Continuous time System Dynamics and Responses Mae 143a

Continuous Time Dynamical System

Simulation of Dynamic Models

Mark M. Meerschaert , in Mathematical Modeling (Fourth Edition), 2013

6.2 Continuous-Time Models

In this section we discuss the fundamentals of simulating continuous-time dynamical systems. The methods presented here are simple and usually effective. The basic idea is to use the approximation

$\frac{dx}{dt}\approx \frac{\Delta x}{\Delta t}$

to replace our continuous-time model (differential equations) by a discrete– time model (difference equations). Then we can use the simulation methods we introduced in the preceding section.

Example 6.2. Reconsider the whale problem of Example 4.2. We know now that starting at the current population levels of B = 5, 000, F = 70, 000, and assuming a competition coefficient of α < 1.25 × 10⁻⁷, both populations of whales will eventually grow back to their natural levels in the absence of any further harvesting. How long will this take?

We will use the five-step method. Step 1 is the same as before (see Fig. 4.3), except that now the objective is to determine how long it takes to get to the equilibrium starting from B = 5, 000, F = 70, 000.

Step 2 is to select the modeling approach. We have an analysis question that seems to require a quantitative method. The graphical methods of Chapter 4 tell us what will happen, but not how long it will take. The analytical methods reviewed in Chapter 5 are local in nature. We need a global method here. The best thing would be to solve the differential equations, but we don't know how. We will use a simulation; this seems to be the only choice we have.

There is some question as to whether we want to adopt a discrete-time or a continuous-time model. Let us consider, more generally, the case of a dynamic model in n variables, x = (x ₁,…, x_n ), where we are given the rates of change F = (f ₁,…, f_n ) for each of the variables x ₁,…, x_n , but we have not yet decided whether to model the system in discrete-time or continuous-time. The discrete-time model looks like

(6.4) $\begin{array}{l}\Delta x_1=f_1\left(x_1,\dots ,x_n\right)\hfill \\ ⋮\hfill \\ \Delta x_n=f_n\left(x_1,\dots ,x_n\right),\hfill \end{array}$

where Δx_i represents the change in x_i over 1 unit of time (Δt = 1). The units of time are already specified. The method for simulating such a system was discussed in the previous section.

If we decided on a continuous-time model instead, we would have

(6.5) $\begin{array}{l}\frac{{dx}_1}{dt}=f_1\left(x_1,\dots ,x_n\right)\hfill \\ ⋮\hfill \\ \frac{{dx}_n}{dt}=f_n\left(x_1,\dots ,x_n\right),\hfill \end{array}$

which we would still need to figure out how to simulate. We certainly can't expect the computer to calculate x(t) for every value of t. That would take an infinite amount of time to get nowhere. Instead we must calculate x(t) at a finite number of points in time. In other words, we must replace the continuous– time model by a discrete-time model in order to simulate it. What would the discrete-time approximation to this continuous-time model look like? If we use a time step of Δt = 1 unit, it will be exactly the same as the discrete-time model we could have chosen in the first place. Hence, unless there is something wrong with choosing Δt = 1, we don't have to choose between discrete and continuous. Then we are done with step 2.

Step 3 is to formulate the model. As in Chapter 4, we let x ₁ = B and x ₂ = F represent the population levels of each species. The dynamical system equations are

(6.6) $\begin{array}{ll}\frac{{dx}_1}{dt}\hfill & =0.05x_1\left(1-\frac{x_1}{150,000}\right)-\alpha x_1{x}_2\hfill \\ \frac{{dx}_2}{dt}\hfill & =0.08x_2\left(1-\frac{x_2}{400,000}\right)-\alpha x_1{x}_2\hfill \end{array}$

on the state space x ₁ ≥ 0, x ₂ ≥ 0. In order to simulate this model we will begin by transforming to a set of difference equations

(6.7) $\begin{array}{l}\Delta x_1=0.05x_1\left(1-\frac{x_1}{150,000}\right)-\alpha x_1{x}_2\hfill \\ \Delta x_2=0.08x_2\left(1-\frac{x_2}{400,000}\right)-\alpha x_1{x}_2\hfill \end{array}$

over the same state space. Here, Δx_i represents the change in population x_i over a period of Δt = 1 year. We will have to supply a value for α in order to run the program. We will assume that α = 10⁻⁷ to start with. Later on, we will do a sensitivity analysis on α.

