- Email: [email protected]
Chaos in a linear wave equation
Journal Pre-proof
Chaos in a linear wave equation Ned J. Corron PII: DOI: Reference:
S2590-0544(19)30012-0 https://doi.org/10.1016/j.csfx.2019.100014 CSFX 100014
To appear in:
Chaos, Solitons & Fractals: X
Received date: Accepted date:
3 October 2019 12 October 2019
Please cite this article as: Ned J. Corron , Chaos in a linear wave equation, Chaos, Solitons & Fractals: X (2019), doi: https://doi.org/10.1016/j.csfx.2019.100014
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Highlights • A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics • The linear partial differential equation is a wave equation with gain that models, for example, the optical gain medium in a laser • The behavior of this linear system is straightforward and relatively simple, which suggests that common definitions of chaos are being stressed to, and possibly beyond, their limits • This model provides a context to judge the significance of linear chaos and assess if rote application of a definition for chaos is even meaningful
1
Chaos in a Linear Wave Equation Ned J. Corron Charles M. Bowden Laboratory U.S. Army CCDC AvMC Redstone Arsenal, Alabama, USA
Abstract A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics. The system comprises a one-dimensional wave equation with gain that operates on a semi-infinite line. A boundary condition enforces that the waves remain finite. It is shown that the resulting solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Definitions of chaos are considered in light of such linear chaos. Keywords chaos, definition of chaos, linear chaos, backward shift, wave equation
Corresponding Author: Dr. Ned J. Corron U.S. Army CCDC AvMC Charles M. Bowden Laboratory FCDD-AMW-GM Redstone Arsenal, AL 35898 USA +1-256-876-1860 [email protected]
2
1. Introduction Chaos is often seen as an impenetrably complex, nonlinear phenomenon that is unyielding to conventional analysis. However, there are many examples that contradict this view. Indeed, a fundamental chaotic dynamical system is the Bernoulli shift, and establishing conjugacy to this system is conclusive evidence for the shift and fold dynamics that are essential for lowdimensional chaos [1]. Yet the dynamics of the Bernoulli shift are simple and transparent, and this chaotic system exhibits an exact analytic solution that can be represented as a linear superposition [2][3][4]. This canonical example is often considered singular and atypical of real chaos, but it is not unique. A class of exactly solvable chaotic maps have been identified and explored, providing many examples of chaos described by an analytic solution [5][6]. Various chaotic piecewise-linear oscillators have also been shown to admit exact analytic solutions, which can be written as a linear convolution of a discrete sequence and a fixed basis function [7][8][9]. And chaos is not exclusively a nonlinear phenomenon. Indeed, several examples of socalled linear chaos have been reported, in which the properties of chaos are present in linear systems [10][11][12][13][14][15][16][17]. Although some may find the idea of linear chaos surprising, we note that a common test for chaos—a positive Lyapunov exponent—analyzes the stability of a dynamical system using linearization and linear eigenvalue methods [18]. In this paper, we demonstrate a new example of linear chaos by considering a wave equation with gain. This dynamical system is a linear partial differential equation that models, for example, the optical gain medium in a laser [19]. The presence of linear chaos in a physical system is important, as it suggests this phenomenon can impact other common physical systems that were previously considered immune to chaos due to their linear nature. This model also provides a physical context in which we can judge the significance of linear chaos and evaluate if rote application of a common definition for chaos is even meaningful. Following this introduction, Section II presents a common mathematical definition of chaos that is used in this paper. The concept of linear chaos is introduced with an example in Section III. Section IV is the heart of the paper, presenting a linear wave equation with gain and showing it satisfies the conditions required by the definition of chaos. Other definitions of chaos are briefly considered in Section V. The paper ends in Section VI with the caveat that mechanical application of existing definitions for chaos can lead to unintended conclusions. 2. Definition of Chaos A widely accepted mathematical definition of chaos is due to Devaney [1]. This definition was originally developed for an iterated, one-dimensional map function; however, the key elements of that definition have been extended and are often applied to other dynamical systems, including higher dimensional maps and differential equations [15]. A generalization of this definition considers a metric space
