unstable and cannot be controlled, see Theorem 1 (1) and (3a). In the remainder of the parameter space, the pulse is unstable but can be stabilized by proportional control in the v-component, as implemented in (7).
(a) The spectrum for (5) for the toy model (2), with parameter choices and . For these parameters, the pulse is unstable. (b) Application of proportional feedback control as in (7), with is (more than) sufficient to stabilize the pulse; this is a direct consequence of Theorem 1, statement 2. (c) Proportional feedback control as in in (7), with stabilizes the pulse through a Hopf bifurcation. The spectral configurations shown in this figure are numerically obtained roots of the Evans function (13).
(a) The spectrum for (5) for the toy model (2), with parameter choices and . For these parameters, the pulse is unstable. (b) Application of proportional feedback control as in (7), with is (more than) sufficient to stabilize the pulse; this is a direct consequence of Theorem 1, statement 2. (c) Proportional feedback control as in in (7), with stabilizes the pulse through a Hopf bifurcation. The spectral configurations shown in this figure are numerically obtained roots of the Evans function (13).
The real part of the complex eigenvalue pair shown in Fig. 4 as a function of the control strength , with proportional feedback control as in (7). A sufficiently strong control will stabilize the pulse through a Hopf bifurcation.
It is worthwhile to note that control on one variable only suffices to control both components of the pulse.
Specific conditions that the control function needs to satisfy to stabilize the pulse can be found in the proof of Lemma 4 in the Appendix. In particular, the essential spectrum is stable if and only if (A4).
Moreover, note that the controllability of the pulse strongly depends on the quantity , which corresponds to the logarithmic derivative of . This quantity represents the u-dependent nonlinearity of (2)(a) at .
While the purpose of this paper is to show the stabilization of pulse that is unstable in the absence of control, it is worthwhile to note that our control scheme can also destabilize an otherwise stable pulse. For example, choosing destabilizes the pulse through a sideband instability, as the essential spectrum is pushed through the imaginary axis (A4). We do not explore such destabilization scenarios in the current paper; the desirability of pulse destabilization through noninvasive control depends on the model context and application.
V. CONCLUSION AND OUTLOOK
In conclusion, we have demonstrated the possibility of stabilizing singularly perturbed pulses in two-component reaction-diffusion equations across significant areas of the parameter plane. Specifically, for the toy model (2), we have identified three regions in the -plane: a stable region where control is unnecessary, an unstable region where control is always insufficient, and—our main interest here—a large region where pulse stability can be controlled through our proposed control term.
Several potential avenues for future investigation emerge. First, it may be worthwhile to explore the use of a “true” Pyragas control, in which the control is not proportional but includes a delay term, and as such does not rely on explicit pre-existing knowledge of the pulse structure. This poses additional problems in the rigorous stability analysis as difficulties may arise concerning nonlinear stability. For the control scheme investigated in this paper, it was not necessary to differentiate between spectral and nonlinear stability. In the non-controlled eigenvalue problem (8), spectral stability implies nonlinear stability; this follows from a classical result by Henry1 because the operator (9) is sectorial. The linear operator associated to the “controlled” eigenvalue problem (12) has the same (sectorial) property. Hence, one can directly infer nonlinear stability from spectral stability. For other control strategies, such as delayed feedback, the sectorial property is generally not retained. In such a case, one needs to be more careful about deducing nonlinear stability; see, e.g., Ref. 25 for generalizations of Henry’s result.
Second, investigating the impact of control on the large scale component u in comparison to the current focus on the small scale component v would provide valuable insights for this two-component system. Additionally, exploring the potential of non-diagonal controls would also be of interest.
Third, our current analysis has focused on relatively simple spatially localized coherent structures. It would be interesting to investigate the potential extension of our control scheme to more complex patterns, such as multi-circuit configurations11 and/or periodic pulse patterns,26 and to determine the conditions under which stabilization can be achieved for these more intricate situations.
ACKNOWLEDGMENTS
F.V. was supported by a Humboldt Fellowship. I.S. was supported by the Deutsche Forschungsgemeinschaft, SFB 910, Project A4 “Spatio-Temporal Patterns: Control, Delays, and Design.” We would like to thank all the members of SFB 910, in particular Sabine Klapp, Eckehard Schöll, and Bernold Fiedler, for their valuable contributions and continuous support. In addition, we would like to express our gratitude to Nigel Higson for insightful discussions on Weyl’s theorem.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
F. Veerman: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). I. Schneider: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
APPENDIX: PROOF OF THEOREM 1
1. Pulse stability and the Evans function
2. An explicit expression for the Evans function
3. Spectral decomposition in the presence of essential spectrum
Let be as in (A29). Then the following statements hold:
-
;
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There exists such that, if , then ;
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For real , is positive and strictly monotonically decreasing;
-
There exist such that, if , then .
-
Denoting , we have for . From this observation, it immediately follows that . For , we estimate , from which it follows that .
-
Using the same notation as in (I), we have for . From this observation, it immediately follows that for all . For , we find that for and any .