Step 4 is to solve the problem by simulating the system in Eq. (6.7) using a computer implementation of the algorithm in Fig. 6.2. We began by simulating N = 20 years, starting with

$\begin{array}{ll}{x}_1\left(0\right)\hfill & =5,000\hfill \\ x_2\left(0\right)\hfill & =70,000.\hfill \end{array}$

Figures 6.9 and 6.10 show the results of our first model run. Both blue whale and fin whale populations grow steadily, but in 20 years they do not get close to the equilibrium values

Figure 6.9. Graph of blue whales x ₁ versus time n for the whale problem: case α = 10⁻⁷, N = 20.

Figure 6.10. Graph of fin whales x ₂ versus time n for the whale problem: case α = 10⁻⁷, N = 20.

$\begin{array}{ll}{x}_1\hfill & =35,294\hfill \\ x_2\hfill & =382,352\hfill \end{array}$

predicted by our analysis back in Chapter 4.

Figures 6.11 and 6.12 show our simulation results when we input a value of N large enough to allow this discrete-time dynamical system to approach equilibrium.

Figure 6.11. Graph of blue whales x ₁ versus time n for the whale problem: case α = 10⁻⁷, N = 800.

Figure 6.12. Graph of fin whales x ₂ versus time n for the whale problem: case α = 10⁻⁷, N = 100.

Step 5 is to put our conclusions into plain English. It takes a long time for the whale populations to grow back: about 100 years for the fin whale, and several centuries for the more severely depleted blue whale.

We will now discuss the sensitivity of our results to the parameter α, which measures the intensity of competition between the two species. Figures 6.13 through 6.18 show the results of our simulation runs for several values of α. Of course, the equilibrium levels of both species change along with α.

Figure 6.13. Graph of blue whales x ₁ versus time n for the whale problem: case α = 3 × 10⁻⁸, N = 800.

Figure 6.14. Graph of blue whales x ₁ versus time n for the whale problem: case α = 10⁻⁸, N = 800.

Figure 6.15. Graph of blue whales x ₁ versus time n for the whale problem: case α = 10⁻⁹, N = 800.

Figure 6.16. Graph of fin whales x ₂ versus time n for the whale problem: case α = 3 × 10⁻⁸, N = 100.

Figure 6.17. Graph of fin whales x ₂ versus time n for the whale problem: case α = 10⁻⁸, N = 100.

Figure 6.18. Graph of fin whales x ₂ versus time n for the whale problem: case α = 10⁻⁹, N = 100.

However, the time it takes our model to converge to equilibrium changes very little. Our general conclusion is valid whatever the extent of competition: It will take centuries for the whales to grow back.

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Analysis of Dynamic Models

Mark M. Meerschaert , in Mathematical Modeling (Fourth Edition), 2013

Abstract

In this chapter, we consider some of the most broadly applicable techniques for the analysis of discrete and continuous time dynamical systems such as Eigenvalue Methods and Phase Portraits. These methods can provide important qualitative information about the behavior of dynamical systems, even when exact analytic solutions are not obtainable. Eigenvalue method is used to analyze nonlinear dynamical systems for stability and is an appropriate application for a computer algebra system. Its methods can be applied to both continuous time dynamical systems and discrete time dynamical systems. The same concept can be used to obtain the phase portrait, which is a graphical description of the dynamics over the entire state space. A phase portrait is mapped by homeomorphism, a continuous function with a continuous inverse.