X,d
and a mapping function
: X X [20]. The function defines a dynamical system by the repeated iteration xn1 xn from an initial condition x0 X . For a continuous function , the associated dynamical system is chaotic on X if it satisfies three requirements. The first is that periodic points are dense in X. This requirement is explicitly written as U X
x U , n 0 : n x x
3
(1)
where we use to imply an open subset. The second is that the iterated function is topologically transitive. This requirement may be stated as U ,V X
x U , n 0 : n x V
(2)
which implies that the space X cannot be decomposed into smaller, disconnected spaces. The third requirement is that the iterated function exhibits sensitive dependence, or x U X 0 : n n y U , n 0 : d y , x
(3)
where d denotes the metric distance between two elements in X. There are rigorous results that show these three conditions are often redundant [21]. In particular, the requirements in equations (1) and (2) often imply equation (3). However, equation (3) embodies the sensitive dependence on initial conditions, which is the famous hallmark of chaotic dynamics. Thus the redundant third requirement is often retained in this definition of chaos. Although equations (1)-(3) are stated for an iterated map with discrete time, these requirements can apply to dynamical systems with continuous time by sampling. As such, the essence of the three conditions—namely dense periodicity, transitivity, and sensitive dependence—may also be used to define chaos in ordinary and partial differential equations with continuous time without sampling. 3. Linear Chaos Although perhaps not well known, it has been recognized that a linear operator can meet the requirements of Devaney’s definition of chaos [10][11][15]. To provide an example, we consider the infinite-dimensional metric space X , d with elements x x0 , x1, x2 , x3 , , where xi R for all i 0 . For the norm such that
x
2
xi 2
(4)
i 0
we define
X x: x
2
(5)
d x, y x y
(6)
and use the metric induced by the norm
for x, y X . On this metric space, we consider the linear operator T defined as
Tx k x1, x2 , x3 , x4 ,
(7)
where k 1 is a scalar constant. The operator T is called a backward shift with gain [15]. We consider the dynamical system
4
xn1 Txn
(8)
with initial condition x0 X . In 1969, which is well before chaos was the subject of intense study, Rolewicz effectively proved this linear dynamical system exhibits transitivity [22]. More recently, others have completed a proof of chaos by showing dense periodic orbits and sensitive dependence [15][23]. A physical manifestation of the backward shift was identified in the unforced quantum harmonic operator, where the annihilation or lowering operator plays the role of the backward shift [24]. This example of linear chaos does not overturn everything we know about linear systems theory. In particular, it remains true that finite-dimensional linear systems cannot exhibit chaotic dynamics. Having an infinite-dimensional state space is the “loophole” that allows a linear system to meet Devaney’s mathematical definition of chaos. We use this loophole in the next section to devise another example of chaos in a linear system. 4. Wave Equation with Gain Interestingly, there are other infinite dimensional linear systems known to meet the conditions for chaos. Examples of linear chaos include the differential operator and certain translation operators [15]. In this section, we present a linear partial differential equation that also meets these conditions while offering a practical physical realization. Here we consider a one-dimensional wave equation with gain. Specifically, we consider the partial differential equation u u u t x
(9)
on the domain 0 x and t 0 . For an initial condition u x,0 f x , the solution of equation (9) is u x, t et f x t
(10)
which is a left-going, exponentially rising wave. We note f x may be discontinuous, implying a weak solution to the partial differential equation (9). Since the domain is restricted to x 0 , a finite initial condition does not grow unbounded: the exponential growth in the solution (10) is squelched as the wave harmlessly reaches the domain boundary at x 0 . To complete the description of the system, we apply the boundary condition
lim e x u x, t 0
x
(11)
which ensures that the solution remains bounded for all time. At any instant of time t 0 , the state of the dynamical system is defined by a function of only the spatial coordinate x. To emphasize the system state at a fixed time, we explicitly denote ut x u x, t . We define a norm for the state as
ut sup ut x x 0
5
(12)
where sup is the supremum of the function over the indicated domain. A metric space of spatial functions is then defined as W , d where
W ut : ut , lim e x ut x 0 x
(13)
