Lagrangian Fuzzy Dynamics of Physical and Non-Physical Systems
In this paper, we show how to study the evolution of a system, given imprecise knowledge about the state of the system and the dynamics laws. Our approach is based on Fuzzy Set Theory, and it will be shown that the Fuzzy Dynamics of a -dimensional system is equivalent to Lagrangian (or Hamiltonian) mechanics in a -dimensional space. In some cases, however, the corresponding Lagrangian is more general than the usual one and could depend on the action. In this case, Lagrange’s equations gain a non-zero right side proportional to the derivative of the Lagrangian with respect to the action. Examples of such systems are unstable systems, systems with dissipation and systems which can remember their history. Moreover, in certain situations, the Lagrangian could be a set-valued function. The corresponding equations of motion then become differential inclusions instead of differential equations. We will also show that the principal of least action is a consequence of the causality principle and the local topology of the state space and not an independent axiom of classical mechanics.
We emphasize that our adaptation of Lagrangian mechanics does not use or depend on specific properties of the physical system being modeled. Therefore, this Lagrangian approach may be equally applied to non-physical systems. An example of such an application is presented as well.
In experiments with complex systems (for example, living cells), many of the system’s parameters remain hidden or out-of-control. This leads to large deviations in experimental results. As a result, small differences in the numerical values of the experimental data lose their significance. Indeed, the state of such a system is better described by a domain of points rather than a single point in the state space of the system. Moreover, these domains are “cloud-like” and do not have crisp boundaries. In order to give mathematical meaning to such domains, L. Zadeh introduced the notion of Fuzzy Sets and proposed Fuzzy Set Theory and Fuzzy Logic .
Zadeh’s basic idea may be illustrated as follows. Assume that the system parameters are described by the variables , such that each state of the system is represented by a point in the abstract space . An ordinary domain or subset of the space can be identified with its Identity function :
A fuzzy set, on the other hand, is identified with its membership function , where . We think of as the possibility that the given point belongs to the set . For , we should take a continuous function which tends to zero outside of some region. Then the points far from the region will have almost no possibility of belonging to the set. On the other hand, due to our imprecise knowledge, a point near the border of will have intermediate value of possibility of belonging to the domain.
The fuzzy approach enables one to handle the imprecision by operating with pairs instead of the parameter by itself. This approach is not new. In stochastic processes and models of quantum particles, for example, we also use probability distributions of the variables’ values or a wave function instead of the variables’ values by themselves. In our situation however, the imprecision does not have a stochastic or quantum nature. Hence, the membership functions need not possess the properties of a probability distribution.
In what follows, we will need to calculate the membership functions for composite statements like “ is Large OR is Moderate” and “ is Large AND is Small”. In other words, given the membership functions and of “ is Large” and “ is Moderate”, respectively, we need to define membership functions and . To insure that our fuzzy logical connectives are compatible with common human logic, we will require them to satisfy certain conditions. For example, the possibility that “X is Large OR is Moderate” and the possibility that “X is Moderate OR X is Large” should be equal, reflecting the symmetry of the “OR” connective. Also, if is certainly Moderate (), then the possibility that “X is Large OR X is Moderate” should equal , while if is certainly not Moderate (), then the possibility that “X is Large OR X is Moderate” should be equal to the possibility that “X is Large”.
Let us denote the possibility that “X is W OR X is M” by . Then, in accordance with the above-mentioned properties, should satisfy the following:
Similarly, if we denote the possibility that“ is W AND is M” by , we require:
Mathematical operations satisfying (2)-(5) are well known and were intensively studied during the last decades (see [8, 7] and references therein). In the mathematics literature, they are called triangular conorms, or t-conorms, for short. An operation satisfying (6)-(8) (with ) is called a triangular norm, or t-norm. In Fuzzy Set Theory, t-norms and conorms define the “intersection” and “union” of fuzzy sets. This is reasonable because the condition “ belongs to AND belongs to ” (, for short) defines the intersection , while “ belongs to OR belongs to ” () defines the union . Many examples of t-norms and conorms can be found in  and . Practically, the generalized Dubois-Prade t-norm ():
where and is a continuous monotonic function with , could be a good choice, because it joins a wide class of Pseudo-Product t-norms: , () with important Min t-norm: , ().
Note that the Min t-norm is the strongest t-norm in the sense that for any t-norm , we have . This means that an arbitrary t-norm is as depicted in Fig. 1. Of course, there are an infinite number of different t-norms and conorms, but some additional conditions on them could lead to a unique choice of . For example, let us assume that . We can write for :
In the opposite case , the same arguments lead to:
Therefore, the only suitable representation of the t-conorm in the case is
This result will be used in the next section.
1 Fuzzy dynamics
Consider a system that is moving in some space with coordinates . We assume that it is not possible to obtain the exact value of the coordinates or of the velocities at any given time . We can say only that there is some possibility that at time , the system is close to the point and its velocity is close to . In such a case, the system’s movement can be described as follows. If we denote by the possible values of the velocity , we can say that:
If, at the time , the system is located in the vicinity of the point , then at the previous time , the system could be near the point , or near the point , or near the point , or …, and so on, for all possible values of the velocity .