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Some New Chaotic Maps With Application in Stochastic

Ezzedine Mliki , ... Sajad Jafari , in Recent Advances in Chaotic Systems and Synchronization, 2019

4 Discussion and Conclusion

In this chapter, we have investigated dynamical properties of two interesting discrete systems. The first one was a map which was proposed as a new method for constructing a one-dimensional chaotic Gaussian distribution. Results have been tested using normal distribution and visual tests. Also, by plotting the autocorrelation coefficient diagram, we have shown that the distribution could have the characteristics of a white noise distribution. So, it can be used as a model of white Gaussian noise in telecommunication systems and electronics. The second studied system was an ADD model. The system has different dynamics. The most interesting behavior of this system was the coexistence of a period-two attractor with different attractors such as two periodic dynamics, two chaotic dynamics or one larger attractor. The system gives us the idea that existence of horizontal asymptotes can help a map with two symmetric lobes show coexistence of periodic behaviors with different periods or coexistence of periodic and chaotic attractors. ADD is one of the most common disorders, and its modeling can help in its diagnosis and treatment. So, investigating dynamical properties of the ADD model can be helpful for future research work on this disorder.

As to some suggestions for future work, we can mention the following:

(a)

We suggest generalizing the results replacing the map x _{k  +   1}  = f(x _k ) by continuous time dynamical system, x _{t  + h}   = f _h (x _t ), t, h  ≥   0, in which f _{s  + t}   = f _s of t; s; t  ≥   0.

(b)

We suggest generalizing the results by replacing the map x _{k  +   1}  = f(x _k ) by a dynamical system with randomness (Markov process), X _{t  + h}   = f _h (X _t ; w _t ^{t  + h} ); t; h  ≥   0 in which X _t is Markov process.

(c)

We suggest investigating stochastic-chaotic systems to study some new chaotic applications [67–71].

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Billiards in Bounded Convex Domains

S. Tabachnikov , in Encyclopedia of Mathematical Physics, 2006

Billiard Flow and Billiard Ball Map

The billiard system describes the motion of a free particle inside a domain with elastic reflection off the boundary. More precisely, a billiard table is a Riemannian manifold M with a piecewise smooth boundary, for example, a domain in the plane. The point moves along a geodesic line with a constant speed until it hits the boundary. At a smooth boundary point, the billiard ball reflects so that the tangential component of its velocity remains the same, while the normal component changes its sign. This means that both energy and momentum are conserved. In dimension 2, this collision is described by a well-known law of geometrical optics: the angle of incidence equals the angle of reflection. Thus, the theory of billiards has much in common with geometrical optics. If the billiard ball hits a corner, its further motion is not defined.

The billiard reflection law satisfies a variational principle. Let A and B be fixed points in the billiard table and let AXB be a billiard trajectory from A to B with reflection at a boundary point X. Then, the position of a variable point X extremizes the length AXB. This is the Fermat principle of geometrical optics.

In this article, we discuss billiards in bounded convex domains with smooth boundary, also called Birkhoff billiards. A related article treats billiards in polygons (see Polygonal Billiards).

The billiard flow is defined as a continuous-time dynamical system. The time-t billiard transformation acts on unit tangent vectors to M which constitute the phase space of the billiard flow, and the manifold M is its configuration space. Thus, the billiard flow is the geodesic flow on a manifold with boundary.

It is useful to reduce the dimensions by one and to replace continuous time by discrete one, that is, to replace the billiard flow by a mapping, called the billiard ball map and denoted by T. The phase space of the billiard ball map consists of unit tangent vectors (x, v) with the foot point x on the boundary of M and the inward direction v. A vector (x, v) moves along the geodesic through x in the direction of v to the next point of its intersection x ₁ with the boundary ∂M, and then v reflects in ∂M to the new inward vector v ₁. Then, one has: T(x, v) = (x ₁,v ₁). For a convex M, the map T is continuous. If M is n-dimensional, then the dimension of the phase space of the billiard ball map is 2n − 2.