and d is the metric induced by the norm. A function ut x is analogous to a point in this space, and its evolution as a function of time is a trajectory through the space. We immediately note that for any initial condition f x W , the resulting solution in equation (10) satisfies u x, t W for all t 0 . We claim the linear partial differential equation (9) is chaotic on W using Devaney’s definition of chaos. To support this claim, we show in the following that the system exhibits dense periodic states, topological transitivity, and sensitive dependence. We also show the system is characterized by a positive quantity that acts like a Lyapunov exponent. We first show that periodic states are dense in W. Addressing equation (1), we consider an arbitrary initial condition f x W . The resulting solution (10) typically is not periodic in time; however, there exists an initial condition arbitrarily close to f x that yields a time-periodic solution. We show the existence of such an initial condition by construction. To wit, we consider the initial condition f x enT f x nT , nT x n 1 T
where T is a fixed period and n
(14)
. By construction, we have f W and
lim f x f x
T
(15)
for all x 0 . Thus, f x can be made arbitrarily close to the original initial condition f x by choosing a sufficiently large period T. The corresponding solution is u x, t et f x t
(16)
u x, t T u x, t
(17)
for which the periodic identity
is readily verified for all x 0 and t 0 . Thus, for any open set U such that f x U W , there exists an initial condition f x U that yields a periodic orbit, which meets the first requirement in equation (1). A similar construction reveals the existence of a transitive orbit, which meets the topological transitivity requirement of equation (2). We assume two arbitrary functions f x and g x with the corresponding open sets U and V, such that f x U W and g x V W . We then consider the initial condition
6
f x, 0 x T T u ( x, 0) e g x T , T x 2T 0, 2T x
(18)
where T 0 is an arbitrary interval length. We note that lim u x, 0 f x
(19)
lim u x, T g x
(20)
T
and, using the solution (10), T
so that u x, t can be made to pass arbitrarily close to both functions f x and g x . Thus, for any open subsets U W and V W , there exists a transitive orbit u x, t that links the two subsets, and equation (2) is satisfied. The last requirement is a sensitive dependence on initial conditions. Indeed, it is reasonable to expect sensitivity due to the exponential factor in the analytic solution (10). To explicitly address the requirement in equation (3), we consider an arbitrary initial condition
u x,0 f x
(21)
where f x W . We also consider a second initial condition
f x , x x v x, 0 x x f x ,
(22)
where x ln , 0 is a finite perturbation, and is an arbitrary constant. By design, the two functions (21) and (22) are nearly the same, differing by a finite amount at only the single spatial point x x 0 . Using the solution (10) for each initial condition, we find that
et , x t x v x, t u x , t x t x 0,
(23)
so that the difference in the resulting solutions is et , t x u x, t v x, t t x 0.
(24)
which explicitly shows exponential growth of a perturbation. Further, the initial deviation is u x, 0 v x, 0
(25)
max u x, t v x, t
(26)
while the maximum deviation is t 0
7
which is attained at the time t x . Thus, by choosing an arbitrarily small initial perturbation , we have constructed an initial condition arbitrarily close to f x that subsequently diverges and meets the requirements of equation (3).
We note that the initial growth of the difference in equation (24) can be characterized as d u, v et , where 1 . As such, 1 acts like a positive Lyapunov exponent for the system, which is also commonly used as an indicator for chaos in a dynamical system [18]. Altogether, we have shown that the linear wave equation with gain exhibits the three requirements for the Devaney definition of chaos, as well as a positive Lyapunov exponent that describes the growth of small perturbations. This system provides an explicit example of linear chaos in a partial differential equation. Importantly, this system has physical significance, since the optical gain medium in a laser or optical amplifier can be modeled as a wave equation with gain [19]. There may be other physical systems that exhibit growing waves, and analysis may reveal that these systems may also meet the requirements for chaos in a linear system. 