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If , then for ; hence, it follows that is strictly monotonically decreasing for real . Moreover, from the proof of (II), we know that is positive for real . For , we calculate for . Moreover, and as ; we conclude that is positive and strictly monotonically decreasing for real .
- For , we find that for and any . For , we use the estimates , and when , to obtain
The claim follows by continuity of in for .
4. Results on proportional feedback control
Using the results obtained in the previous sections, we can formulate the following lemma on solutions to Eq. (A18):
Consider Eq. (A18) for with and , where is the unique bounded solution to (A16), and , cf. (5). Then the following statements hold:
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If , then all solutions to (A18) lie on the real axis.
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If and , then there always exists a real, positive solution to (A18) for any .
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If , then for any , there exists a real, positive solution to (A18).
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If , then there exists a value such that all solutions to (A18) obey .
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If and , then there exists a value such that (A18) has no solutions.
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If , then there exists such that, when , all solutions to (A18) lie to the left of the vertical line , with independent of and .
1. If , then all solutions to (A18) lie on the real axis. For the principal complex square root , which is defined for all z away from the negative real line, it holds that and . It follows that, if , can only equal if both are zero, by Lemma 3 (I).
2. If and , then there always exists a real, positive solution for any . By statement 1, we take . Both and are positive for sufficiently large by Lemma 3 (II); moreover, as . It follows that for sufficiently large , since . Moreover, from the observation that as , combined with the fact that is continuous for and as , it follows by continuity of for that there exists a real, positive solution to (A18), that lies to the right of the point and to the right of the point .
3. If , then for any , there exists a real, positive solution to (A18). Since is positive for real by Lemma 3 (III), the intersection of the graphs and lies above the horizontal axis. Since and , the graph of is strictly monotonically increasing on its domain and intersects the horizontal axis before intersecting the graph of , as increases. Denote the -value for which the two graphs intersect as . From the monotonicity of the graphs, it follows that is the largest solution to (A18). Moreover, as if and only if , we conclude that since .
4. If , then there exists a value such that all solutions to (A18) obey . We calculate and . Hence, the largest solution to (A18) is contained in the interval if and only if by the monotonicity and continuity of and . From Lemma 3 (III), it follows that for sufficiently large .
5. If and , then there exists a value such that (A18) has no solutions. By Lemma 3 (III), there exists such that for all . Choosing , so that , we see that (A18) cannot have any real solutions, which implies that (A18) does not have any (complex) solutions by statement 1.
6. If , then there exists such that, when , all solutions to (A18) lie to the left of the vertical line , with independent of , and .
Suppose , and consider . We can choose , making sure that the domain of lies entirely to the right of . On this domain, we have , while , from which we conclude that no real solutions to (A18) exist. The same reasoning can be extended to complex solutions of (A18), under the same assumption that : because for all z away from the negative real line, we have that for all on its domain, while there exists a such that for all to the right of the line by Lemma 3 (II).
Now suppose . For , the same reasoning as in the case can be applied, choosing . The reasoning for complex is also analogous to the case , now with , again using Lemma 3 (II).
5. Finishing the proof of Theorem 1
The results in Lemma 4 can now be used prove Theorem 1, restated here for convenience:
Let be sufficiently small, and assume that is a nondegenerate solution to (4). Consider the symmetric singular pulse solution to (2), which is to leading order in given by (5), and introduce .
The pulse solution is spectrally stable if and only if, for the shifted eigenvalue (A7), the condition (A8) is satisfied. Eigenvalues correspond to zeroes of the Evans function (A10). These zeroes are, to leading order in , given by the solutions to (A17).
Claim 1 follows from the observation that the only solution to (A14) is the trivial solution when . Hence, the slow transmission function (A15) does not have a pole at ; therefore, no zero-pole cancellation in the Evans function (A10)—as detailed in Ref. 12—takes place. It follows that the positive real zero near of the fast transmission function is also a zero of the full Evans function , which means the pulse solution is unstable.
If , then (A17) is of the form (A18), with and .
When , it follows from Lemma 4 (6) that there is a such that all solutions of (A18) lie to the left of the line . Since is, in particular, independent of , and the statement holds for all , we can choose , which implies that all solutions of (A18) obey the stability criterion . Since if and only if , this proves claim 2.
When and , it follows from Lemma 4 (5) that (A18) does not have any solutions when is negative and is sufficiently large. As , the zeroes of near and are canceled by the poles of ; the remaining zero is stable for . It follows that all zeroes the Evans function (A10) obey , which implies that is spectrally stable. The condition is equivalent to , while is equivalent to . Since both and by assumption, the latter condition is equivalent to . This allows us to rewrite the -condition as , thereby proving the second part of claim 3 for .
When , it follows from Lemma 4 (4) that all solutions to (A18) obey the stability criterion . As before, the condition is equivalent to , while the condition can be rewritten as , which finalizes the proof of claim 3 b).
When , it follows from Lemma 4 (3) that there always exists a solution to (A18) that does not satisfy the stability criterion , regardless of the value of . This implies that, under the condition , the pulse is unstable for any choice of the control function . The condition is equivalent to , while the condition can be rewritten as , which proves claim 3 a).