Let us denote the possibility that the system is in a small domain around the point at the time by . We denote the possibility that near the point and at the time the system’s velocity is approximately by . Then the above expression can be symbolically written as
Expression (11) is nothing more than the previous natural language expression, written in symbolic form. In order to translate it into an equation of the system’s dynamics, we should define mathematical representations of the expressions , and the logical connectives and . It is understood that corresponds to some measure of the domain 111 In accordance with Fuzzy Logic paradigm, can be considered as a truth value of the fact that the system is in the domain .. If we are interested in distances much more than the characteristic size of the domain , it is reasonable to consider a limit where the domain collapses to point:
Theoretically, there are two cases:
The first one is the main case of our study, while the second one is equivalent to the probabilistic approach to dynamical problems 222In the first case, corresponds to the so called atomic measure of the domain, while in the second one can be considered as a common additive measure.. Further, we will assume that and are continuous, bounded functions: , where the value corresponds to the minimal possibility, and the value to corresponds to the maximal one.
The connectives , can be represented by the various t-norms and t-conorms. It is remarkable, however, that the natural properties of the state space’s local topology drastically restrict the available choice of the representations of the connective . To demonstrate this, let us consider two nearest-neighbor domains , of the system’s state space. It is obvious that the possibility that the system is in the joint domain is equal to the possibility that it is in the domain OR it is in the domain . Hence, we can write
Now, if both domains are collapsed to the same point: , we have
which implies that:
As shown above, equation (15) implies that
This result is crucial for our study, and it is important that this representation for is dictated by the local topology of the space rather than our mathematical taste, convenience, etc. Similar arguments, however, cannot be employed to the connective . The reason is that this connective can include membership functions which depend on variables that belong to different spaces. For example, in the expression (11), the possibility depends on domain of the system’s state space, while the possibility depends on the domain of its tangential space. In this case, collapsing both of the domains to the same point is impossible, and, therefore, the above-mentioned argumentation becomes invalid. Thus, the explicit form of the t-norm remains arbitrary 333In the case (14), however, unique representation of the connective can be found by using topological properties of the system’s trajectories ..
which is the Master Equation of Fuzzy Dynamics -. (Note that (17) is a particular case of Zadeh’s so-called Extension Principle ). The system’s evolution is described by the function , which reflects the possibility that the system’s variables have the values at the time . The function should be found by solving Eq. (17) with the initial condition
where is the possibility that the state of the system was at the time . The function is determined by the system’s dynamics law 444In the case (14), “Master-Equation” of the fuzzy dynamics is equivalent to an ordinary master-equation of the stochastic dynamics  .
If we interested in a time interval much more than , it is reasonable to take the limit . To do this, refer to Fig.1 and note that
where is the velocity corresponding to the maximal value of the right side of (17). Thus, we can write
For small , we can expand with respect to . In the limit , one obtains 
Note that in order for to be maximal, should be minimal. On the other hand, for , we have
Therefore, can be found by minimization of
under the restriction
where is a solution of the equation . For example, for the Dubois-Prade t-norm, we have
where is an arbitrary number. (Note that doesn’t depend on concrete choice of the function in (9)).
where so that will correspond to the minimum of (23).
It should be emphasized that the functions and cannot be identified with any probability density , because they have different mathematical features. and are pointwise limited: , while the integral of or over all space could be infinite. On the other hand, , while can be infinite at some points 555Actually, is a function, while is a functional.
2 Classical mechanics as fuzzy dynamics in -dimensional space
Consider a dynamical system, whose behavior is described by variables: coordinates and an additional scalar variable (S-variable). We will assume that our knowledge about the system’s location and velocity is imprecise, so that the system’s dynamics should be described by its membership function
and by a function
We can solve Eq.(33) with respect to and obtain
Solution of (35) with respect to gives
where and . Equation (38) leads to a system of ordinary differential equations:
Introducing a new variable
we can rewrite (39) as
Since is conserved along the trajectories, the system’s trajectories are on a surface in the -space, which is defined by equation 666In the fuzzy set’s literature the set is called -cut of .. Consider variations on this surface. We have
which leads to
This implies that
where is a fixed number and should be found from Eq.(54).
It is easily seen that is stationary along the trajectories (50). Indeed, we have
Using (50), one obtains
If doesn’t explicitly depend on , then and do not depend on , either. In this case, equations (41)-(43) and (50) become the well-known Hamiltonian and Lagrangian equations of classical mechanics, while in (51) becomes the classical action. As a result, we will call the -variable an “action,” even in the general case. As we can see in (28), the action can be considered as added dimension of the state space. In this approach, however, this dimension is not equivalent to the other ones and plays an exclusive role (see, however, Appendix A).
It should be emphasized that equations (50) and (51) were obtained independently by using Eqs.(30) and (26), which follow from the Master Equation (17). This means that stationarity of (that reflects principle of least action) is, in fact, a consequence of the causality principle and the local topology of the state space and is not an independent axiom of classical mechanics.