5. Other Definitions The recognition that the linear partial differential equation (9) meets the common definition of chaos does not mean this linear system suddenly exhibits the impenetrably complex behavior that we are accustomed to seeing in chaotic systems. In fact, the behavior of this wave equation is simple and transparent: its general solution (10) is a left-going, exponentially increasing wave. Indeed, this straightforward solution is not the complicated dynamic that inspired Li and Yorke to coin the term “chaos” in their landmark paper [25]. A number of definitions of chaos have been proposed and may be considered as alternatives to Devaney’s three requirements. A popular approach is to define chaos for an orbit: for example, an orbit is chaotic if it is not eventually periodic and exhibits a positive Lyapunov exponent [26]. In contrast, Devaney’s three requirements define chaos for a dynamical system. This distinction is subtle but, for example, enables one to characterize observed data without a system model [20]. Loosely, defining a chaotic orbit can be seen as more general, since it encompasses typical waveforms generated by dynamical systems that meet Devaney’s definition. As such, a typical solution of the linear system (9) is also chaotic by the orbit definition. Some alternative definitions for chaos are based on information concepts, including a recently proposed criterion that uses expansion entropy [27]. This definition directly senses extreme sensitivity by detecting the growth of uncertainty in a dynamical system. The resulting quantity is an entropy, and a positive value indicates chaos. An advantage of expansion entropy is that it allows the characterization of certain systems automatically excluded by Devaney’s requirements; for example, a nonautonomous system with quasi-periodic forcing does not admit periodic states, but it may still exhibit sensitive dependence on initial conditions. Since the linear system (9) exhibits a sensitive dependence characterized by exponential divergence, it also exhibits a positive expansion entropy and is still chaotic by this newer definition. 6. Conclusion A linear wave equation with gain is shown to satisfy the three conditions in Devaney’s definition of chaos, as well as appearing to meet alternative definitions based on Lyapunov
8
exponents and expansion entropy. Since the analytic solution to this linear system is straightforward and its behavior is relatively simple, this result suggests that the common definitions of chaos are being stressed to, and possibly beyond, their limits. From a functional point of view, it is fair to conclude that these definitions cannot be mechanically applied without potentially devaluing the label of chaos. We acknowledge that Devaney offered his widely adopted definition in the context of a simple dynamical system, which is an iterated one-dimensional map on the unit interval [1]. It is perhaps surprising that this definition has served admirably well when applied to more complicated systems, including continuous flows described by differential equations. However, the recent examples of linear chaos, including the linear wave equation with gain shown here, as well as a linear wave equation without gain [28], may actually reveal that this definition has its limitations. Indeed, rather than blindly attributing chaos to systems that simply meet a technical definition, it may be better to reconsider the definition when its application leads to an unintended conclusion.
Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements The author acknowledges Dr. Shangbing Ai, Dr. Erik Bollt, Dr. Daniel Hahs, and Dr. Shawn Pethel for helpful discussions regarding the development and presentation of the research results. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] R. L. Devaney, Introduction to Chaotic Dynamical Systems (Addison-Wesley, 1989). [2] D. F. Drake, Information’s Role in the Estimation of Chaotic Signals, Ph.D. dissertation (Georgia Institute of Technology, 1998). [3] S. T. Hayes, “Chaos from linear systems: Implications for communicating with chaos, and the nature of determinism and randomness,” J. Phys. Conf. Ser. 23, 215 (2005). [4] D. F. Drake and D. B. Williams, “Linear, random representations of chaos,” IEEE Trans. Signal Process. 55, 1379 (2007). [5] S. Katsura, W. Fukuda, “Exactly solvable models showing chaotic behavior,” Physica A 130, 597 (1985). [6] K. Umeno, “Method of constructing exactly solvable chaos,” Phys. Rev. E 55, 5280 (1997).
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[7] N. J. Corron, “An exactly solvable chaotic differential equation,” Dyn. Contin. Discrete Impuls. Syst. A 16, 777 (2009). [8] N. J. Corron, J. N. Blakely, M. T. Stahl, “A matched filter for chaos,” Chaos 20, 023123 (2010). [9] N. J. Corron, J. N. Blakely, “Exact folded-band chaotic oscillator,” Chaos 22, 023113 (2012). [10] C. R. MacCluer, “Chaos in linear distributed systems,” J. Dynam. Syst. Measure. Control 114, 322 (1992). [11] R. deLaubenfels, H. Emamirad, V. Protopopescu, “Linear chaos and approximation,” J. Approx. Theory 105, 176 (2000). [12] Y. Hirata, K. Judd, “Constructing dynamical systems with specified symbolic dynamics,” Chaos 15, 033102 (2005). [13] N. J. Corron, S. T. Hayes, S. D. Pethel, J. N. Blakely, “Chaos without nonlinear dynamics,” Phys. Rev. Lett. 97, 024101 (2006). [14] N. J. Corron, S. T. Hayes, S. D. Pethel, J. N. Blakely, “Synthesizing folded band chaos,” Phys. Rev. E 75, 045201R (2007). [15] K.