Question

Could you please explain them in your words very shortly to understand logically? Thanks.

Briefly describe the applications of the following mathematical tools.

Answer 1

Liapanov function: A Liapunor function is a scalar function defined on the phase space, which can be used to prove the stability, of various differential equations and systems or an equilibrium point. Definition of a Liap unor function: A Liap unor function for an autonomous dynamical system. ş g: RR with an equilibrium point at yoo is a scalar function V R R that is continuous has continuous first derivatives. is locally positive definite, and for which nov.g is also locally! positive definite, The Condition that -TV, g in locally positive definite is some times stated as tvog is locally megative definite. Example: Consider the following differential equation with solution x on R. Å= n Considering that ² is always positive around the origin it is a natural candidate to be a Liapunor function to help us study n. So let v(n) = n² or R, Then, I The Notation v(n) to denote the time derivate of the t'apunor candidate function vil va)= v(n) ☆ = 2n. (n) = + 2n² <o This correctly shows that the abone differential

equation, n, is asymptotically stable about the ongin. Note that ceoing the same trapanov candidate one com shood that the equilibrium is abo globally asymptotically stable. 2) Hartman - Grobman Theorem: In Mathematics, the study of dynamical system, the Hartman - Grobman theorena or linean'sation theorem is a theorem about the local behaviour of dy mamical systems in the neighbourhood of a hyperbolic equilibrium point Main Theorem: Consider a system evolving in time with stale UCE E RR that satisfies the differential equation dult - file) for some smath map f: R2 Rh suppose the map has a hyperbolic equilibrium state ukern, that is f("* )= 0 and the Jacobian matn's A= dfi/ani off at state ut has no eigen value with real part epeal to hero, Then there exists a neighbourhood of the equilibrium ut and a homeomorphism hi M R such that h(4") = 0 and such that in the Neighbourhood I the How of der/dt = fee) is topologically conjugate By the continuoun map o=h(") to the flow of its linearis atton du/dt = AU

Examples consider the 2D system in variablex w = (4,2) evolving according to the poir of coupled differential Eqreations. de = - 3y + y2 and da = 2+y2. By Direct Computation it can be seen that the only equilibrium of this syatem lies at the origin, that to aho. The Co-ordinate transform u=ht(u) where U=(Y,Z) given by Yo Y+Y2 + 2 y 3 + ŹY 2² 22 Z - fy² - I Y Z is a smooth map between the original u=(4,2) and new u = (Y,Z) Co-ordinates, at least near the equilitosium at the origin. In the new co-ordinates the dynamical system transforms to ito linearisation, dy=-37 and d2 - 2. That is, a distorted verkion of the Wnearisation gines the onginal dynamics in some finite meishbowehood. ** Invariant Manifolds : . In dynamical systems, an invariant manifold is a topological manifold that is innariant under the action of the dynamical system.

Difinition: Consider the differential equation du/dt = f(n): XEIR with flow n(t) = det (no) being the Solution of the differential equation with no nos A set se Rm. in called an invariant set for the differential equation, if for each nots, the solution t d c (26) defined on its maximal internal of existence, has its image in s Alternatively, the orbit passing through each mes les ins. In this case six called an invarciant manifold if sis a manifold. Example: for a fixed parameter a, Corsider the variables n(t), Y Ct governed by the pair of coupled differential equations. dedt = anany and dy/dt = -y + n - 242 The origin is the equilibrium. The's system has two invariant manifolds of interest through the origin. The vertical line no is invariant, when no the n-equation Becomes dadto , which enures n remains zero, The's invariant manifold, The parabola yan/C142a) in invariant for all parameter a. -