一定期間にわたってスケールが徐々に変化する動作を表現することができます。
import 'package:flutter/material.dart';

void main() {
  runApp(const MyApp());
}

class MyApp extends StatelessWidget {
  const MyApp({Key? key}) : super(key: key);

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      theme: ThemeData(
        primarySwatch: Colors.blue,
      ),
      home: const MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  const MyHomePage({Key? key, required this.title}) : super(key: key);

  final String title;

  @override
  State<MyHomePage> createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  double scale = 1.0;

  void _changeScale() {
    setState(() => scale = scale == 1.0 ? 3.0 : 1.0);
  }

  @override
  Widget build(BuildContext context) {
    return Column(
      mainAxisAlignment: MainAxisAlignment.center,
      children: <Widget>[
        ElevatedButton(
          child: const Text('Scale Logo'),
          onPressed: _changeScale,
        ),
        Padding(
          padding: const EdgeInsets.all(50),
          child: AnimatedScale(
            scale: scale,
            duration: const Duration(seconds: 2),
            child: const FlutterLogo(),
          ),
        ),
      ],
    );
  }
}