-G. Grosse-Erdmann, A. P. Manguillot, Linear Chaos (Springer, 2011). [16] D. W. Hahs, N. J. Corron, J. N. Blakely, “Synthesizing antipodal chaotic waveforms,” J. Franklin Inst. 351, 2562 (2014). [17] N. J. Corron, “Koopman operators and linear chaos,” in: A. Buscarino, L. Fortuna, R. Stoop, eds., Advances on Nonlinear Dynamics of Electronic Systems, 14 (World Scientific, 2019). [18] E. Ott, Chaos in Dynamical Systems (Cambridge University Press, 1993). [19] B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, 2007). [20] E. M. Bollt, N. Santitissadeekorn, Applied and Computational Measurable Dynamics (SIAM, 2013). [21] J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacy, “On Devaney’s definition of chaos,” Amer. Math. Monthly 99, 332-334 (1992). [22] S. Rolewicz, “On orbits of elements,” Studia Math. 32, 17 (1969). [23] G. Godefroy, J. H. Shapiro, “Operators with dense, invariant, cyclic vector manifolds,” J. Funct. Anal. 98, 229 (1991). [24] A. Gulisashvili, C. R. MacCluer, “Linear chaos in the unforced quantum harmonic oscillator,” J. Dynam. Syst. Measure. Control 118, 337 (1996). [25] T.-Y. Li, J. A. Yorke, “Period three implies chaos,” Amer. Math. Monthly 82, 985 (1975). [26] K. T. Alligood, T. D. Sauer, J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, 1996). [27] B. R. Hunt, E. Ott, “Defining chaos,” Chaos 25, 097618 (2015).
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[28] N. J. Corron, “Linear chaos in a tape recorder,” in: V. In, P. Longhini, A. Palacios, eds., Proc. 5th International Conference on Applications in Nonlinear Dynamics, 54 (Springer, Cham; 2019).
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Chaos in a linear wave equation Ned J. Corron PII: DOI: Reference:
S2590-0544(19)30012-0 https://doi.org/10.1016/j.csfx.2019.100014 CSFX 100014
To appear in:
Chaos, Solitons & Fractals: X
Received date: Accepted date:
3 October 2019 12 October 2019
Please cite this article as: Ned J. Corron , Chaos in a linear wave equation, Chaos, Solitons & Fractals: X (2019), doi: https://doi.org/10.1016/j.csfx.2019.100014
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Highlights • A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics • The linear partial differential equation is a wave equation with gain that models, for example, the optical gain medium in a laser • The behavior of this linear system is straightforward and relatively simple, which suggests that common definitions of chaos are being stressed to, and possibly beyond, their limits • This model provides a context to judge the significance of linear chaos and assess if rote application of a definition for chaos is even meaningful
1
Chaos in a Linear Wave Equation Ned J. Corron Charles M. Bowden Laboratory U.S. Army CCDC AvMC Redstone Arsenal, Alabama, USA
Abstract A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics. The system comprises a one-dimensional wave equation with gain that operates on a semi-infinite line. A boundary condition enforces that the waves remain finite. It is shown that the resulting solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Definitions of chaos are considered in light of such linear chaos. Keywords chaos, definition of chaos, linear chaos, backward shift, wave equation
Corresponding Author: Dr. Ned J. Corron U.S. Army CCDC AvMC Charles M. Bowden Laboratory FCDD-AMW-GM Redstone Arsenal, AL 35898 USA +1-256-876-1860 [email protected]
2
1. Introduction Chaos is often seen as an impenetrably complex, nonlinear phenomenon that is unyielding to conventional analysis. However, there are many examples that contradict this view. Indeed, a fundamental chaotic dynamical system is the Bernoulli shift, and establishing conjugacy to this system is conclusive evidence for the shift and fold dynamics that are essential for lowdimensional chaos [1]. Yet the dynamics of the Bernoulli shift are simple and transparent, and this chaotic system exhibits an exact analytic solution that can be represented as a linear superposition [2][3][4]. This canonical example is often considered singular and atypical of real chaos, but it is not unique. A class of exactly solvable chaotic maps have been identified and explored, providing many examples of chaos described by an analytic solution [5][6]. Various chaotic piecewise-linear oscillators have also been shown to admit exact analytic solutions, which can be written as a linear convolution of a discrete sequence and a fixed basis function [7][8][9]. And chaos is not exclusively a nonlinear phenomenon. Indeed, several examples of socalled linear chaos have been reported, in which the properties of chaos are present in linear systems [10][11][12][13][14][15][16][17]. Although some may find the idea of linear chaos surprising, we note that a common test for chaos—a positive Lyapunov exponent—analyzes the stability of a dynamical system using linearization and linear eigenvalue methods [18]. In this paper, we demonstrate a new example of linear chaos by considering a wave equation with gain. This dynamical