2.1 Uncertainty as an external field
Consider a particle which is certainly free far from an origin, but not certainly free in the vicinity of the origin. In this case, the possibility function could be approximated as
where is any monotonically decreasing function with and , is Lagrangian of the free particle and . The corresponding Lagrangian is
(the sign of is chosen such that in (26) is positive). We see that uncertainty influences as a “ghost” field, which disappears for and increases with decreasing of .
The situation becomes more complicated, however, if we assume that
In this case,
where is an arbitrary function of . The Lagrangian (59) is a set-valued function because it corresponds to a set of functions and not to a unique function as in (57). This means that Eqs.(50), with Lagrangian (59), and (41)-(43), with corresponding Hamiltonian (37), become differential inclusions instead of differential equations (see - and references therein for more information about differential inclusions). Solving differential inclusions is more complicated than solving differential equations because inclusions describe the dynamics of a set rather than the dynamics of a point. Fortunately, in our particular case, the solution of inclusion with Lagrangian (59) can be found in a simple way by considering (59) in polar coordinates
For small velocities, we can write
Since is a cyclic variable, we have
The Lagrangian (62) does not depend on time. Therefore, the energy
is conserved. The Lagrangian (62) leads to the equations of motion:
These equations can be easily solved in an implicit form:
Since the solution (66)-(67) depends on an arbitrary function , it describes a set of equally possible trajectories rather than a single trajectory as in the usual (non-fuzzy) case. In order to understand and to interpret the evolution of inclusion, we need to know the border of the set (66)-(67). In general, this is a nontrivial and complicated task, but in our case, this border can be found quite simply. Indeed, it is seen from (66) that its upper and lower borders correspond to , respectively. The solution of inclusion (66)-(67) for is shown in Fig. 2.
2.2 Cost of memory
Consider now the general case, in which the velocity of a system is determined not only by the system’s current state, but by its action as well. This means that the possibility of values of the system’s velocity depends on . Hence, in accordance with (34), the Lagrangian and the Hamiltonian depend on as well. We will call such a Lagrangian an S-Lagrangian. It follows from (51) that the -variable depends on a system’s history. Therefore, such a system should “remember” its history. This memory, however, has a certain cost.
Since the system is closed, we have
Setting , we obtain
This means that even in closed systems with a time-independent Hamiltonian (), neither total moment nor energy are conserved 777On the other hand for some exotic S-Hamiltonian: with the energy is conserved. (however, the notions of “energy”, “moment” and “action” of systems with an S-Lagrangian might be different from the usual notions of energy, moment and action (see Appendix D)). This result is reasonable, because the term in (69) can be considered as an effective friction coefficient. Note that for closed systems with a time-independent S-Hamiltonian, the quantity is conserved:
It should be noted, that for the systems with an -Lagrangian
3 Lagrangian mechanics of non-physical system
Consider an organism which acquires a particular resource by moving on a surface. It spends this resource in order to maintain the system’s activity (in our case, its movement). In addition, the acquired resource spontaneously decays. During the time , the system obtains an amount of the resource equal to
where is the velocity, is proportional to the density of the resource, is the path of the system during the time , is the rate of spending of the resource in maintaining the system’s activity, and is rate of spontaneous decay. Equation (75) admits different semantics. For example, it could be a simple model of real biological systems such as a caterpillar on a plant or a whale in a plankton field. It could also model the selling of products. In this case, is income, describes the distribution of the buyers, is travel and other expenses, and is the tax obligation.
For small and , we can expand and with respect to and :
where are constants.
It is seen in Fig. 3(a) that a system without memory () demonstrates a random walk on the surface (see Fig. 4(a)). The system with memory (), on the other hand, very quickly finds the place to maximize the enriching of its resources and enters this place (see Fig. 3(b,c)). The power spectrum of the trajectory (Fig. 4(b)) implies that in the last case, the system’s trajectory becomes a strange attractor, which is concentrated in the resource enriching area.
It should be emphasized that we did not introduce any special equipment for memorization of the system’s history. In fact, the dependence of the Lagrangian on the “action” (acquired resource in our case) itself creates the system’s memory.
Interestingly, the considered model is an example of “rational behavior without mind.” It is seen in Fig. 5 that in the absence of decay of the resource (), the random walk is an effective behavior. However, if we omit the “memory term” in the equations of motion (right-hand side of equation (50)), but keep the decay of the resource in the equation for , (simply set in equation (78), but keep it in equation (79)), the random walk will be ineffective, while the strange attractor in the vicinity of the resource-enriching area becomes an optimal strategy.
4 Quantum mechanics of systems with -Lagrangian
This Section does not directly relate to fuzzy dynamics, but rather exploits the notion of -Lagrangian introduced above. Consider a quasi-particle with Lagrangian
where is a constant and is an action. If, at time , the quasi-particle’s state was , then at time , the state of the particle is
where is Feynman’s propagator 
where is a path integral over all trajectories from to . It is useful to consider a more general propagator
with initial condition: