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Dynamical systems: An overview
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
State variables are those that offer a complete description of the state of a dynamical system while the state space corresponds to the set of all the possible values of the state variables. The dimension of the dynamical system is given by the number of state variables. Phase space normally refers to the state space which is continuous and finite-dimensional. The phase plane consists of trajectories obtained from the solutions of (5.1). For instance, taking m = 1, a phase plane is guided by the pair (x,dxdt) while for m = 2, the phase plane corresponds to (x1, x2) or simply (x, y) where x and y are a pair of Cartesian coordinates. An ensemble of trajectories for a given system in the phase plane or a phase space is called a phase portrait (see Figure 5.1). An isocline refers to a curve in a phase plane or a phase space on which the trajectories have a fixed gradient. In general, the set of initial conditions (x1(0), x2(0), …xm(0)) dictates how a dynamical system evolves ultimately toward the final state (which may be a point or an area or a curve depending on the manifold the dynamical system operates in) in the phase space or state space. Such a destination is called an attractor.
Geometric Approaches and Applications of Systems of Differential Equaions
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
Just as with linear systems in Section 6.1, we are interested in finding the equilibria1 of the system (6.8) of first-order differential equations. We know that the solution (x, y) is at equilibrium when it is not changing, i.e., when x′ = 0 and y′ = 0. In order to find equilibria, we need to consider the curves f(x, y) = 0 and ɡ(x, y) = 0 in the phase plane. Any curve of the form h(x, y) = k, where k is a constant, is called an isocline or level curve of the function h. In other words, a curve is an isocline or level curve of a function if the function takes the same value at every point on the curve. In the special case when k = 0, the curve h(x, y) = k = 0 is called a nullcline. In the system (6.8), f(x, y) = 0 and ɡ(x, y) = 0 are the nullclines since they take the value zero. From the discussion above we see that the intersection point of the two nullclines is an equilibrium. More formally, (x*, y*) is an equilibrium of (6.8) if f(x*, y*) = 0 and ɡ(x*, y*) = 0.
Properties of Nonlinear Systems
Published in Zoran Vukiæ, Ljubomir Kuljaèa, Dali Donlagiæ, Sejid Tešnjak, Nonlinear Control Systems, 2003
Zoran Vukiæ, Ljubomir Kuljaèa, Dali Donlagiæ, Sejid Tešnjak
In order to construct the trajectory between two adjoining isoclines, from the point M0 two straight lines with slopes N1 and N2 are drawn in the direction of the isocline N2. As the trajectory at point M0 has slope N1, and the next isocline must intersect the trajectory with the slope N2, the intersection point with isocline N2 must be inside the angle of the lines N1 and N2. It can be approximately taken that the point B, the intersection of the isocline N2 and the phase trajectory, is situated in the middle of the isocline section between the lines N1 and N2. An analogous procedure is valid for the points C and D, as well as for all other points. By choosing other initial points on the same isocline, the other phase trajectories are found. The isocline method is appropriate to find phase portraits when initial conditions for a given section are known.
The Complex Dynamical Behavior of a Prey-Predator Model with Holling Type-III Functional Response and Non-Linear Predator Harvesting
Published in International Journal of Modelling and Simulation, 2022
Prahlad Majumdar, Surajit Debnath, Susmita Sarkar, Uttam Ghosh
In this section, we shall discuss the existence of various feasible equilibrium points of the system (4), which are the points of intersection of the prey zero growth isocline and predator zero growth isocline lying in the first quadrant of the plane (see Figure 1). The local stability of the equilibrium points and the global stability of the interior equilibrium points will be analyzed here.
Vehicle control synthesis using phase portraits of planar dynamics
Published in Vehicle System Dynamics, 2019
Carrie G. Bobier-Tiu, Craig E. Beal, John C. Kegelman, Rami Y. Hindiyeh, J. Christian Gerdes
The movement of the equilibria described in the previous section is determined by the isocline geometry. The isoclines of a state derivative are defined as lines on a phase portrait where that state derivative is a constant value; nullclines are lines of zero value. Figure 6 shows the yaw acceleration isocline geometry for P1 at 10 m/s, and varying steer angle: the dotted lines in the figure denote several isoclines of yaw acceleration. Additionally, the horizontal black lines indicate the maximum and minimum steady-state yaw rates, which are found by solving for the equilibrium state of Equation (1) and assuming all four tyres simultaneously reach (positive or negative) saturation. The positively sloped black lines are determined by the rear slip angles at full (positive or negative) tyre saturation, where of Equation (3) is defined by of Equation (7), which is the slip angle corresponding to peak tyre force. The negatively sloped black lines are determined by the front slip angles at full tyre saturation in a similar fashion. The resulting equations for these lines are as follows: From the equations, it is apparent that for a given speed and friction coefficient, the only lines that move are those associated with the extreme front slip angles. These lines change with steering angle, as shown in Figure 6, where the steering angle increases from 0 to 11. As the steering angle increases, the overall shape of the isoclines does not change, but their placement in the plane shifts towards values of higher sideslip and yaw rate. This shift is most noticeable by observing the line between points A and B (the intersections of the front and rear slip angle lines). The line between A and B is a linear approximation of the nullcline, denoting the change from positive to negative yaw rates that surrounds the stable equilibrium. This nullcline intersects the origin when and moves upwards and downwards with positive and negative steer angles.