system is a linear partial differential equation that models, for example, the optical gain medium in a laser [19]. The presence of linear chaos in a physical system is important, as it suggests this phenomenon can impact other common physical systems that were previously considered immune to chaos due to their linear nature. This model also provides a physical context in which we can judge the significance of linear chaos and evaluate if rote application of a common definition for chaos is even meaningful. Following this introduction, Section II presents a common mathematical definition of chaos that is used in this paper. The concept of linear chaos is introduced with an example in Section III. Section IV is the heart of the paper, presenting a linear wave equation with gain and showing it satisfies the conditions required by the definition of chaos. Other definitions of chaos are briefly considered in Section V. The paper ends in Section VI with the caveat that mechanical application of existing definitions for chaos can lead to unintended conclusions. 2. Definition of Chaos A widely accepted mathematical definition of chaos is due to Devaney [1]. This definition was originally developed for an iterated, one-dimensional map function; however, the key elements of that definition have been extended and are often applied to other dynamical systems, including higher dimensional maps and differential equations [15]. A generalization of this definition considers a metric space
X,d
and a mapping function
: X X [20]. The function defines a dynamical system by the repeated iteration xn1 xn from an initial condition x0 X . For a continuous function , the associated dynamical system is chaotic on X if it satisfies three requirements. The first is that periodic points are dense in X. This requirement is explicitly written as U X
x U , n 0 : n x x
3
(1)
where we use to imply an open subset. The second is that the iterated function is topologically transitive. This requirement may be stated as U ,V X
x U , n 0 : n x V
(2)
which implies that the space X cannot be decomposed into smaller, disconnected spaces. The third requirement is that the iterated function exhibits sensitive dependence, or x U X 0 : n n y U , n 0 : d y , x
(3)
where d denotes the metric distance between two elements in X. There are rigorous results that show these three conditions are often redundant [21]. In particular, the requirements in equations (1) and (2) often imply equation (3). However, equation (3) embodies the sensitive dependence on initial conditions, which is the famous hallmark of chaotic dynamics. Thus the redundant third requirement is often retained in this definition of chaos. Although equations (1)-(3) are stated for an iterated map with discrete time, these requirements can apply to dynamical systems with continuous time by sampling. As such, the essence of the three conditions—namely dense periodicity, transitivity, and sensitive dependence—may also be used to define chaos in ordinary and partial differential equations with continuous time without sampling. 3. Linear Chaos Although perhaps not well known, it has been recognized that a linear operator can meet the requirements of Devaney’s definition of chaos [10][11][15]. To provide an example, we consider the infinite-dimensional metric space X , d with elements x x0 , x1, x2 , x3 , , where xi R for all i 0 . For the norm such that
x
2
xi 2
(4)
i 0
we define
X x: x
2
(5)
d x, y x y
(6)
and use the metric induced by the norm
for x, y X . On this metric space, we consider the linear operator T defined as
Tx k x1, x2 , x3 , x4 ,
(7)
where k 1 is a scalar constant. The operator T is called a backward shift with gain [15]. We consider the dynamical system
4
xn1 Txn
(8)
with initial condition x0 X . In 1969, which is well before chaos was the subject of intense study, Rolewicz effectively proved this linear dynamical system exhibits transitivity [22]. More recently, others have completed a proof of chaos by showing dense periodic orbits and sensitive dependence [15][23]. A physical manifestation of the backward shift was identified in the unforced quantum harmonic operator, where the annihilation or lowering operator plays the role of the backward shift [24]. This example of linear chaos does not overturn everything we know about linear systems theory. In particular, it remains true that finite-dimensional linear systems cannot exhibit chaotic dynamics. Having an infinite-dimensional state space is the “loophole” that allows a linear system to meet Devaney’s mathematical definition of chaos. We use this loophole in the next section to devise another example of chaos in a linear system. 4. Wave Equation with Gain Interestingly, there are other infinite dimensional linear systems known to meet the conditions for chaos. Examples of linear chaos include the differential operator and certain translation operators [15]. In this section, we present a linear partial differential equation that also meets these conditions while offering a practical physical realization. Here we consider a one-dimensional wave equation with gain. Specifically, we consider the partial differential equation u u u t x