Equivalently, and more in the spirit of geometrical optics, one considers $\mathcal{L}$ , the space of oriented geodesics (rays of light) that intersect the billiard table. This space of lines is in one-to-one correspondence with the phase space of the billiard ball map: to an inward unit vector (x, v) there corresponds the oriented line through x in the direction v ( Figure 1 ).

Figure 1. Billiard ball map.

The space of rays $\mathcal{L}$ carries a canonical symplectic structure, that is, a closed nondegenerate differential 2-form. In the Euclidean case, this symplectic structure ω is defined as follows. Given an oriented line ℓ in R ⁿ , let q be the unit vector along ℓ and p be the vector obtained by dropping the perpendicular from the origin to ℓ. Then, ω = dp ∧ dq = ∑dp _i ∧ dq _i . This construction identifies $\mathcal{L}$ with the cotangent bundle of the unit sphere: q is a unit vector and p is a (co)tangent vector at q, and ω identifies with the canonical symplectic structure of T*S ⁿ − 1. In the general case of a Riemannian manifold M, the symplectic structure on the space of oriented geodesics is obtained from that on T*M by symplectic reduction.

One has an important result: the billiard ball map preserves the symplectic structure T*(ω) = ω. As a consequence, T is also measure preserving. In the planar case, one has the following explicit formula for this measure. Let t be an arc length parameter along the boundary of the billiard table and let α ∈ [0, π] be the angle made by the unit vector with this boundary. Then, (α, t) are coordinates in the phase space, identified with the cylinder, and the invariant measure is sin α dα dt.

As a consequence, the total area of the phase space equals 2L where L is the perimeter length of the boundary of the billiard table, and the mean free path equals πA/L, where A is the area of the billiard table. In the general n-dimensional case, the mean free path equals

$\frac{\text{vol}\left(S^{n-1}\right)}{\text{vol}\left(B^{n-1}\right)}\frac{\text{vol}\left(M\right)}{\text{vol}\left(\partial M\right)}$

where S ⁿ⁻¹ and B ⁿ⁻¹ are the unit sphere and the unit disk in Euclidean spaces.

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On pinning control of some typical discrete-time dynamical networks

Haifeng Zhang , ... Xinchu Fu , in Communications in Nonlinear Science and Numerical Simulation, 2010

5 Extending the interval for the feedback gain d

For a dynamical network, its synchronizability without changing its topology structure? Many researchers have considered this question. For a continuous-time dynamical system

(5.1) $\dot{x_i}=f(x_i)+\frac{c}{k_i^{\beta }}\sum _{i=1}^N{A}_{\mathit{ij}}h(x_j)\text{,}i=1\text{,}2\text{,}\dots \text{,}N\text{,}$

where $h(x)$ is the inner-coupling function and $k_i$ is the degree of the node i. In [16], A.E. Motter et al. suggested that the synchronizability of the network (5.1) is maximum when $\beta =1$ . So, we may enhance the ability of feedback control by the similar method used in [16].

We rewrite Eq. (2.4) as follows:

(5.2) $\begin{array}{ccc}{x}_{i_l}(t+1)\hfill & =\hfill & f(x_{i_l}(t))+\frac{c}{k_{i_l}}\sum _{j=1}^N{a}_{i_{l}j}f(x_j(t))+{\mu }_{i_l}\text{,}l=1\text{,}2\text{,}\dots \text{,}m\text{,}\hfill \\ x_{i_l}(t+1)\hfill & =\hfill & f(x_{i_l}(t))+\frac{c}{k_{i_l}}\sum _{j=1}^N{a}_{i_{l}j}f(x_j(t))\text{,}l=m+1\text{,}m+2\text{,}\dots \text{,}N.\hfill \end{array}$

With this new coupling mechanism introduced into above four models, the intervals of d are extended to about $d\in [26\text{,}76]$ , $d\in [18\text{,}50]$ , $d\in [26\text{,}72]$ , and $d\in [52\text{,}145]$ for star-shaped network, complete network, BA network, and variant small-world network, respectively, and the results are shown in Fig. 6(a)–(d).