(9)
on the domain 0 x and t 0 . For an initial condition u x,0 f x , the solution of equation (9) is u x, t et f x t
(10)
which is a left-going, exponentially rising wave. We note f x may be discontinuous, implying a weak solution to the partial differential equation (9). Since the domain is restricted to x 0 , a finite initial condition does not grow unbounded: the exponential growth in the solution (10) is squelched as the wave harmlessly reaches the domain boundary at x 0 . To complete the description of the system, we apply the boundary condition
lim e x u x, t 0
x
(11)
which ensures that the solution remains bounded for all time. At any instant of time t 0 , the state of the dynamical system is defined by a function of only the spatial coordinate x. To emphasize the system state at a fixed time, we explicitly denote ut x u x, t . We define a norm for the state as
ut sup ut x x 0
5
(12)
where sup is the supremum of the function over the indicated domain. A metric space of spatial functions is then defined as W , d where
W ut : ut , lim e x ut x 0 x
(13)
and d is the metric induced by the norm. A function ut x is analogous to a point in this space, and its evolution as a function of time is a trajectory through the space. We immediately note that for any initial condition f x W , the resulting solution in equation (10) satisfies u x, t W for all t 0 . We claim the linear partial differential equation (9) is chaotic on W using Devaney’s definition of chaos. To support this claim, we show in the following that the system exhibits dense periodic states, topological transitivity, and sensitive dependence. We also show the system is characterized by a positive quantity that acts like a Lyapunov exponent. We first show that periodic states are dense in W. Addressing equation (1), we consider an arbitrary initial condition f x W . The resulting solution (10) typically is not periodic in time; however, there exists an initial condition arbitrarily close to f x that yields a time-periodic solution. We show the existence of such an initial condition by construction. To wit, we consider the initial condition f x enT f x nT , nT x n 1 T
where T is a fixed period and n
(14)
. By construction, we have f W and
lim f x f x
T
(15)
for all x 0 . Thus, f x can be made arbitrarily close to the original initial condition f x by choosing a sufficiently large period T. The corresponding solution is u x, t et f x t
(16)
u x, t T u x, t
(17)
for which the periodic identity
is readily verified for all x 0 and t 0 . Thus, for any open set U such that f x U W , there exists an initial condition f x U that yields a periodic orbit, which meets the first requirement in equation (1). A similar construction reveals the existence of a transitive orbit, which meets the topological transitivity requirement of equation (2). We assume two arbitrary functions f x and g x with the corresponding open sets U and V, such that f x U W and g x V W . We then consider the initial condition
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f x, 0 x T T u ( x, 0) e g x T , T x 2T 0, 2T x
(18)
where T 0 is an arbitrary interval length. We note that lim u x, 0 f x
(19)
lim u x, T g x
(20)
T
and, using the solution (10), T
so that u x, t can be made to pass arbitrarily close to both functions f x and g x . Thus, for any open subsets U W and V W , there exists a transitive orbit u x, t that links the two subsets, and equation (2) is satisfied. The last requirement is a sensitive dependence on initial conditions. Indeed, it is reasonable to expect sensitivity due to the exponential factor in the analytic solution (10). To explicitly address the requirement in equation (3), we consider an arbitrary initial condition
u x,0 f x
(21)
where f x W . We also consider a second initial condition
f x , x x v x, 0 x x f x ,
(22)
where x ln , 0 is a finite perturbation, and is an arbitrary constant. By design, the two functions (21) and (22) are nearly the same, differing by a finite amount at only the single spatial point x x 0 . Using the solution (10) for each initial condition, we find that
et , x t x v x, t u x , t x t x 0,
(23)
so that the difference in the resulting solutions is et , t x u x, t v x, t t x 0.
(24)
which explicitly shows exponential growth of a perturbation. Further, the initial deviation is u x, 0 v x, 0
(25)
max u x, t v x, t
(26)
while the maximum deviation is t 0
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which is attained at the time t x . Thus, by choosing an arbitrarily small initial perturbation , we have constructed an initial condition arbitrarily close to f x that subsequently diverges and meets the requirements of equation (3).
We note that the initial growth of the difference in equation (24) can be characterized as d u, v et , where 1 . As such, 1 acts like a positive Lyapunov exponent for the system, which is also commonly used as an indicator for chaos in a dynamical system [18]. Altogether, we have shown that the linear wave equation with gain exhibits the three requirements for the Devaney definition of chaos, as well as a positive Lyapunov exponent that describes the growth of small perturbations. This system provides an explicit example of linear chaos in a partial differential equation. Importantly, this system has physical significance, since the optical gain medium in a laser or optical amplifier can be modeled as a wave equation with gain [19]. There may be other physical systems that exhibit growing waves, and analysis may reveal that these systems may also meet the requirements for chaos in a linear system. 5. Other Definitions The recognition that the linear partial differential equation (9) meets the common definition of chaos does not mean this linear system suddenly exhibits the impenetrably complex behavior that we are accustomed to seeing in chaotic systems. In fact, the behavior of this wave equation is simple and transparent: its general solution (10) is a left-going, exponentially increasing wave. Indeed, this straightforward solution is not the complicated dynamic that inspired Li and Yorke to coin the term “chaos” in their landmark paper [25]. A number of definitions of chaos have been proposed and may be considered as alternatives to Devaney’s three requirements. A popular approach is to define chaos for an orbit: for example, an orbit is chaotic if it is not eventually periodic and exhibits a positive Lyapunov exponent [26]. In contrast, Devaney’s three requirements define chaos for a dynamical system. This distinction is subtle but, for example, enables one to characterize observed data without a system model [20]. Loosely, defining a chaotic orbit can be seen as more general, since it encompasses typical waveforms generated by dynamical systems that meet Devaney’s definition. As such, a typical solution of the linear system (9) is also chaotic by the orbit definition. Some alternative definitions for chaos are based on information concepts, including a recently proposed criterion that uses expansion entropy [27]. This definition directly senses extreme sensitivity by detecting the growth of uncertainty in a dynamical system. The resulting quantity is an entropy, and a positive value indicates chaos. An advantage of expansion entropy is that it allows the characterization of certain systems automatically excluded by Devaney’s requirements; for example, a nonautonomous system with quasi-periodic forcing does not admit periodic states, but it may still exhibit sensitive dependence on initial conditions. Since the linear system (9) exhibits a sensitive dependence characterized by exponential divergence, it also exhibits a positive expansion entropy and is still chaotic by this newer definition. 6. Conclusion A linear wave equation with gain is shown to satisfy the three conditions in Devaney’s definition of chaos, as well as appearing to meet alternative definitions based on Lyapunov
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exponents and expansion entropy. Since the analytic solution to this linear system is straightforward and its behavior is relatively simple, this result suggests that the common definitions of chaos are being stressed to, and possibly beyond, their limits. From a functional point of view, it is fair to conclude that these definitions cannot be mechanically applied without potentially devaluing the label of chaos. We acknowledge that Devaney offered his widely adopted definition in the context of a simple dynamical system, which is an iterated one-dimensional map on the unit interval [1]. It is perhaps surprising that this definition has served admirably well when applied to more complicated systems, including continuous flows described by differential equations. However, the recent examples of linear chaos, including the linear wave equation with gain shown here, as well as a linear wave equation without gain [28], may actually reveal that this definition has its limitations. Indeed, rather than blindly attributing chaos to systems that simply meet a technical definition, it may be better to reconsider the definition when its application leads to an unintended conclusion.
Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements The author acknowledges Dr. Shangbing Ai, Dr. Erik Bollt, Dr. Daniel Hahs, and Dr. Shawn Pethel for helpful discussions regarding the development and presentation of the research results. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] R. L. Devaney, Introduction to Chaotic Dynamical Systems (Addison-Wesley, 1989). [2] D. F. Drake, Information’s Role in the Estimation of Chaotic Signals, Ph.D. dissertation (Georgia Institute of Technology, 1998). [3] S. T. Hayes, “Chaos from linear systems: Implications for communicating with chaos, and the nature of determinism and randomness,” J. Phys. Conf. Ser. 23, 215 (2005). [4] D. F. Drake and D. B. Williams, “Linear, random representations of chaos,” IEEE Trans. Signal Process. 55, 1379 (2007). [5] S. Katsura, W. Fukuda, “Exactly solvable models showing chaotic behavior,” Physica A 130, 597 (1985). [6] K. Umeno, “Method of constructing exactly solvable chaos,” Phys. Rev. E 55, 5280 (1997).
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