Fig. 6. For (a) star-shaped network, (b) complete network, (c) the BA network and (d) the variant small-world network, we find that the maximum of d can be extended to $d=76\text{,}d=50\text{,}d=72$ , and $d=145$ , respectively. Then the homogenous state $\overline{x}=0.6314$ can be achieved when the dynamical behavior is described by Eq. (5.2).

We remark here that although our results are based on the Hénon chaotic map, they can also be generalized to other chaotic maps. To show this, we added one more simulation with the Logistic map $x(t+1)=4x(t)(1-x(t))$ on small-world network with $N=200$ , the simulation results are shown in Figs. 7(a) and (b) and 8.

Fig. 7. Simulation with the Logistic map $x(t+1)=4x(t)(1-x(t))$ on small-world network with $N=200$ and the coupling strength $c=0.01$ . (a) When d is outside the range [50,   120], then the homogenous state $\overline{x}=0.75$ cannot be achieved; (b) when $d\in [50\text{,}120]$ the system can be pinned to the homogenous state $\overline{x}=0.75$ .

Fig. 8. For the Logistic map when the dynamical behavior is described by Eq. (5.2), the maximum of d can be extended to $d=133$ , and the homogenous state $\overline{x}=0.75$ can be achieved.

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Numerical characterization of nonlinear dynamical systems using parallel computing: The role of GPUs approach

Filipe I. Fazanaro , ... Romis Attux , in Communications in Nonlinear Science and Numerical Simulation, 2016

3 Characterizing nonlinear dynamical systems through the Lyapunov exponents

Nonlinear dynamical systems can be characterized according to distinct manners, with different aspects and measures. A particular interesting one is provided by the Lyapunov spectrum, i.e., the set of all Lyapunov exponents, since they define not only the topological structure of the attractor for continuous-time dynamical systems (i.e., if there is convergence to a fixed point, limit cycle, torus or even to a strange attractor), but also the information generated by the dynamics (or lost by an external observer), its fractal dimension or even a lower bound estimator for the Kolmogorov–Sinai entropy. This explain why the Lyapunov spectrum is commonly taken as a general measure of complexity [63,68–71].

Classically, Lyapunov exponents can be defined in terms of the mean divergence (or convergence) rate of initially close initial conditions in the phase space, given the successive application of the motion equations. Their origins can be traced to Lyapunov's seminal work and the definition of the first Lyapunov method or Lyapunov indirect method [72]. In this approach, the stability of the dynamics is studied by means of a "step-by-step" linearization of the system, being the long term temporal average of the real part of the eigenvalues of the Jacobian matrix (calculated for every step in time) the key attributes for defining the system stability [73]. Its numerical evaluation is subject to more recent works [74,75], and can be reviewed in Refs. [63,70,71].

Briefly, the calculation of the exponents for an F(x, t) n-dimensional dynamical system can be performed with the tangent map approach, in which an n-dimensional identity matrix (I _n ) anchored in the fiducial trajectory (the solution of the dynamical system) is successively transformed by the tangent map applications, and the resulting stretching (or contracting) effect is measured and average for a long-term evolution. The principal axes of the tangent map are determined by the variational equations:

(1) $\begin{array}{cc}\hfill \dot{\mathbf{\Phi }}(\mathbf{x},t)& =\mathbf{J}(\mathbf{x},t)\mathbf{\Phi }(\mathbf{x},t)\hfill \end{array}$

where J(x, t) is the Jacobian of F(x, t). Details concerning the method can be found in Refs. [63,70,71], and a MATLAB code for the Duffing oscillator (Eqs. (3) and (4)) can be found in the supplementary material of the present work.

Given the relevance of the Lyapunov spectrum, an alternative way to compute it – called the Cloned Dynamics (ClDyn) approach – was developed, based on perturbation theory [63]. This approach is particularly convenient for non-smooth dynamical systems or even for state equations whose Jacobian would require a laborious process [76,77], since it does not require the construction of the tangent space and the solution of the variational equations as required by the classical approach [70,71].

In essence, the ClDyn approach consists in estimating the Lyapunov exponents monitoring the difference state vector between the fiducial (referential) trajectory – defined by the original dynamical system – and copies (clones) of these state equations departing from initial conditions very close to the fiducial solution, as illustrated in Fig. 4 (modified from Ref. [63]).

Fig. 4. Illustration of the ClDyn approach. For simplicity, the vectors P and ${\mathbf{v}}_2^{\left(1\right)}$ are omitted in panel C.

In other words, the ClDyn approach starts by obtaining the fiducial trajectory and defining close initial conditions in linearly independent directions for the clones, and, consequently, defining the initial perturbation vectors. Each clone will be responsible for the time evolution of a perturbation applied to a specific and orthogonal direction in the phase space, corresponding to a Lyapunov exponent to be estimated, i.e., the mean divergence (or convergence rate) to the fiducial trajectory in a given direction.

For the sake of simplicity, and without loss of generality, consider a two-dimensional system, which would require only two clones for evaluating the time evolution of the small orthogonal perturbations, denoted by ${\mathit{\delta }}_{1x}^{\left(0\right)}$ and ${\mathit{\delta }}_{2x}^{\left(0\right)}$ in panel A of Fig. 4. These perturbations are propagated by the motion equations for a time interval T and the respective difference state vectors are updated taking the difference from the final point of the fiducial trajectory x(t) and each clone x _c1(t) and x _c2(t), giving rise to the vectors ${\mathit{\delta }}_{1x}^{\left(1\right)}$ and ${\mathit{\delta }}_{2x}^{\left(1\right)},$ in which the superscript index denoted the current iteration of the algorithm.

The second stage of the ClDyn approach, as illustrated in panel B of Fig. 4, consists in applying the Gram–Schmidt Reorthonormalization (GSR) procedure [70,71] in order to correct the tendency of alignment of the difference state vectors in the most expansive direction (if there is any), and, moreover, maintaining the orthogonality between these vectors. Therefore, a new set of numerically corrected vectors ${\mathbf{v}}_1^{\left(1\right)}$ and ${\mathbf{v}}_2^{\left(1\right)}$ are obtained. Finally, the clones are anchored in the same directions of the numerically corrected vectors (denoted by ${\mathbf{u}}_1^{\left(1\right)}$ and ${\mathbf{u}}_2^{\left(1\right)}$ ) but close to the fiducial trajectory again, in order to provide the initial condition for the next iteration of the algorithm, as illustrated in the panel C of Fig. 4.

In this case, if a sufficient small perturbation with magnitude δ _x0 is considered, after the Kth iteration and for K sufficiently large, the ith Lyapunov exponent (with $i=1,2,\dots ,n$ ) can be computed by Eq. (2),

(2) $\begin{array}{cc}\hfill {\lambda }_i& =\frac{1}{KT}\sum _{k=1}^K\ln \parallel \frac{{\mathbf{v}}_i^{\left(k\right)}}{{\delta }_{x0}}\parallel .\hfill \end{array}$

For a complete mathematical description and more details regarding the estimation of the algorithm parameters, see Ref. [63]. In the following, such metric is employed for determining both the local behavior of the vector field and the global (mean) characteristic of the attractor. Local and global behavior are obtained by setting the number of algorithm iterations (K parameter). For a practical introduction, a MATLAB code concerning the ClDyn approach applied to the Duffing dynamical system (Eqs. (3) and (4)) can also be found in the supplementary material.

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